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1040 ([HUFLVHV 3UDFWLFDO3UREOHPV $SSHQGL[$.H\WRµ7HVW<RXU8QGHUVWDQGLQJ¶ $SSHQGL[%6WDWLVWLFDODQG2WKHU7DEOHV $SSHQGL[&$QVZHUVWR3UDFWLFDO3UREOHPV %LEOLRJUDSK\ ,QGH[ ĞĐŝƐŝŽŶͲDĂŬŝŶŐĂŶĚYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐഩഩ 1 'HFLVLRQ0DNLQJ DQG4XDQWLWDWLYH 7HFKQLTXHV &KDSWHU2YHUYLHZ Decision-making is an integral part of the process of management. It is the competence as a decision-maker that distinguishes a good manager from others. While traditionally, decision-making has been considered as an art, the growing complexity and competitive nature of environment in which managements have to operate now necessitate that decisions be made more systematically using as much quantitative information as possible, along with a consideration of qualitative factors. This book is about the use of quantitative techniques in managerial decision-making. Broadly speaking, decision-making involves choosing a course of action from various available alternatives. The job of a manager, in the process of selecting from among available alternatives, is facilitated in a large measure by the application of appropriate quantitative techniques when, and to an extent, a problem is quantified. This approach to decision-making is known by several names like operations research, management science, quantitative analysis, so on. The contents of this chapter will help a manager to understand questions like the following: What is operations research and how has it evolved? What are its characteristic features? What is the methodology used in operations research? In this context, what is problem formulation, model building, acquisition of input data, solution and interpretation of the results obtained, model validation and implementation of the solution? What are different types of models, and what is the use of mathematical models in operations research? What are different classifications of solutions–feasible and infeasible; optimal and non-optimal; and unique and multiple optimal solutions? What is sensitivity analysis? How quantitative analysis is an integral part of the modern computer-based information systems and how are quantitative tools used in each of the subsystems? Thus, this introductory chapter gives some details about the decision-making process and an idea about the nature and methodology of the quantitative analysis. Finally, a plan of the book is presented which contains a brief account of the contents of each of the chapters to follow. ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ Learning Objectives After reading this chapter, you should be able to: LO 1 LO 2 LO 3 LO 4 LO 5 LO 6 /ĚĞŶƟĨǇƚŚĞĞůĞŵĞŶƚƐŽĨĂĚĞĐŝƐŝŽŶĂŶĚǀĂƌŝŽƵƐĚĞĐŝƐŝŽŶͲŵĂŬŝŶŐƐŝƚƵĂƟŽŶƐ <ŶŽǁƚŚĞƌŽůĞŽĨƋƵĂŶƟƚĂƟǀĞĂŶĂůǇƐŝƐŝŶĚĞĐŝƐŝŽŶͲŵĂŬŝŶŐ ZĞĐĂůůƚŚĞŚŝƐƚŽƌŝĐĂůĚĞǀĞůŽƉŵĞŶƚŽĨŽƉĞƌĂƟŽŶƐƌĞƐĞĂƌĐŚ ǆƉůĂŝŶƚŚĞŶĂƚƵƌĞĂŶĚĐŚĂƌĂĐƚĞƌŝƐƟĐĨĞĂƚƵƌĞƐŽĨŽƉĞƌĂƟŽŶƐƌĞƐĞĂƌĐŚ hŶĚĞƌƐƚĂŶĚƚŚĞŵĞƚŚŽĚŽůŽŐǇŽĨŽƉĞƌĂƟŽŶƐƌĞƐĞĂƌĐŚ ĞƐĐƌŝďĞŽƉĞƌĂƟŽŶƐƌĞƐĞĂƌĐŚĂƐĂŶŝŶƚĞŐƌĂůƉĂƌƚŽĨĐŽŵƉƵƚĞƌͲďĂƐĞĚ ŝŶĨŽƌŵĂƟŽŶƐǇƐƚĞŵƐ ,1752'8&7,21 'HFLVLRQPDNLQJ LV DQ HVVHQWLDO SDUW RI WKH PDQDJHPHQW SURFHVV$OWKRXJK DXWKRULWLHV GLIIHU LQ WKHLU GH¿QLWLRQVRIWKHEDVLFIXQFWLRQVRIPDQDJHPHQWHYHU\ERG\DJUHHVWKDWRQHLVQRWDPDQDJHUXQOHVVKHRU VKHKDVVRPHDXWKRULW\WRSODQRUJDQLVHDQGFRQWUROWKHDFWLYLWLHVRIDQHQWHUSULVHDQGEHKDYLRXURIRWKHUV 7KXV GHFLVLRQPDNLQJ SHUYDGHV WKH DFWLYLWLHV RI HYHU\ EXVLQHVV PDQDJHU )XUWKHU VLQFH WR FDUU\ RXW WKH 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This is all the more true at the micro level. Thus, there is always a problem as to how to allocate the given resources in the best possible manner. Linear programming is a technique which provides the answer in a wide variety of cases. Some situations where a manager can use linear programming include the following: How to allocate the advertising budget among various alternate advertising media which have different degrees of effectiveness in reaching audiences and involve different costs? In case of a multi-product firm, what product-mix will yield the maximum profit, when different products are known to have different profitability coefficients and different resource requirements? How should the given funds be allocated between different investment opportunities that yield varying returns and involve different degrees of risk? How should a dietician decide about the foods that contain varying proportions of ingredients like carbohydrates, vitamins, proteins, etc. to be given to the patients so that their nutrition requirements are met with at the minimum cost? How should the land on an agricultural farm be allocated between different crops which involve different costs on account of labour, manure, seeds, etc. and have different yields, resulting in unequal profitability of the agricultural products produced? How should the HR manager of a hospital decide about the employment of nurses that involves lowest cost and yet meets the requirements at different times of the 24-hour day? The next few chapters are devoted to a detailed account of linear programming. It must be kept in mind that the most important thing is to develop the skill and ability to translate a given real-life situation into a linear programming format, keeping in mind its assumptions and limitations. This chapter illustrates this with examples. Later in the chapter, the graphic solution to some such problems is provided. The chapter heavily uses the inequalities of ‘less than’ and ‘greater than’ types and equations. You should be conversant with two-dimensional graphs and their use. Plotting of equations and inequalities on a graph and the ability to determine the space on the graph over which they are satisfied, called the feasible area, holds the key to successful graphic solution to the problems. ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ Learning Objectives After reading this chapter you should be able to: LO 1 LO 2 LO 3 LO 4 LO 5 LO 6 LO 7 LO 8 ĞĮŶĞůŝŶĞĂƌƉƌŽŐƌĂŵŵŝŶŐĂŶĚŝƚƐĂƉƉůŝĐĂƟŽŶƐ <ŶŽǁƚŚĞĐŽŵƉŽŶĞŶƚƐĂŶĚĨŽƌŵƵůĂƟŽŶŽĨůŝŶĞĂƌƉƌŽŐƌĂŵŵŝŶŐƉƌŽďůĞŵ;>WWͿ ǆƉƌĞƐƐƚŚĞŐĞŶĞƌĂůŵĂƚŚĞŵĂƟĐĂůŵŽĚĞůŽĨ>WW ǆƉůĂŝŶƚŚĞĂƐƐƵŵƉƟŽŶƐŽĨĂŶ>WWŵŽĚĞů ^ŚŽǁŚŽǁƚŽƐŽůǀĞ>WWŐƌĂƉŚŝĐĂůůǇ /ůůƵƐƚƌĂƚĞĐŽŶǀĞǆƐĞƚƐĂŶĚƚŚĞŝƌƌĞůĞǀĂŶĐĞƚŽ>WWƐ ŝƐĐƵƐƐƚŚĞĐŽŶĐĞƉƚƐŽĨďŝŶĚŝŶŐ͕ŶŽŶͲďŝŶĚŝŶŐĂŶĚƌĞĚƵŶĚĂŶƚĐŽŶƐƚƌĂŝŶƚƐ ĞƚĞƌŵŝŶĞƵŶŝƋƵĞĂŶĚŵƵůƟƉůĞŽƉƟŵĂůƐŽůƵƟŽŶƐ͕ŝŶĨĞĂƐŝďŝůŝƚǇĂŶĚ ƵŶďŽƵŶĚĞĚŶĞƐƐŝŶ>WWƐ ,1752'8&7,21 LO 1 Define linear programming and its applications ! 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( ! @AAD? /2 68 ) & > 65 > 6596£56J596£58JB59:6≥=J 56≥8 65B=6£8 B5966≥B6 B:5B:6≥B67 :5966 5≥66≥8 ./ & 585B76. /& 85B76./& 75986./& 75986 /2 66 .!5/ * .!6/ .15 16/ " !5 !6 568 6,8 15 16 ::8 5;= & * * >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ/͗&ŽƌŵƵůĂƟŽŶĂŶĚ'ƌĂƉŚŝĐ^ŽůƵƟŽŶഩഩ $ 13 !5 !6 15 16 : : : ? 5: F 7 , 568 6,8 ::8 5;= ;88 56; 6; K $ * .! ' ( 688F/ /2 6: )5 6 " ) 2 ) 5 ./ > C688 6 C678 ./ N C7; C;8 ./ N) N FgC6L 58gC6L N 58gC66;L =gC66;L N+ 7gC6;L 56gC6;L ./ U C=;8 C55;8 ' EC6F;888 ! )NE68JNE5;J N'E5F ! L L#E78 ! # E;8 ./ ' ./ ./ . / ' d d /2 67 # C5=888 0 C F88 :8* " C7888 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ # "U 1 d1 "U d /2 6; > & > 2 5659:6 759=6≥67888 596≥;888 F5966≥5=888 56≥8 6= & :5976 6596£8J5976£:=J5966£58J5≥ ; 6≥, 6, ./ 0 d . / & E=5B66 > F5B76£F 8£5£: 6≥8 ) > /2 6F > & > ) EB759:6 6? ' 5B66≥B7J659:6≥5:J5B6≥7 56≥8 :5B66≥= B659,6≥, 65B:6£= 56≥8 EF59;6 ) /2 :8 G " ) >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ/͗&ŽƌŵƵůĂƟŽŶĂŶĚ'ƌĂƉŚŝĐ^ŽůƵƟŽŶഩഩ & < ?6 ' J ;88 5; 78 :88 68 :8 ' 5888 :8 6; ( 688 :; 68 < ) , 6 % = % @ " '< ?6 , 5) & 858 78 568 868 :8 6=8 6) & 855 78 578 866 :8 6;8 ' <! ( ( =>>C? :5 ' N # C7888888 $ U 58* * :;# 0 CF8888U 68* * :8# 0 C5:8888U' K $ 5F C5;8888U $ 5F U ' $ 5F ' 65 " 5;8 & :81 *# d <! ( 5 =>>F? :6 " 1N . 1N/ C;888888 " .N5 N ,/ #1 * 1 :8 " * D * " ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ J ' N5 C788888* 6888888 N, C688888* 5888888 C:88888 5688888 #1 " ' <! ( @A=@? :: "K # 0 C5; " ) "$ 1 1 5=f 858 56f 88= 57f 88; 56;f 88, 55f 88: + + " " # V 78 0 # * ' C5 ;; ' - :7 C:8888 > 56: 7 ) ! 5 C6F888 C:=888 6 C;6888 C:5888 : C67888 C78888 7 C66888 C68888 1 0 8; J 58 J 5; 5F ' ; >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ/͗&ŽƌŵƵůĂƟŽŶĂŶĚ'ƌĂƉŚŝĐ^ŽůƵƟŽŶഩഩ :; " "&0 + # CF8# $ " # "U " 68 + "U * # # 8 588 " 78 ;8 "U F81 d' := 0 " ' CFC58 C57 + ' C; = 58" + C68 C:8 C:8 ". / ) '$ ! ' N 6; 78 6; > 6; 68 68 78 :8 78 " # 4 ' <! ( 0 =>>L? :, # ' + ) + & & 5F8 ! ;8 5=; ' " 5;; ! 6; ! 6; ! ;8 ! ) ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ ' '$ <;? ;88 5?; ;88 57; :=8 5;5 0 ' :F ) ! % . / G . / % .C58888L / G .C58888L / = @ C 5 6 :; :? 6 6 7: 7, : 6 78 7; C:888 " ' <! ( =>BI? :? 0 $ =; F8 5:; ,;" C5= C68 " " $ C6" $ D ' < ? / 1 0 / $ 5 F8 58 =; 6 ?8 58 F8 : ?; 68 5:; 7 ,8 58 ,; ' <! ( ! @AAD? 78 <--"Q < & + ) & < N < $ " $ ) >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ/͗&ŽƌŵƵůĂƟŽŶĂŶĚ'ƌĂƉŚŝĐ^ŽůƵƟŽŶഩഩ "$ 7 : ; ! " * ( > .C/ 7588 ,;88 57=88 - 6;88 7;88 ?888 .C/ " " " C78888 * # # 5 5??:""!+ # C=8888 " > :5&5??: ) <$ + 5;888 K 558888 h L - - - " 5; 56 57 " :888 5688 $ < ? & < N < ! < ? 5;8888 ?8888 ,8888 h 588888 ' h 5;888 588888 U :8888 "!+ # =8888 0 ;888 + * # 78888 % ;8888 * 6;888 - + 578888 ;88888 " ;88888 " C58888 0 * 67 0 "!+ + * # # C5688 & N < # " #"> & # ' " .' !$ =>>C/ 3 /LQHDU3URJUDPPLQJ,, 6LPSOH[0HWKRG &KDSWHU2YHUYLHZ Once a linear programming problem is formulated, the next issue is to solve it for optimal values of the decision variables, which leads to maximisation (say, of profit) or minimisation (may be, of cost). The graphic method is indeed available but, as we know, it has its limitations. When only two items are produced by a firm, two investment opportunities are available to an investor, two advertising media are available to an advertising manager—in other words when only two decision variables are involved — then we can think of a graphic solution to an LPP. In case of more than two variables, however, the manager has to turn to the Simplex method. The Simplex method is a very powerful tool for solving linear programming problems. The Simplex and its variants can handle any complexities in the LPPs irrespective of the number and nature of the decision variables and constraints. The application of Simplex method for solving a linear programming problem helps a manager to know answers to questions like the following: What mix of the products involved would yield the maximum profit? What maximum profit can be obtained from the optimal product-mix? Whether all resources would be fully utilised by the optimal mix, or are there any unutilised resources and to what extent? If the problem is of the minimisation type, then what mix of the products/items would minimise the cost and what is the cost involved? Does the problem have an alternate solution which is as good as the optimal solution obtained? If yes, what is that solution? Is any of the given restrictions of no consequence? To illustrate, a particular raw material may be available in plenty in a given situation. Does the problem have no solution that can meet all the requirements? While the solution to relatively small or moderately large problems may be found while working manually with the method, large-scale problems can be solved with the help of computers where software are available. Real life problems obviously require a computer for solution. Knowledge of simple algebraic manipulations and a good hand at arithmetic calculations are the prerequisites for applying Simplex algorithm. A command over arithmetic operations on fractional values is necessary and desirable for Simplex calculations. Unfortunately, an ordinary electronic calculator is not of much help in this case. >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ Learning Objectives After reading this chapter, you should be able to: LO 1 LO 2 LO 3 LO 4 LO 5 LO 6 LO 7 LO 8 /ĚĞŶƟĨǇ^ŝŵƉůĞǆŵĞƚŚŽĚĂŶĚĐŽŶĚŝƟŽŶƐĨŽƌŝƚƐĂƉƉůŝĐĂƟŽŶ <ŶŽǁŚŽǁƚŽƚƌĂŶƐĨŽƌŵĂŐŝǀĞŶ>WWŝŶƚŽĂƐƚĂŶĚĂƌĚŝƐĞĚĨŽƌŵ ĞƐĐƌŝďĞƚŚĞƐƚĞƉƐĨŽƌŽďƚĂŝŶŝŶŐƐŽůƵƟŽŶƚŽĂŶ>WWƵƐŝŶŐ^ŝŵƉůĞǆŵĞƚŚŽĚ hŶĚĞƌƐƚĂŶĚƚŚĞũƵƐƟĮĐĂƟŽŶĂŶĚƐŝŐŶŝĮĐĂŶĐĞŽĨƚŚĞĞůĞŵĞŶƚƐ ŝŶĂ^ŝŵƉůĞǆdĂďůĞĂƵ ŝƐĐƵƐƐƚŚĞŝŐͲDŵĞƚŚŽĚĨŽƌƐŽůǀŝŶŐůŝŶĞĂƌƉƌŽŐƌĂŵŵŝŶŐƉƌŽďůĞŵƐ ^ŚŽǁƚŚĞƚǁŽͲƉŚĂƐĞŵĞƚŚŽĚĨŽƌƐŽůǀŝŶŐůŝŶĞĂƌƉƌŽŐƌĂŵŵŝŶŐƉƌŽďůĞŵƐ ^ƵŵŵĂƌŝƐĞŵƵůƟƉůĞŽƉƟŵĂůƐŽůƵƟŽŶƐ͕ŝŶĨĞĂƐŝďŝůŝƚǇ͕ƵŶďŽƵŶĚĞĚŶĞƐƐĂŶĚ ĚĞŐĞŶĞƌĂĐǇ ^ŽůǀĞůŝŶĞĂƌƉƌŽŐƌĂŵŵŝŶŐƉƌŽďůĞŵƐƵƐŝŶŐD^^ŽůǀĞƌ ,1752'8&7,21 ! " # $ % # & " % % & ' 6,03/(;0(7+2' & " ( LO 1 Identify Simplex ' method and conditions for its application & ' & " ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ & % ! % ) ' " & % * & +, % * & # & & & &RQGLWLRQVIRU$SSOLFDWLRQRI6LPSOH[0HWKRG ' " + &-./ $ " $ 0+) ' 1+02,≥03 01+42,£35 % , 6 $ / 7 8 7 8 $ & $ * 9 ': ;' <=+4>,0,2 / ,+4=,422 £1? =+4,,0,2 £2? ,432 £@, +, ≥?2 / 2 $ @=/ 2<@0= ;' <=+4>,0,@4,= / ,+4=,42@02= £1? =+4,,0,@4,= £2? ,43@03= £@, +,@= ≥? ! @= 2 dŚĞĂƉƉůŝĐĂƟŽŶŽĨƐŝŵƉůĞǆŵĞƚŚŽĚƌĞƋƵŝƌĞƐƚŚĂƚ;ŝͿĂůůƚŚĞǀĂƌŝĂďůĞƐĂƌĞŶŽŶͲŶĞŐĂƟǀĞĂŶĚ;ŝŝͿŶŽŶĞŽĨƚŚĞZ,^ ǀĂůƵĞƐŽĨƚŚĞĐŽŶƐƚƌĂŝŶƚƐŝƐŶĞŐĂƟǀĞ͘ZĞƉůĂĐĞĂŶǇƵŶƌĞƐƚƌŝĐƚĞĚǀĂƌŝĂďůĞďǇƚŚĞĚŝīĞƌĞŶĐĞŽĨƚǁŽŶŽŶͲŶĞŐĂƟǀĞ ǀĂƌŝĂďůĞƐ͘ >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ & # ( " ) 2, ' 62/87,21720$;,0,6$7,21352%/(06 & & 6',+ ([DPSOH 0D[LPLVH = [[ 3UR¿W LO 2 Know how to transform a given LPP into a standardised form 6XEMHFWWR [[ £ 5DZPDWHULDOFRQVWUDLQW [[ £ /DERXUKRXUVFRQVWUDLQW [[ ≥ & % % &" % ,+42,£3?& % + $ ,+42,4 +<3? 5 ,+42,<3? +<? ,+42,A3? + % & + ? 3??£ +£3? ,+42,& + # #$ $ % % ' + # , % % @+42,4 ,<B3. , ?B3?£ ,£B3 @+4 2, ( ' # ( + , C " & " ( "& " * & # % C : ;' <@?+42=, / ,+42, £3? @+42, £B3 +, ≥? ;' <@?+42=,4? +4? , / ,+42,4 + <3? @+42,4 , <B3 +, + , ≥? ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ & ' $ . ' # & +?= +?=,=@+ & +<+?,<=# # +<,= ,<@+& ' $ # ! ' $ # & ' % @0,<, % ' # * < % ≥ 0 % * / D 0 % * % & * ! . 0 $ & " C % $ ! < $ & " % ' ' ! ' $ ) % % @",<3 ' ) 2+ O )LJXUH *UDSKLF3UHVHQWDWLRQRI &RQVWUDLQWV >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ & : # ! $ + , +, + <? , <? , , + + , <? , <? + <,@ , + + , , <? + <? + , , + + <? + + , , + <? +, + , + <+1 , <1 & % E F + <+, E - + <2? , <0,@ 5 & , <? , <2, + <023 5 G + <? , <,? , <23 E + <? , <? + <3? , <B3 E D D ' G& 6 ' & ' # & ' " /ŶĂďĂƐŝĐƐŽůƵƟŽŶ͕ƐŽŵĞǀĂƌŝĂďůĞƐĂƌĞďĂƐŝĐǁŚŝůĞŽƚŚĞƌƐĂƌĞŶŽŶͲďĂƐŝĐ͘dŚĞŶƵŵďĞƌŽĨďĂƐŝĐǀĂƌŝĂďůĞƐŝƐĞƋƵĂůƚŽ ƚŚĞŶƵŵďĞƌŽĨĐŽŶƐƚƌĂŝŶƚƐ;ŽƌĞƋƵĂƟŽŶƐͿ͕ǁŚŝůĞƚŚĞŶƵŵďĞƌŽĨŶŽŶͲďĂƐŝĐǀĂƌŝĂďůĞƐŝƐĞƋƵĂůƚŽƚŚĞƚŽƚĂůŶƵŵďĞƌ ŽĨǀĂƌŝĂďůĞƐŵŝŶƵƐƚŚĞŶƵŵďĞƌŽĨĞƋƵĂƟŽŶƐ͘dŚĞŶŽŶͲďĂƐŝĐǀĂƌŝĂďůĞƐĂƌĞĂůůƐĞƚĞƋƵĂůƚŽǌĞƌŽĂŶĚƚŚĞǀĂůƵĞƐŽĨ ďĂƐŝĐǀĂƌŝĂďůĞƐĂƌĞŽďƚĂŝŶĞĚďǇƐŽůǀŝŶŐƚŚĞĞƋƵĂƟŽŶƐƐŝŵƵůƚĂŶĞŽƵƐůǇ͘/ĨƚŚĞǀĂůƵĞƐŽĨĂůůƚŚĞďĂƐŝĐǀĂƌŝĂďůĞƐĂƌĞ ŶŽŶͲŶĞŐĂƟǀĞ͕ƚŚĞƐŽůƵƟŽŶŝƐƐĂŝĚƚŽďĞbasic feasible solution͘ 6WHSVLQWKH6LPSOH[0HWKRG & ' : D & ' D LO 3 Describe the steps for obtaining solution to an LPP using Simplex method & ' % 6 / ' & ( 2EWDLQLQJWKH,QLWLDO7DEOHDXDQG6ROXWLRQ H & ' &2+ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 3.1 Simplex Tableau Coefficient values from constraint equations Basic variables with their coefficients in the objective function Basis x1 x2 S1 S2 bi S1 2 3 1 0 60 S2 Contribution per unit 0 Constraint values 0 cj 4 3 0 1 40 35 0 0 Solution 0 0 60 96 Dj = cj – zj 40 35 0 0 96 Values of basic and non-basic variables & " * . :+, + ,5' " % " # % & " # # * % ' & " ,+42,4 +4? , " +, + , ,2+? ! " $ ) ' " . ' ' ( ' & * ' +8& % * & * % " & " % * 8 & ! ! 0 $ I& # ) + , . + , +, $ & +, * + ,% 3?B3 ) + ! ' ' ! $ " & " D' ' ' 8QGHU FHUWDLQ FRQGLWLRQV VRPH EDVLF YDULDEOHV PD\ DOVR KDYH ]HUR YDOXHV 7KLV FRQGLWLRQ LV FDOOHG GHJHQHUDF\DQGLVGLVFXVVHGODWHU >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ ) +<?,<? +<3? ,<B3/ <@? ¥ ?42= ¥ ?4? ¥ 3?4? ¥ B3<? % * $ * " dŚĞŝŶŝƟĂůƐŽůƵƟŽŶŝƐŽďƚĂŝŶĞĚďǇƐĞƫŶŐĂůůƚŚĞĚĞĐŝƐŝŽŶǀĂƌŝĂďůĞƐĞƋƵĂůƚŽǌĞƌŽĂŶĚŽďƚĂŝŶŝŶŐƚŚĞƐŽůƵƟŽŶǁŝƚŚ ƚŚĞŚĞůƉŽĨƚŚĞƐůĂĐŬǀĂƌŝĂďůĞƐƚŚĂƚĨŽƌŵĂŶŝĚĞŶƟƚǇŝŶƚŚĞƐŝŵƉůĞǆƚĂďůĞĂƵ͘ 7HVWLQJWKH2SWLPDOLW\ & ' + D' % '0)' &2+& )' " " & )'& )' " +, + , ') + ,# @ / * ".)+<? ¥ ,4? ¥ @<? ( ' " +<@?D'<'0)' " '0 / +C@? " D+<@?0?<@? 5 K K K K K K K K K K K K K K K K K K &D' ! ! + , + , )+<? ¥ ,4? ¥ @<? \D+<@?0? < @?J ),<? ¥ 24? ¥ 2<? \D,<2=0? < 2=J )2<? ¥ +4? ¥ ?<? \D2<?0? < ?J )@<? ¥ ?4? ¥ +<? \D@<?0? < ? (!* 56- 7KH7HVW! '+ ' D' L D'£? ' L D'≥? 5 # D' ' + 'HULYLQJD5HYLVHG7DEOHDXIRU,PSURYHG6ROXWLRQ & D' 6 D' " ( D' +& + &2, &2+ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 3.2 Simplex Tableau 1: Non-optimal Solution $ + , + , ,' + ? , 2 + ? 3? 2? , ? @I 2 ? + B3 ,@ ¨D # ' @? 2= ? ? / ? ? 3? B3 2= ? ? D' @? ≠ # & ( 5' # M' N ' * % (* & # # ( &2, # # ,&# @ # #I H # # # N # -# & * ) ' + 2M@ ? +M@ ,@ ) # Ê Row element in the 5 <D 0 Á key column Ë ! * , 2 + ? 3? ¥ Corresponding replacment ˆ ˜¯ row value .* ( ¥ " * / #* 0 0 0 0 0 , , , , , ¥ ¥ ¥ ¥ ¥ + 2M@ ? +M@ ,@ < < < < < ? 2M, + 0+M, +, >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ & ' &22 TABLE 3.3 Simplex Tableau 2: Non-optimal Solution $ + , + , ,' + ? ? 2M,I + 0+M, +, 1 , @? + 2M@ ? +M@ ,@ 2, ¨D ' @? 2= ? ? / ,@ ? +, ? D' ? = ? 0+? ≠ '7 8 ! +<,@,<? +<+, ,<?/ <@? ¥ ,@42= ¥ ?4? ¥ +,4? ¥ ?<B3? & ,@ $ ! +, # " CB3?5 . ) 2+ 5 ) D' . % * & 5HYLVHG7DEOHDXIRU)XUWKHU,PSURYHG6ROXWLRQ & D # # #& D'% = $ ,& # #( H ' # 12,! " (* + & # # #2M, # # &22 ( # # # ' & &2@ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 3.4 Simplex Tableau 3: Optimal Solution $ + , + , , 2= ? + ,M2 0+M2 1 + @? + ? 0+M, +M, +1 ' @? 2= ? ? / +1 1 ? ? D' ? ? 0+?M2 0,=M2 & ' 0 ) 2+& +<+1 ,<1 @? ¥ +142= ¥ 14? ¥ ?4? ¥ ?<+???C *& D' £? C D' " " & ' &ŽƌĂƐŽůƵƟŽŶĨŽƵŶĚƚŽďĞŶŽŶͲŽƉƟŵĂůŝŶƚĞƌŵƐŽĨDjǀĂůƵĞƐ͕ĂŶĞǁƐŽůƵƟŽŶŵĂǇďĞŽďƚĂŝŶĞĚďǇůŽĐĂƟŶŐŬĞǇ ĐŽůƵŵŶĂŶĚŬĞǇƌŽǁ͘/ŶƐĞůĞĐƟŶŐŬĞǇƌŽǁ͕ĚŽŶŽƚĐŽŶƐŝĚĞƌbiͬaijƌĂƟŽƐǁŚŽƐĞaijǀĂůƵĞƐĂƌĞǌĞƌŽŽƌŶĞŐĂƟǀĞ͘ -XVWLÀFDWLRQDQG6LJQLÀFDQFHRI(OHPHQWVLQ6LPSOH[7DEOHDX ( ' " ' D' " / '& , / ' & 2 LO 4 Understand the justification and significance of the elements in a Simplex Tableau )RU6LPSOH[7DEOHDX & &2= Table 3.5 Simplex Tableau 2 $ + , + , , ? ? 2M, + 0+M, +, + @? + 2M@ ? +M@ ,@ ' @? 2= ? ? / ,@ ? +, ? D' ? = ? 0+? EL9DOXHV ' + &2, + " C@? $, C2= . + ' ( % Number of units of raw material available 60 = <2? Raw material required per unnit of product 2 >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ Number of labour hours available 96 = <,@ Labour hours required per unit of prooduct 4 & ,@ % & ,@ ' % ,@ ¥ ,<@1# 3?0@1<+,# & +, +,@ + ' , 6XEVWLWXWLRQ5DWHV " ,:& , / % B3 & % ) ' , +M@ + ++M@ 5 % / % ,# , ++M@ +M@ ¥ ,<+M,# & 0+M,+M@ ++ , " +:!* + +# & # +# ( +,# #+# + . 7+8 + +# + +,# " +:!7+8 + ' / ' ,@ # % # ) % & * + " ,:& 2M@2M, + + & $, 2M@ + $ % % @& $ 2M@ ! $ 2M,# + ' # ! $ 2M@ % ,# & 2M@ , ¥ 2M@<2M,# 5 2# % $ 2M,# % 202M,<2M,# & M ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ DM(OHPHQWV ( ' D' ' / D' "'0)'" )' !)' " ' ,)' +, + , @?2??+? 1 +:! + ? +& " C@?. + % @? 1 ,:& $ , :+ + 2M, ,+ 2M@ 5 " + + ". " + C@? 2M@ + @? ¥ 2M@<C2?. " , ), ?42?<2? 1 +:& + +& " 1 ,:! , :+ +M, +, +M@ +& +M, + " " + * . +M@ + @? ¥ +M@<C+? C@? " & , C+? " ( D' &' " & @?2=?? +, + ,& + C@? " " C@? )' & " + D'<'0)'<@?0@?<?/ , " C2= " C2? ! " , $ 2=02?<C=5 + * , C+? " " C+? & D' Y " )RU6LPSOH[7DEOHDX & &23' : TABLE 3.6 Simplex Tableau 3 $ + , + , , 2= ? + ,M2 0+M2 1 + @? + ? 0+M, +M, +1 ' @? 2= ? ? / +1 1 ? ? D' ? ? 0+?M2 0,=M2 >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ & $ 1+1 & D' : & ?+ + + $ ' & " +<C@? " ' ' % C@?! "D'<@?0@?<? ! +?D'<? , & + ,M2 ,0+M, +& +# ,M2 $ +M, $ % 2# ,M2 2 ¥ ,M2<,# +M, + +# % ,# & ,M2 , +M, + C+?M2& D'<0+?M2 2= ¥ ,M20@? ¥ +M,< 0+M2+M, , ,+ ( +M, +M2 $& +M, @ ¥ +M,<, +M2 $ 2 ¥ +M2<+ & C,=M2 " @? ¥ +M,<C,? 2= ¥ +M2<C2=M2 D' ([DPSOH $¿UPSURGXFHVWKUHHSURGXFWV$%DQG&HDFKRIZKLFKSDVVHVWKURXJKWKUHHGHSDUWPHQWV IDEULFDWLRQ¿QLVKLQJDQGSDFNDJLQJ(DFKXQLWRISURGXFW$UHTXLUHVDQGDXQLWRISURGXFW%UHTXLUHV DQGZKLOHHDFKXQLWRISURGXFW&UHTXLUHVDQGKRXUVUHVSHFWLYHO\LQWKHWKUHHGHSDUWPHQWV(YHU\GD\ KRXUVDUHDYDLODEOHLQWKHIDEULFDWLRQGHSDUWPHQWKRXUVLQWKH¿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£ 3? @+4@,4@2£ >, ) ) ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ ,+4@,4=2£ +?? +,2≥ ? # # ;' <=+4?,4124? +4? ,4? 2 / 2+4=,4,24 + <3? @+4@,4@24 , <>, ,+4@,4=24 2 <+?? +,2 + , 2 ≥? & ' TABLE 3.7 $ &2>02B Simplex Tableau 1: Non-optimal Solution + , 2 + , 2 ,' + ? 2 =I , + ? ? 3? +, ¨D , ? @ @ @ ? + ? >, +1 2 ? , @ = ? ? + +?? ,= ' = +? 1 ? ? ? / ? ? ? 3? >, +?? D' = +? 1 ? ? ? ≠ TABLE 3.8 $ Simplex Tableau 2: Non-optimal Solution + , 2 + , 2 ,' + +? 2M= + ,M= +M= ? ? +, 2? , ? 1M= ? +,M=I 0@M= + ? ,@ +? ¨D 2 ? 0,M= ? +>M= 0@M= ? + =, ,3?M+> ' = +? 1 ? ? ? / ? +, ? ? ,@ =, D' 0+ ? @ 0, ? ? ≠ >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ TABLE 3.9 Simplex Tableau 3: Optimal Solution $ + , 2 + , 2 , +? +M2 + ? +M2 0+M3 ? 1 2 1 ,M2 ? + 0+M2 =M+, ? +? 2 ? 01M2 ? ? +M2 0+>M+, + +1 ' = +? 1 ? ? ? ? 1 +? ? ? +1 0++M2 ? ? 0,M2 0=M2 ? / D' ! / '& 2 + < ? , < 1 2 < +?& 1 +? $ " & ' = ¥ ?2+? ¥ 121 ¥ +?<C+3? 2 % +1% # 6 % ' & % +M2 $,,M2 "2 ( " +?1 " +? ¥ +M241 ¥ ,M2<,3M2- ,2 1M2 # : 3 4 5- " ( 5 - , + , $ @ +M2 @M2 " = ,M2 +?M2 5 1M2 . " " . " <+?M24+3M24?<,3M2( " C= % ++M2 D & : & 3 # + 6 " #* , 1 0 M2¥3 < 3 2 +? 0 ,M2¥3 < 3 2 +1 0 01M2¥3 < 2@ +3? 0 ++M2¥3 < +21 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ & 3 "++M2 ¥ 3< C,, "% C+21 $, "@ D # +3 ! + 6 " #* , 1 4 +M2¥+, < +, 2 +? 4 0+M2¥+, < 3 2 +1 4 +M2¥+, < ,, +3? 4 ,M2¥+, < +31 & +, $@ "@ @ # & "C1 62/87,21720,1,0,6$7,21352%/(06 & ' ' & 6',, ([DPSOH 0LQLPLVH 6XEMHFWWR = [[ 7RWDOFRVW [[≥ 3KRVSKDWHUHTXLUHPHQW [[≥ 1LWURJHQUHTXLUHPHQW [[≥ ) " % % & % 7 8 % % ' ./ % % -./ ( + , " ; <@?+4,@,4? +4? , / ,?+4=?,0 +< @1?? 1?+4=?,0 ,< >,?? +, + ,≥ ? 5 '' # ( ' % * 5 +,% * +<0@1?? ,<0>,?? >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ ! ( ' # # " + , 0+ %,*00(7+2' ' C $; + . # " & LO 5 Discuss the Big-M method for solving linear programming problems ( " ,?+4=?,0 +4+< @1?? 1?+4=?,0 ,4,< >,?? " " % 7 8 % 7≥8 7<8 ' D " % " & ' Z " " ) " $!& " 4 " D ' " " 75 " " : ' ' ' ) ' ; <@?+4,@,4? +4? ,4+4, " " ' " " & ' &2+? ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 3.10 $ Simplex Tableau 1: Non-optimal Solution + , + , + , ,' + ,? =?I 0 ? + ? @1?? , 1? =? ? 0+ ? + >,?? +@@ ' @? ,@ ? ? / ? ? ? ? @1?? >,?? D' @?0?? ,@0?? ? B3 ¨ ? ≠ ) D' * & D' tŚĞŶĞǀĞƌĂŶ>WWŚĂƐƐŽŵĞĐŽŶƐƚƌĂŝŶƚƐŝŶǀŽůǀŝŶŐ͚≥͛Žƌ͚с͛ƐŝŐŶƐ͕ǁĞŶĞĞĚƚŽŝŶƚƌŽĚƵĐĞĂƌƟĮĐŝĂůǀĂƌŝĂďůĞƐƚŽ ŚĂǀĞŝĚĞŶƟƚǇŝŶƚŚĞƐŝŵƉůĞǆƚĂďůĞĂƵ͕ƚŽĞŶĂďůĞƵƐƚŽŽďƚĂŝŶĂŶŝŶŝƟĂůƐŽůƵƟŽŶ͘dŚĞĂƌƟĮĐŝĂůǀĂƌŝĂďůĞƐĂƌĞůŝŬĞ ĐĂƚĂůǇƐƚƐƚŚĂƚŽŶůǇĞŶĂďůĞƵƐƚŽďĞŐŝŶǁŝƚŚĂƐŽůƵƟŽŶ͘dŚĞǇĂƌĞĞǆƉĞĐƚĞĚƚŽďĞĞůŝŵŝŶĂƚĞĚŝŶƚŚĞƐƵĐĐĞƐƐŝǀĞ ŝƚĞƌĂƟŽŶƐŽĨŽďƚĂŝŶŝŶŐŝŵƉƌŽǀĞĚƐŽůƵƟŽŶƐ͘dŽĞŶƐƵƌĞƚŚŝƐ͕ǁĞĂƐƐŝŐŶŝŶƚŚĞŽďũĞĐƟǀĞĨƵŶĐƟŽŶĂĐŽĞĸĐŝĞŶƚĞƋƵĂů ƚŽM;ďŝŐͲMͿƚŽĞĂĐŚĂƌƟĮĐŝĂůǀĂƌŝĂďůĞĨŽƌĂŵŝŶŝŵŝƐĂƟŽŶƉƌŽďůĞŵĂŶĚʹMŝŶĐĂƐĞŽĨĂŵĂǆŝŵŝƐĂƟŽŶƉƌŽďůĞŵ͘ 5HYLVHG6ROXWLRQ & ' D' # & # ' ' & ' M' ' * % ) ' ( . , + & &2++ &2+2 TABLE 3.11 Simplex Tableau 2: Non-optimal Solution $ + , + , + , ,' , ,@ ,M= + 0+M=? ? +M=? ? B3 ,@? , 3?I ? + 0+ 0+ + ,@?? @? ' @? ,@ ? ? / B3 ? 12 ,0 25 ,@?? D' ? 12 0 25 ? ? 152 03? 5 ≠ ? ? ¨ >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ TABLE 3.12 Simplex Tableau 3: Non-optimal Solution $ + , + , + , ,' , ,@ ? + 0,M>= +M+=? ,M>= 0+M+=? 1? 02??? + @? + ? +M3?I 0+M3? 0+M3? +M3? @? ,@?? ' @? ,@ ? ? / @? 1? ? ? D' ? ? 0,M>= 21M>= ≠ ? 38 4 75 ? 2 4 75 Table 3.13 ¨ Simplex Tableau 4: Optimal Solution $ + , + , + , , ,@ 1M= + ? 0+M=? ? +M=? +@@ + ? 3? ? + 0+ 0+ + ,@?? ' @? ,@ ? ? / ? +@@ ,@?? ? ? ? D' 1M= ? ? +,M,= 0+,M,= &2+?2++ & &2+? +<?,<? &2++ +<?,<B3 " & &2+, " &2+2 ! <@? ¥ ?4,@ ¥ +@@4? ¥ ,@??4? ¥ ?4? ¥ 4?¥<C2@=3 & +<,@?? ' 3UREOHPZLWK0L[HG&RQVWUDLQWV ' ([DPSOH 6ROYHWKHIROORZLQJ/33 0D[LPLVH = [[ 6XEMHFWWR [[ £ [[ ≥ [[ [[ ≥ ! + : ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ ;' <,+4@,4? +4? ,0+0, / ,+4,4 + <+1 2+4,,0 ,4+ <2? +4,,4, <,3 +, + ,+, ≥? & &2+@02+3 TABLE 3.14 Simplex Tableau 1: Non-optimal Solution $ + , + , + , ,' + ? , + + ? ? ? +1 +1 + 0 2 , ? 0+ + ? 2? += , 0 + ,I ? ? ? + ,3 +2 ¨ ' , @ ? ? 0 0 / ? ? +1 ? 2? ,3 ? 0 ? ? D' @4, @4@ TABLE 3.15 ≠ Simplex Tableau 2: Non-optimal Solution $ + ? + 2M, , + , + ? + ? ? ,I , ,' 0+M, = +?M2 + 0 ? ? 0+ + 0+ @ , , @ +M, + ? ? ? +M, +2 ,3 ' , @ ? ? 0 0 ? / ? +2 = ? @ D' , ? ? 0 ? 0,0, ≠ TABLE 3.16 Simplex Tableau 3: Optimal Solution $ + ? + , , ' @ + ? + , ? ? + + , 2M@ + 02M@ ? 0+M, +M, , +M@ , 0+M, , +, ? + ? +M@ 0+M@ 2M@ , @ ? ? 0 0 / , +, , ? ? ? D' ? ? ? ? 0 00, ¨ & +<,,<+, +<, <?& ,¥,4@¥+,<=, >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ START Add necessary slack, surplus and artificial variables to convert inequalities into equations Write the initial tableau and obtain the solution corresponding to the identity Calculate Dj = cj – zj Is the problem maximisation or minimisation? Maximisation Minimisation Select the most negative Dj. Designate the column as ‘key column' Select the largest Dj. Designate the column as ‘key column’ Divide the coefficients in the key column into RHS elements, bi’s Select the row with smallest non-negative quotient and call it ‘key row’ Designate the intersection of key row and key column as ‘key element' Divide all elements of the key row by key element to get replacement row, for the revised simplex tableau Solve for each remaining row in the matrix Element in Corresponding the key element in New Row i = Old Row i column the replacement and row i row Calculate Dj = cj – zj Maximisation No Are all Dj 0 or - ve? Yes Is the problem maximisation or minimisation? The solution is optimal Minimisation Yes Are all Dj 0 or + ve? STOP )LJXUH 6FKHPDWLF3UHVHQWDWLRQRI 6LPSOH[0HWKRG No ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 7:23+$6(0(7+2' ! $! LO 6 Show the two-phase " & *! method for solving linear programming problems " " " ' & / : n < Â ' ' ; j =+ / n Â ''≥ <+, j =+ '≥ ? '<+, & 3KDVH, 6WHS C % " 6WHS ! * " ' J " " ' " " 0+& " & " ' : ; / n m m j =+ i =+ i =+ 8< Â ?'4 Â ? 4 Â + ; S n Â ''0 4< <+, j =+ ' ≥ ? ' 6WHS / ' ' % * " +,% * & ' " " * " " & ' ' " >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ 3KDVH,, " ' - ' ' ) +,`! ' & ([DPSOH 6ROYHWKHSUREOHPJLYHQLQ([DPSOHE\XVLQJWKHWZRSKDVHPHWKRG7KHSUREOHPLV = [[ 0LQLPLVH 7RWDOFRVW 6XEMHFWWR [[≥ 3KRVSKDWHUHTXLUHPHQW [[≥ 1LWURJHQUHTXLUHPHQW [[≥ 3KDVH , ) " * " " + " & ; <?+4?,4? +4? ,4+4, / ,?+4=?,0 +4+<@1?? 1?+4=?,0 ,4,<>,?? +, + ,+,≥? & ' &2+>02+B TABLE 3.17 Simplex Tableau 1: Non-optimal Solution $ + , + , + , ,' + + =? 0 ? + ? @1?? ,@? , + 1?I =? ? 0 ? + >,?? B? ' ? ? ? ? + + / ? ? ? ? @1?? >,?? D' 0+?? 0+?? + + ? ? ¨ ≠ TABLE 3.18 ,? $ Simplex Tableau 2: Non-optimal Solution + , + , + , ,' + + ? >=M,I 0+ +M@ + 0+M@ 2??? 1? + ? + =M1 ? 0+M1? ? +M1? B? +@@ ' ? ? ? ? + + / B? ? ? ? 2??? ? D' ? 0>=M, ? 0+M@ ? =M@ ¨ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 3.19 Simplex Tableau 3: Optimal Solution $ + , + , + , , ? ? + 0,M>= +M+=? ,M>= 0+M+=? 1? + ? + ? +M3? 0+M3? 0+M3? +M3? @? ' ? ? ? ? + + / @? 1? ? ? ? ? D' ? ? ? ? + + 3KDVH,, &/ '& 2 ' " +,& ' &2,?2,+ TABLE 3.20 Simplex Tableau 4: Non-optimal Solution $ + , + , ,' , ,@ ? + 0,M>= +M+=? 1? 02??? + @? + ? +M3?I 0+M3? @? ,@?? ' @? ,@ ? ? / @? 1? ? ? D' ? ? 0,M>= 21M>= ≠ TABLE 3.21 ¨ Simplex Tableau 5: Optimal Solution $ + , + , , ,@ 1M= + ? 0+M=? +@@ + ? 3? ? + 0+ ,@?? ' @? ,@ ? ? / ? +@@ ,@?? ? D ' 1M= ? ? +,M,= & / '& = +<?,<+@@ 620(63(&,$/723,&6 0XOWLSOH2SWLPDO6ROXWLRQV ( % $ " $ >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ % ' LO 7 Summarise multiple optimal solutions, infeasibility, unboundedness and degeneracy & ' ' 6',@ ([DPSOH 0D[LPLVH = [[ 6XEMHFWWR [[ £ [ £ [[ £ [[ ≥ / ' TABLE 3.22 &2,,02,@ Simplex Tableau 1: Non-optimal Solution $ + + ? + + + ? ? , ? ? +I ? + 2 ? 2 3 ? ? 1 +3 ? ? ? / ? ? ,?? +,= B?? 1 +3 ? ? ? ' D ' , TABLE 3.23 + , M' ,?? ,?? ? +,= +,= + B?? +=? 2 ≠ Simplex Tableau 2: Non-optimal Solution $ + , + , 2 M' + ? + ? + 0+ ? >= >= , +3 ? + ? + ? +,= • 2 ? 2I ? ? 03 + +=? =? ' 1 +3 ? ? ? / ? +,= >= ? +=? ? ? 0+3 ? D ' 1 ≠ ¨ ¨ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 3.24 Simplex Tableau 3: Optimal Solution $ + , + , 2 + ? ? ? + + 0+M2 ,= , +3 ? + ? + ? +,= + 1 + ? ? 0, +M2 =? ' 1 +3 ? ? ? / =? +,= ,= ? ? D ' ? ? ? ? 01M2 & D ' * % 1¥=?4+3¥+,=<,@?? ' D '<? $ ( D' $ * 9 ' ( $ * D' % ' 5 &2,@ $ , * D' & * ( # ,# & ,= +,=7,= " # + & TABLE 3.25 Simplex Tableau 4: Optimal Solution $ , ? ? ? + + 0+M2 ,= , +3 ? + 0+? ? +M2 +?? + 1 + ? , ? 0+M2 +?? ' 1 +3 ? ? ? / +?? +?? ? ,= ? ? ? ? ? 01M2 D ' + , + , 2 . % 1 ¥ +?? 4 +3 ¥ +?? <,@?? $ + * D ' % & $ " * >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ ,QIHDVLELOLW\ ! 9 , ' # . # & ' ' " " ( 6',> ([DPSOH 0D[LPLVH = [[ 6XEMHFWWR [[£ [±[£ [≥ [[≥ # " : /,?+42?,4? +4? ,4? 20+ ;' / ,+4,4 +<@? @+0,4 ,<,? + 0 24+<2? +, + , 2+ ≥ ? & &2,302,1 TABLE 3.26 $ Simplex Tableau 1: Non-optimal Solution + , + , 2 + M ' + ? , + + ? ? ? @? ,? , ? @I 0+ ? + ? ? ,? = ¨ 2? 2? + 0 + ? ? ? 0+ + ' ,? 2? ? ? ? 0 / ? ? @? ,? ? 2? ,?4 2? ? ? 0 ? D ' ≠ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 3.27 Simplex Tableau 2: Non-optimal Solution $ + ? ? 2M,I + 0+M, ? + ,? + 0+M@ ? +M@ + 0 ? +M@ ? 0+M@ ' ,? 2? ? / = ? 2? ? 0=0 @ + , + D ' ? 2=4 @ TABLE 3.28 $ , 2 + ,' ? 2? ,? ¨D ? ? = 0,? 0+ + ,= +?? ? ? 0 ? ? ,= 0 ? ≠ Simplex Tableau 3: Final, Non-optimal Solution + , + , 2 + 2? ? + ,M2 0+M2 ? ? ,? , + ,? + ? +M3 +M3 ? ? +? + 0 ? ? 0+M3 0+M3 0+ + ,? ' ,? 2? ? ? ? 0 / +? ,? ? ? ? ,? D ' ? ? 70 3 6 20 3 6 0 ? - 7ZRSKDVH0HWKRGDQG,QIHDVLELOLW\ ( $ % * ' " " &2,B022+ " & % 0,? " + ,?. ' 3KDVH, ;' 8<?+4?,2? +2? ,2? 27+ / ,+4,4 +< @? @+0,4 ,< ,? +0 22+< 2? +, + , 2+≥ ? >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ TABLE 3.29 $ Simplex Tableau 1: Non-optimal Solution + , + , 2 + ,' + ? , + + ? ? ? @? ,? , ? @I 0+ ? + ? ? ,? = + 0+ + ? ? ? 0+ + 2? 2? ' ? ? ? ? ? 0+ / ? ? @? ,? ? 2? D ' + ? ? ? 0+ ? ¨ ≠ TABLE 3.30 $ + ? Simplex Tableau 2: Non-optimal Solution + , + , 2 + M' ? 2M,I + 0+M, ? ? 2? ,? + ? + 0+M@ ? +M@ ? ? = 0,? + 0+ ? +M@ ? 0+M@ 0+ + ,= +?? ' ? ? ? ? ? 0+ / = ? 2? ? ? ,= ? +M@ ? 0+M@ 0+ ? ≠ D ' TABLE 3.31 Simplex Tableau 3: Final, Non-optimal Solution $ + , + , 2 + , ? ? + ,M2 0+M2 ? ? ,? + ? + ? +M3 +M3 ? ? +? + 0+ ? ? 0+M3 0+M3 0+ + ,? ' ? ? ? ? ? 0+ / +? ,? ? ? ? ,? D ' ? ? 0+M3 0+M3 0+ ? ¨ 8QERXQGHGQHVV ! 9 , ' 5 ' !# ' $ $ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ M' $ % • % * & ' # * ' 9 ' ([DPSOH 0D[LPLVH = [[ 6XEMHFWWR [[≥ [[≥ [[≥ & &22,022= " " <+?+4,?,4? +4? ,0+0, ;' / TABLE 3.32 $ ,+4@,0 +4+ <+3 +4=,0 ,4, <+= +, + ,+, ≥? Simplex Tableau 1: Non-optimal Solution + , + + , , M' + 0 , @ 0+ ? + ? +3 @ , 0 + =I ? 0+ ? + += 2 ' +? ,? ? ? 0 0 / ? ? ? ? +3 += +?42 ,?4B 0 0 ? ? D ' ≠ TABLE 3.33 $ Simplex Tableau 2: Non-optimal Solution + , + , + , M' + 0 3M=I ? 0+ @M= + 0@M= @ +?M2 , ,? +M= + ? 0+M= ? +M= 2 += ' +? ,? ? ? 0 0 / ? 6 34 5 ≠ 2 ? @ ? 0 ? 4 @4 5 ? 9 0@0 5 D ' ¨ ? ¨ >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ TABLE 3.34 $ Simplex Tableau 3: Non-optimal Solution + , + + +? + ? , ,? ? + ' +? / +?M2 D ' TABLE 3.35 $ , 0=M3 + , M' ,M2 =M3 0,M2 +?M2 0@ +M3I 0+M2 0+M3 +M2 >M2 ,? ? ? 0 0 >M2 ? ? ? ? ? ? = ? 00= 0 ≠ +@ ¨ Simplex Tableau 4: Unbounded Solution + , + , + , ,' + +? + = ? 0+ ? + += 0+= + ? ? 3 + 0, 0+ , +@ 0> ' +? ,? ? ? 0 0 / += ? +@ ? ? ? D ' ? 02? ? +? 0 00+? ≠ ) / '& @ D ' ! , 9 & , " * & 'HJHQHUDF\ N * ! $ $ 0 $ . % * & ( $* ([DPSOH 0D[LPLVH = [[ 6XEMHFWWR [[ £ [[ £ [[ £ [[ ≥ & " ) 22 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ x2 8 3x 1 + x2 7 =8 A 6 Degenerate, optimal solution 6x 1 +3 5 x2 8 =1 4 4x 1 + 3 2 5x 2 = 30 B 1 0 O 1 2 C x1 3 4 5 6 7 8 9 )LJXUH *UDSKLF6ROXWLRQ'HJHQHUDWH2SWLPDO & $"&' $ + ? ? , , ? 3 , <,1+42?, ? +1?I ++3 " 1M2 ? >@ 2 3 & ! # " + % * & # % * & % * ,# 2+4 ,£1 / $* ! @+4=,£2? % 5 ' # : ;' <,1+42?,4? +4? ,4? 2 / 3+42,4 +< +1 2+4,4 ,< 1 >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ @+4=,4 2< 2? +, + , 2≥ ? & &223 TABLE 3.36 Simplex Tableau 1: Non-optimal Solution $ + , + , 2 M' + ? 3 2 + ? ? +1 3 , ? 2 + ? + ? 1 1 2 ? @ = ? ? + 2? 3 ' ,1 2? ? ? ? / ? ? +1 1 2? D ' ,1 2? ? ? ? ≠ & " + 2 # ( C# + &22> TABLE 3.37 $ Simplex Tableau 2: Optimal Solution + , + , 2 , + +M2 ? ? 3 , 2? , ? + ? 0+M2 + ? , 2 ? 03 ? 0=M2 ? + ? ' ,1 2? ? ? ? / ? 3 ? , ? D ' 02, ? 0+? ? ? & +<?,<3 % +1?5 2 ' &221 TABLE 3.38 $ + ? Simplex Tableau 3: Non-optimal Solution + , +1M=I ? + + , ? 2 02M= ? M' ? ¨ , ? ++M= ? ? + 0+M= , +?M++ , 2? @M= + ? ? +M= 3 +=M, ' / ,1 2? ? ? ? ? 3 ? , ? ? ? ? 03 D ' @ ≠ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ & $ D . ,1¥?42?¥3<+1?& &22B TABLE 3.39 $ Simplex Tableau 4: Optimal Solution + , + , 2 + ,1 + ? =M+1 ? 0+M3 ? , ? ? ? 0++M+1 + +M3 , , 2? ? + 0,MB ? +M2 3 ' ,1 2? ? ? ? / ? 3 ? , ? D ' ? ? 0+?MB ? 0+3M2 & D ' * & +<?,<3 <,1¥?2?¥3<+1? / &22> 2 * &22B + % * & 2 " + ( ' ( # ' ' . ' & &22122B& 7 $ 8 '7 8 . % j& " ( 6'2+? ([DPSOH 6ROYHWKHSUREOHPJLYHQEHORZERWKJUDSKLFDOO\DQGXVLQJWKHVLPSOH[PHWKRG 0D[LPLVH = [[ 6XEMHFWWR [[ £ [[ £ [±[ £ [[ ≥ >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ & ) 2@ x2 & $"&' 10 + , <=+4,, 6 $ " ? ? + 2 ? 1 3 ? ? +3 +>I += 4 Non-degenerate optimal solution 8 A B 4x 1 3x 0 x2 1 + FEASIBLE REGION 2 +2 x2 =1 6 =9 1 + = –9 C 3 2 1 3x x2 x1 4 –2 & $ " " ' " % & . ' ' " +<2,<? & $ $ Degenerate, non-optimal solution –4 –6 –8 – 10 )LJXUH *UDSKLF6ROXWLRQ1RQGHJHQHUDWH 2SWLPDO & ' ;' <=+4,,4? +4? ,4? 2 / @+4,,4 + <+3 2+4,4 , <B 2+0,4 2 <B +, + , 2 ≥? & &2@? TABLE 3.40 $ + Simplex Tableau 1: Non-optimal Solution + ? @ , + ? ? +3 @ , ? 2 + ? + ? B 2 2 ? 2 0+ ? ? + B 2 ' = , ? ? ? / ? ? +3 B B D ' = , ? ? ? ≠ , + , 2 M' & ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ / " # , # 2 :KHQ6LV(OLPLQDWHG ( , &2@+2@, TABLE 3.41 Simplex Tableau 2: Non-optimal Solution $ + + ? ? ,M2I + 0@M2 ? @ 3 ¨ + = + +M2 ? +M2 ? 2 B 2 ? ? 0, ? 0+ + ? 0 ' = , ? ? ? / 2 ? @ ? ? D ' ? +M2 ? 0=M2 ? , + , 2 M' ≠ + + * & +<2,<?<+= 2 ! $ & # , 3 + 2 * # & &2@, TABLE 3.42 $ Simplex Tableau 3: Optimal Solution + , 2 , 2 , , ? + 2M, 0, ? 3 + = + ? 0+M, + ? + 2 ? ? ? 2 7= + +, ' = , ? ? ? / + 3 ? ? +, D ' ? ? 0+M, 0+ ? & +<+,<3<+> ! :KHQ6LV(OLPLQDWHG C &2@? 2 &2@202@= >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ TABLE 3.43 $ Simplex Tableau 4: Non-optimal Solution + , + , 2 ,' : ? ? +?M2 + ? 0@M2 @ 3M= , ? ? ,I ? + 0+ ? ? + = + 0+M2 ? ? +M2 2 0+ ' = , ? ? ? / 2 ? @ ? ? D ' ? ++M2 ? ? 0=M2 ≠ ¨ ) & 2@2 + < 2 , < ? < += & $ & $ % ? , & &2@@ TABLE 3.44 $ + Simplex Tableau 5: Non-optimal Solution + , + , 2 ? ? ? + 0=M2 +M2I @ , , ? + ? +M, 0+M, ? 0 + = + ? ? +M3 +M3 2 +1 ' = , ? ? ? / 2 ? @ ? ? D ' ? ? ? 0++M3 +M3 M' +, ¨ ≠ & &2@@ +<2,<? <+= & $ D ' & 2 +5 , * ## ' # <0+M, TABLE 3.45 $ Simplex Tableau 6: Optimal Solution + , + , 2 2 ? ? ? 2 0= + +, , , ? + 2M, 0, ? 3 + = + ? 0+M, + ? + ' = , ? ? ? / + 3 ? ? +, D ' ? ? 0+M, 0+ ? ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ & &2@= &2@, $ 6 &\FOLQJ ! $ . # D ' D # % # ( ' 9 . ' ' ; " C & # & " % ' ) 2= ŶŽƉƟŵĂůƐŽůƵƟŽŶŝŶǁŚŝĐŚĂůůƚŚĞŶŽŶͲďĂƐŝĐǀĂƌŝĂďůĞƐŚĂǀĞDjπϬŝƐƵŶŝƋƵĞ͘ ŶŽƉƟŵĂůƐŽůƵƟŽŶŝŶǁŚŝĐŚƐŽŵĞŶŽŶͲďĂƐŝĐǀĂƌŝĂďůĞŚĂƐ DjсϬŝƐŶŽƚƵŶŝƋƵĞ͘dŚĞƌĞĂƌĞmultipleŽƉƟŵĂů ƐŽůƵƟŽŶƐƚŽƚŚĞƉƌŽďůĞŵ͘ ƐŽůƵƟŽŶ͕ ĮŶĂů ŝŶ ƚĞƌŵƐ ŽĨ ƚŚĞ Dj ǀĂůƵĞƐ͕ ǁŝƚŚ ƚŚĞ ƉƌĞƐĞŶĐĞ ŽĨ ĂŶ ĂƌƟĮĐŝĂů ǀĂƌŝĂďůĞ ŝŶ ƚŚĞ ďĂƐŝƐ ŝŶĚŝĐĂƚĞƐ infeasibility͘ dŽŝŵƉƌŽǀĞĂŶŽŶͲŽƉƟŵĂůƐŽůƵƟŽŶ͕ŝĨŶŽŶĞŽĨƚŚĞĞůĞŵĞŶƚƐŝŶƚŚĞŬĞǇĐŽůƵŵŶŝƐƉŽƐŝƟǀĞ͕ƚŚĞƐŽůƵƟŽŶƚŽƚŚĞ ƉƌŽďůĞŵŝƐunbounded͘ ƐŽůƵƟŽŶŝŶǁŚŝĐŚƐŽŵĞďĂƐŝĐǀĂƌŝĂďůĞŚĂƐĂƐŽůƵƟŽŶǀĂůƵĞĞƋƵĂůƚŽǌĞƌŽŝƐĂdegenerateƐŽůƵƟŽŶ͘ĚĞŐĞŶĞƌĂƚĞ ƐŽůƵƟŽŶŵĂǇďĞŶŽŶͲŽƉƟŵĂůŽƌŽƉƟŵĂů͘ 06([FHODQG/LQHDU3URJUDPPLQJ / #;/6' & LO 8 Solve linear " programming problems ./ " using MS Solver C % # ) ' ' 6'/ & ' &6' # )RUPXODWLRQRIDQ/33RQ([FHO6SUHDGVKHHW & 6' 6'2,& 6' ) 23 >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ Start Obtain initial solution to the problem Any basic variable with solution value =0? No Yes Solution is non-degenerate Solution is degenerate Is the problem maximisation or minimisation ? Maximisation Minimisation Are all D j ≥ 0? 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( " % =9,<+? $ % * - $ & / : $ $ ( 8 / : $ / : / : # $ ([DPSOH 0D[LPLVH = [[ 6XEMHFWWR [[£ [[£ [≥ [≥ / & " ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ + % 1, +?! # " " / +≥1 +02 <12 +<12,< +?@ : <31424,?+?4@ ;' <324,?@4,@1 / ,1424+?4@£2, 21424@+?4@£1? # ;' < 324,?@4,@14? +4? , / ,24@4 + <3 224@@4 , <+3 ,24@ £3 224@@ £+3 2@ ≥? 2@ + , ≥? & &2@32@> TABLE 3.46 $ Simplex Tableau 1: Non-optimal Solution 2 @ + , ,' + ? , + + ? 3 3 , ? 2 @I ? + +3 @ ' 3 ,? ? ? / ? ? 3 +3 D' 3 ,? ? ? ≠ TABLE 3.47 $ + ? @ ' ¨ Simplex Tableau 2: Optimal Solution 2 @ + , =M@ ? + 0+M@ , ,? 2M@ + ? +M@ @ 3 ,? ? ? / ? @ , ? D ' 0B ? ? 0= 9 &2@> 2<?@<@ % 3¥?4,?¥@4,@1<2,1! +<142<1,<+?4@<+?4@<+@ >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ ([DPSOH &KHPLFR /LPLWHG PXVW SURGXFH H[DFWO\ NJ RI PL[WXUH RI FKHPLFDOV $ DQG % WR EH GHOLYHUHGWRDFXVWRPHU&KHPLFDO$FRVWVCSHUNJDQGFKHPLFDO%FRVWVCSHUNJ,WLVVWLSXODWHGWKDWQR PRUHWKDQNJRIFKHPLFDO$DQGQROHVVWKDQNJRIFKHPLFDO%PXVWEHXVHGLQSUHSDULQJWKLVPL[WXUH )RUPXODWHWKLVDVDOLQHDUSURJUDPPLQJSUREOHPDQGGHWHUPLQHWKHOHDVWFRVWEOHQGRIWKHWZRLQJUHGLHQWV XVLQJWKHVLPSOH[PHWKRG'HWHUPLQHWKHPLQLPXPWRWDOFRVWLQYROYHG # 9 $+, & : <3?+41?, ;' / +4, <+,?? + £@?? , |@?? +, |? # " : <3?+41?,4? +4? ,4+4+ ;' / +4,4+ <+,?? +4 + <@?? ,0 ,4, <@?? +, + ,+, |? & &2@102=+ TABLE 3.48 $ Simplex Tableau 1: Non-optimal Solution ' ' D' ¨ ≠ TABLE 3.49 $ Simplex Tableau 2: Non-optimal Solution ' ' D' ≠ ¨ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 3.50 $ Simplex Tableau 3: Non-optimal Solution ' ' D' ¨ ≠ TABLE 3.51 Simplex Tableau 4: Optimal Solution $ ' D ' !""#$%&'("(! %&'("( $'('C ([DPSOH $¿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¿UP":K\" LLL &DOFXODWHWKHSHUFHQWDJHRIFDSDFLW\XWLOLVDWLRQLQWKHRSWLPDOVROXWLRQ LY :KDWDUHWKHVKDGRZSULFHVRIWKHPDFKLQHKRXUV" Y ,VWKHRSWLPDOVROXWLRQGHJHQHUDWH" )* +!#"#'#!($!"#( ,-""& ( &(.! ,"+//($#!'01 2$ 3 4 >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ 5( £ 6 £ £ 6 ≥ 2("7 2("77 2("777 )* "#&"#$"!08!((% ! ! $#"#!!''01 2$ 5( 3 36 3 36 ≥ "& TABLE 3.52 $ Simplex Tableau 1: Non-optimal Solution 6 6 ' 6 6 D' Simplex Tableau 2: Non-optimal Solution $ ' ,' ¨ 3 ≠ TABLE 3.53 D' ≠ ,' ¨ 36 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 3.54 $ Simplex Tableau 3: Optimal Solution ' D' 3 !"#"#133!3 0"3"!("#!("!#!(!,"8"( !(!,"D' ). *! ("'#!(!0!!(" #8,CC )* "(#(, !"(#(,!##& 01 " ( 7 " ( 4 A " ( 4 4 6 6 9 77 9 777 6 : :9 "& ," '"(% ) * "!0#('"(""& ,"D ' )&&&*'"(% ;((!&,", 2("7 2("77 2("777 CC:#" C#" < ) * "#!&('"( " . = ([DPSOH &RQVLGHUWKHIROORZLQJ/33 0D[LPLVH = [[ 6XEMHFWWR [[ £E [[ £E [[ ≥ 7KHIROORZLQJLVWKHRSWLPDOVROXWLRQWDEOHFRUUHVSRQGLQJWRWKHVSHFL¿FYDOXHVRIEDQGE >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ %DVLV [ [ 6 6 EL [ $ ± [ ± % &M = M & ' D M ( D )LQGWKHYDOXHVRIWKHHOHPHQWV$%&'DQG( E )LQGWKHULJKWKDQGVLGHYDOXHVEDQGE F :KDWKDSSHQVZKHQWKHGLUHFWLRQRILQHTXDOLWLHVLVFKDQJHGIURP£WRLQERWKWKHFRQVWUDLQWV":KDW LVWKHQDWXUHRIWKLVSUREOHPDQGZKDWGLI¿FXOW\ZRXOG\RXIDFHLQVROYLQJWKLVQHZ/33" )* >3'"!,)*$0"( $ 36( 6¥$3 \$3)6¥*36 " 3¥ ¥ 3 ? 3¥ ¥ 3 % 3 3 )* ("(% ".0" 0""# 3¥ 10 7 10 7 ¥ 3 !3¥ ¥ 3 3 3 3 3 )(* 7'"&'"(("&!& "#( 2$ 3 5( ≥ ≥ ≥ 70!.!(&#!8( !"#( !"("0!!! ([DPSOH 6ROYHWKHIROORZLQJ/33 = [[[ 0D[LPLVH 6XEMHFWWR [±[[ ±[[ ≥± [[ ≥ [[ ≥ DQG[XQUHVWULFWHGLQVLJQ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ "& #0%"'0&("&1 )* 2#,"(!(, )* 3 )(* 7!((%#!8( !"#!'01 2$ 3 3 3 3 ≥ 5( ""#& : TABLE 3.55 Simplex Tableau 1: Non-optimal Solution $ ' D ' ≠ ' ¨ TABLE 3.56 Simplex Tableau 2: Non-optimal Solution $ ' D ' ≠ ' ¨ >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ TABLE 3.57 Simplex Tableau 3: Non-optimal Solution $ 6 ' D ' : !""#'!333!3;((!&, #"& #'01333 3!3 ([DPSOH $ODG\ZDQWVWRLQYHVWXSWRDQDPRXQWRICLQ¿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£ ≥ £ ≥ ≥ 7!(&"(,(%#!8( "#,0 2$ 5( 3 7 1 100 10 3 3 3 3 ≥ D&"#$"!"#(!& ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 3.58 Simplex Tableau 1: Infeasible Solution $ ,' ' : 7 100 ≠ 1 10 D ' TABLE 3.59 ¨ Simplex Tableau 2: Infeasible Solution $ : ' : D' 17 100 7 100 7 100 ≠ Table 3.60 ,' $ : : 3 17 100 ' D' ¨ Simplex Tableau 3: Non-optimal Solution ≠ 17 100 1 10 ,' 6 ¨ >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ TABLE 3.61 Simplex Tableau 4: Non-optimal Solution $ ,' : : 3 17 100 7 100 ≠ ' D' TABLE 3.62 - ¨ 7 100 Simplex Tableau 5: Optimal Solution $ : ' : D' : "& !,# (D'E ".= x2 30 0 = + x1 24 + x2 = 0 ,0 18 0 (in '000 Rs) 30 ;(#'"!&#"(,! ,#$&""0"#$ (#!#$)0"33 * #$(#!#")0"3 3*0"#$("0"" (#?)0"3 !3* 7 ! " " " & ,(( (#! 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FCCG- >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ//͗^ŝŵƉůĞǆDĞƚŚŽĚഩഩ /2 ;(#,#!(0#!(!$" '#!( #(' "'"0#!(Q""#!("0'0"(""!, ,!/!(!$""'# !##( ,"#('"0#!(C!C# )* 4(#&&''"# )* !"#,#$"! )(* ! #', 5" = :DDI- /2 6 "'0&+//1 2$ 5( 3 ≥ ≥6 ≥ /2 "'0&+//1 2 3 5( ≥6 ≥ ≥ -"0!"##'"5( '('"#0'"V$E,#T /2 M 0"5( '("(!"8#$'#F &&#!(F$#1 S5( '(1 2$ 41 3 £6 )"!#7* £ £ ≥ )"!#77* )"!#777* ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ )LQDO6LPSOH[7DEOHDX "' "' : ' "' ' 6 0 ( )* -"##!(F$!"#8("0," )* 7"'T-",TM ')* )(* R"#" , #T7'"0(" )!* 7!(0"""& "!&!7',0"(" !&F !T )* 7'"(#,#!($'"0,'!"" !(! ', M 0"#$'$,#'#&&#1 "' Æ $ 0 ( >!#"'#!('""#!($!"#(F ,0" ! #"#( (%"(! ;00""'0&.""1 )* 7'T )* 7#T )* 7!!T ) * 7!&T ) * R"#" ##T7',& # ) * -"""!0#('""(T ) * -"("'"#!(&#!(!!0",T ) * -""5( '('"#T )$* 7'!!#!($'#!(0"''(0" "#'" #!(!""(T )$* 7' !(0,#!(,"0(""!"#('"#!( (!''((T 4 /LQHDU3URJUDPPLQJ,,, 'XDOLW\DQG6HQVLWLYLW\ $QDO\VLV &KDSWHU2YHUYLHZ The application of simplex method allows not only the solution to linear programming problems but also provides a fund of information which can be usefully employed by the manager in decision-making. The numbers generated in the course of applying simplex provide inputs to obtain answers to many ‘what-if’ questions. Further, every linear programming problem has a mirror image problem which is also a linear programming problem. Using a set of rules, the mirror image problem, called the dual, can be obtained for a given linear programming problem, known as primal problem in this context. This ‘primal-dual’ relationship is very important. Since the two problems are connected to each other, there is obviously a relationship to be expected between their solutions; there indeed is. This chapter explores and explains this, and does more. Apart from a mathematical connection between primal and dual, it explains the economic significance of the dual and helps a manager to answer questions like the following: Is it advisable to produce all the items that can be produced in a given situation? If some item is left out in the optimal product-mix, then under what condition will it be advisable to produce it? What is the marginal profitability of each of the resources of the firm? In turn, this means by how much the profit will increase if more quantity of a particular resource is added or how much reduction in profit will result if its availability is reduced? To what extent will additional quantities of a resource (or its reduction) cause an increase (or decrease) at a uniform rate? Will the optimal product-mix need to change it the profitability of the various products changes? Obviously, when some small changes (increases or decreases) occur in the profitability of the products it would not cause changes in the quantities of the products being produced, but then what are the limits of these price changes? Is it advisable to introduce a new product given the amounts of resources required for its production and its profitability? How would the product-mix change, if at all, its technological changes cause the resource requirements of a product to change? This chapter involves handling of inequalities; arithmetical operations on fractional values; plotting of equalities and inequalities on graphs, and their understanding. Further, a knowledge of the concepts of matrices and their transpose is needed. ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ Learning Objectives After reading this chapter, you should be able to: LO 1 LO 2 LO 3 LO 4 LO 5 LO 6 LO 7 LO 8 <ŶŽǁƚŚĞĐŽŶĐĞƉƚŽĨĚƵĂůŝƚǇ /ĚĞŶƟĨǇƚŚĞĐŽŶĚŝƟŽŶƐŶĞĐĞƐƐĂƌǇĨŽƌŽďƚĂŝŶŝŶŐĚƵĂůƚŽĂŶ>WW ŽŶƐƚƌƵĐƚƚŚĞĚƵĂůƚŽĂŐŝǀĞŶ>WW ŽŵƉĂƌĞƚŚĞŽƉƟŵĂůƐŽůƵƟŽŶƐŽĨƉƌŝŵĂůĂŶĚĚƵĂůƉƌŽďůĞŵƐ ŝƐĐƵƐƐƚŚĞĞĐŽŶŽŵŝĐŝŶƚĞƌƉƌĞƚĂƟŽŶŽĨĚƵĂůǀĂƌŝĂďůĞƐ /ůůƵƐƚƌĂƚĞƐĞŶƐŝƟǀŝƚǇĂŶĂůǇƐŝƐ ĂƌƌǇŽƵƚƐĞŶƐŝƟǀŝƚǇĂŶĂůǇƐŝƐƵƐŝŶŐŐƌĂƉŚŝĐĂƉƉƌŽĂĐŚ͕ƐŝŵƉůĞǆŵĞƚŚŽĚĂŶĚƐŽůǀĞƌ ǆĞĐƵƚĞϭϬϬƉĞƌĐĞŶƚƌƵůĞŝŶƐĞŶƐŝƟǀŝƚǇĂŶĂůǇƐŝƐǁŝƚŚŵƵůƟƉůĞƉĂƌĂŵĞƚĞƌĐŚĂŶŐĞƐ ,1752'8&7,21 ! "## $ GXDO "## % & % ' "## '8$/,7<,1/,1($5352*5$00,1* $ LO 1 Know the concept of duality $ $ SULPDO GXDO ( ) % * + , - $ >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ + , $ + , + , + , . + , . + , - "## * LO 2 Identify the conditions necessary for obtaining dual to an LPP + , / $ + , / $$ $' $ !+£, "## $$ $' $ !+≥, % :KHQDOOYDULDEOHVLQWKH/33DUHQRWQRQQHJDWLYH "## ! $ $ [0≥1 $ [2 3 - [24[5([6 [5[6 $ 7 $ 3 :KHQDOOFRQVWUDLQWVDUHQRWLQWKHµULJKW¶GLUHFWLRQ ' (0 ' * 8[0(9[2≥6 ' £ (8[0:9[2£(6 ' (0 01 ;9 (01 <(9 = ' ' ' $ 21 $£21 $ ≥21 $£21$≥21 % $' 21/ 8[0:9[2462 8[09[2£62 = 8[0:9[2≥62 (0 ' ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ :ULWLQJWKH'XDO = & > ? > . LO 3 Construct the dual to a given LPP =4 F[ D[£ E [≥1 F4 % . [4 D4 % E4 @-> % ? > . *4E¢\ D¢\≥F¢ \≥1 E¢4 E D¢4 % F¢4 . % \4 "## Q P P Q ([DPSOH )RUWKH/33JLYHQLQ([DPSOHUHSURGXFHGEHORZZULWHWKHGXDO 0D[LPLVH = [[ 6XEMHFWWR [[£ [[£ [[≥ ? > . * 4A1\0:BA\2 2\0:6\2≥61 5\0:5\2≥59 \0\2≥1 >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ ǀĞƌǇůŝŶĞĂƌƉƌŽŐƌĂŵŵŝŶŐƉƌŽďůĞŵŚĂƐĂŵŝƌƌŽƌͲŝŵĂŐĞƉƌŽďůĞŵ͕ĐĂůůĞĚŝƚƐĚƵĂů͘dŚĞƚǁŝŶĐŽŶĚŝƟŽŶƐĨŽƌǁƌŝƟŶŐ ƚŚĞĚƵĂůƚŽĂŐŝǀĞŶƉƌŝŵĂůƉƌŽďůĞŵĂƌĞ;ŝͿĂůůƚŚĞǀĂƌŝĂďůĞƐĂƌĞŶŽŶͲŶĞŐĂƟǀĞĂŶĚ;ŝŝͿĂůůƚŚĞĐŽŶƐƚƌĂŝŶƚƐƐŚŽƵůĚ ďĞ͚£͛ƚǇƉĞŝĨŝƚŝƐĂŵĂǆŝŵŝƐĂƟŽŶƉƌŽďůĞŵĂŶĚ͚≥͛ƚǇƉĞŝĨŝƚŝƐĂŵŝŶŝŵŝƐĂƟŽŶƉƌŽďůĞŵ͘ĐŽŶƐƚƌĂŝŶƚƐŝŶǀŽůǀŝŶŐ ƚŚĞ͚с͛ƐŝŐŶƐŚŽƵůĚďĞƌĞƉůĂĐĞĚďǇĂƉĂŝƌŽĨŝŶĞƋƵĂůŝƟĞƐŝŶŽƉƉŽƐŝƚĞĚŝƌĞĐƟŽŶƐ;£ĂŶĚ≥Ϳ͕ǁŝƚŚŝĚĞŶƟĐĂů>,^ĂŶĚ Z,^ǀĂůƵĞƐ͘ = % +, +, A1BA % \0 \2 . C % . +, % % % +, ' ' £ ≥ ŶnͲǀĂƌŝĂďůĞmͲĐŽŶƐƚƌĂŝŶƚƉƌŝŵĂů>WWǁŝůůŚĂǀĞĂŶmͲǀĂƌŝĂďůĞnͲĐŽŶƐƚƌĂŝŶƚĚƵĂů>WW͕ƐŝŶĐĞĂĚƵĂůƉƌŽďůĞŵŝƐ ŽďƚĂŝŶĞĚďǇƚƌĂŶƐƉŽƐĞŽĨƚŚĞƚŚƌĞĞŵĂƚƌŝĐĞƐŽĨƚŚĞƉƌŝŵĂůƉƌŽďůĞŵ͘ŵĂǆŝŵŝƐĂƟŽŶ>WWǁŝůůŚĂǀĞĂŵŝŶŝŵŝƐĂƟŽŶ ĚƵĂů>WW͕ĂŶĚǀŝĐĞͲǀĞƌƐĂ͘ D 60 PRIMAL DUAL c x1 Maximise Z = [40 35] y1 y2 Minimise G = [60 96] x2 Subject to Subject to a 2 4 y b¢ x 3 3 x x1 x2 ≥ 60 96 y a¢ b 2 3 4 3 y1 y2 c¢ ≥ 40 35 &KDUW 7KH3ULPDODQG'XDO5HODWLRQVKLS 'XDORI/33ZLWK0L[HG&RQVWUDLQWV > "## ' ' > ' ' ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ ([DPSOH :ULWHWKHGXDORIWKHIROORZLQJ/33 = [[ 0LQLPLVH 6XEMHFWWR [[≥ [[≥ [±[£ [[≥ - % ' + ≥ . , ? ( 0 (2[0:[2≥(AE 3ULPDO ? > . 'XDO =401[0:21[2 5[0:2[2≥0F [0:5[2≥ F (2[0:[2≥(A [0[2≥ 1 ([DPSOH ? > . *4F\0:F\2(A\5 5\0\2(2\5£01 2\0:5\2:\5£21 \0\2\5≥1 2EWDLQWKHGXDORIWKH/33JLYHQKHUH 0D[LPLVH = [[[ 6XEMHFWWR [±[£ [[£ [[[≥ [[±[ [[[≥ % &RQVWUDLQWVDQG > £ &RQVWUDLQW ≥ £ ( ([0([2([5£ (2 &RQVWUDLQW ' /' ' £ ≥ 5[0:2[2([5£F 5[0:2[2([5≥F £ (0 ( 5[0(2[2:[5£(F >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ E 3ULPDO ? > . 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D 62 35,0$/ '8$/ 0D[LPLVDWLRQ 0LQLPLVDWLRQ 1RRIYDULDEOHV 1RRIFRQVWUDLQWV 1RRIFRQVWUDLQWV 1RRIYDULDEOHV £W\SHFRQVWUDLQW 1RQQHJDWLYHYDULDEOH 8QUHVWULFWHGYDULDEOH W\SHFRQVWUDLQW 8QUHVWULFWHGYDULDEOH W\SHFRQVWUDLQW 2EMHFWLYHIXQFWLRQFRHI¿FLHQWIRUMWKYDULDEOH 5+6FRQVWDQWIRUWKHL WKFRQVWUDLQW 5+6FRQVWDQWIRULWKFRQVWUDLQW 2EMHFWLYHIXQFWLRQFRHI¿FLHQWIRUMWKYDULDEOH &RHI¿FLHQW DLM IRUMWKYDULDEOHLQLWKFRQVWUDLQW &RHI¿FLHQW DLM IRUL WKYDULDEOHLQMWKFRQVWUDLQW &KDUW 6\PPHWULFDO5HODWLRQVKLSEHWZHHQ3ULPDODQG'XDO3UREOHPV >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ &RPSDULQJWKH2SWLPDO6ROXWLRQVRI3ULPDODQG'XDO > + , LO 4 Compare the optimal solutions of primal and dual problems 3ULPDO 'XDO ? > . =461[0:59[2 ? > . *4A1\0:BA\2 2[0:5[2£A1 2\0:6\2≥61 6[0:5[2£BA 5\0:5\2≥59 [0[2≥1 \0\2≥1 % +, * 60 +, * 60 & y2 x2 12 P 32 11 28 10 9 24 20 Q 8 B Feasible region 7 Feasible region 6 16 5 12 4 C 3 8 2 4 A 1 D 0 4 8 12 16 (a) 20 24 28 32 x1 0 2 R 4 6 8 10 12 14 16 18 20 22 y1 (b) )LJXUH *UDSKLF6ROXWLRQWR3ULPDODQG'XDO3UREOHPV ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 3RLQW 3ULPDO [ [2 = 3RLQW \ 'XDO \2 * $ 1 1 1 3 1 59G5 0021 % 1 21 811 4 01G5 29G5 & 0F F 5 21 1 0211 ' 26 1 BA1 7 ' 0111 . E + 56, 60 TABLE 4.1 [ 2 Simplex Tableau: Optimal Solution %DVLV [ 0 [ 2 6 0 6 2 EL 59 1 0 2G5 ( 0G5 F [ 0 61 0 1 ( 0G2 0G2 0F FM 61 59 1 1 > 0F F 1 1 DM 1 1 ( 01G5 ±29G5 E % ? > . *4A1\0:BA\2:160:162:0$00$2 2\06\2±60$0 461 5\0:5\2(62:$2 459 \0\26062$0$2 ≥1 62(66 [0 60 [2 62 \0\2 & 6062 >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ TABLE 4.2 Simplex Tableau 1: Non-optimal Solution %DVLV \ 0 \ 2 60 6 2 $ 0 $2 E L EL DLM 01 ¨ $0 0 2 6H (0 1 0 1 61 $2 0 5 5 1 (0 1 0 59 59G5 FM A1 BA 1 1 0 0 > 1 1 1 1 61 59 DM A1(90 BA(80 0 0 1 1 TABLE 4.3 ≠ Simplex Tableau 2: Non-optimal Solution %DVLV \ 0 \ 2 6 0 62 $ 0 $ 2 E L EDLM \2 BA 0G2 0 (0G6 1 0G6 1 01 21 $2 0 5G2H 1 5G6 (0 (5G6 0 9 01G5 ¨ FM A1 BA 1 1 0 0 > 1 01 1 1 1 9 DM 30 4 0 70 (26 4 1 TABLE 4.4 %DVLV 02( 30 2 ≠ 1 26( Simplex Tableau 3: Optimal Solution \ 0 \ 2 6 0 6 2 $ 0 $ 2 EL \2 BA 1 0 (0G2 0G5 0G2 (0G5 29G5 \0 A1 0 1 0G2 (2G5 (0G2 2G5 01G5 FM A1 BA 1 1 0 0 > 01G5 29G5 1 1 1 1 DM 1 1 0F F 0(0F 0(F / +, . [040F[24F=' 61¥0F:59¥F40111> \0401G5\2429G5 * A1¥01G5 :BA¥29G540111 +, ' DM [040F [24F 6040F624F+ DM , > ' DM \0401G5 \2429G5 60401G562429G5+ , ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ = % % D . ' / /ĨƚŚĞƉƌŝŵĂůƉƌŽďůĞŵŚĂƐĂŶŽƉƟŵĂůƐŽůƵƟŽŶ͕ƚŚĞŶƚŚĞĚƵĂůƉƌŽďůĞŵĂůƐŽŚĂƐĂŶŽƉƟŵĂůƐŽůƵƟŽŶǁŝƚŚŝĚĞŶƟĐĂů ŽďũĞĐƟǀĞĨƵŶĐƟŽŶǀĂůƵĞ͘/ĨƚŚĞƉƌŝŵĂůŚĂƐĂŶƵŶďŽƵŶĚĞĚƐŽůƵƟŽŶ͕ƚŚĞŶƚŚĞĚƵĂůǁŽƵůĚŚĂǀĞŶŽĨĞĂƐŝďůĞƐŽůƵƟŽŶ͕ ĂŶĚǀŝĐĞͲǀĞƌƐĂ͘ * * % ' = dŚĞŽƉƟŵĂůƐŽůƵƟŽŶƚŽƚŚĞĚƵĂů>WWĐĂŶĂůƐŽďĞŽďƚĂŝŶĞĚĨƌŽŵƚŚĞŽƉƟŵĂůƐŽůƵƟŽŶŽĨƚŚĞƉƌŝŵĂů>WW͘dŚĞ ĂďƐŽůƵƚĞǀĂůƵĞƐŽĨD jĐŽƌƌĞƐƉŽŶĚŝŶŐƚŽƚŚĞƐůĂĐŬͬƐƵƌƉůƵƐǀĂƌŝĂďůĞƐŽĨĂƉƌŝŵĂůƉƌŽďůĞŵƌĞƉƌĞƐĞŶƚŽƉƟŵĂůǀĂůƵĞƐ ŽĨƚŚĞĚƵĂůǀĂƌŝĂďůĞƐ͘ ([DPSOH ! 1 =#[$% [& 2 " '! ([$%&[& ≥& TABLE 4.5 %DVLV )[$%&[& ≥$* [$)[& ≥$& [$+[& ≥, Simplex Tableau: Optimal Solution [0 [2 60 62 65 $0 $ 2 $5 EL [0 0 0 1 (0G5 0G5 1 0G5 (0G5 1 2 65 1 1 1 8GA ( FG5 0 (8GA FG5 (0 F [2 0G2 1 0 0G2 (0 1 (0G2 0 1 A FM 0 0G2 1 1 1 0 0 0 > 2 A 1 1 F 1 1 1 DM 1 1 0G02 0GA 1 0(0G02 0(0GA 0 >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ + , ? > . *426\00F\202\5 2\0:2\2:5\5 £ \0\2\5 ≥1 A\05\2\5 £ 1 2 + , \0\2\5 DM 606265 \040G02\240GA\541E . 1 2 * = 42: ¥ A49 * * 426¥0G02:0F¥0GA:02¥149 (&2120,&,17(535(7$7,212)'8$/ E N% LO 5 Discuss the economic interpretation of dual variables 7KH0D[LPLVDWLRQ3UREOHP D 760 ? > . =461[0: 59[2 # % 2[0:5[2£ A1 @ 6[0:5[2£ BA " [0[2≥1 0F $F % & %' C0111 E % & % &% A1& BA \0 & \2 % ' A1\0:BA\2 % & & $%& $ ' 2& ' % C61 2\0:6\2≥61 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ % & ' C59 % % 5\0:5\2≥59 \0\2 $ ? > . *4A1\0:BA\2 2\0:6\2≥61 5\0:5\2≥59 \0\2≥1 N \0\2 \0401G5\2429G5/ D M / . % ' % \0\2 & VKDGRZSULFHV LPSXWHGSULFHV % = $%& $ C61 % C01G5 &C29G5 & 2&OC01G5 & 4C 21G5 6 OC29G5 4C 011G5 4C021G5 4C61 $' % > % 5&OC1G5 & 4C01 5 4C 29 OC29G5 4C 59 ' % % PDUJLQDOYDOXHSURGXFWV PDUJLQDOSUR¿WDELOLW\ & % & C01G5 & C29G5 >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ & C01G5 % & %' > % C29G5 %C29G5 % ?#0 % ? > . *4A1?#0: BA?#/ 2?#0:6?#/≥61 5?#0:5?#/≥59 ?#0?#/≥1 ?#/ E % 6062 D M 60 $ 62 % * $ % % C61 % 2¥01G5:6¥29G54C61 61(6141> % ' 3 * DM ' 0F 60 % $ C0F % CF> % $% C0FCF 7KH0LQLPLVDWLRQ3UREOHP / 755 ? > . =461[0:26[2 21[0:91[2≥6F11 # ' F1[0:91[2≥8211 E ' [0[2≥1 * 6A 505 ' C569A 066 % TABLE 4.6 Simplex Tableau: Optimal Solution %DVLV [ 0 [ 2 6 0 6 2 $0 $ 2 EL [2 26 FG9 0 1 (0G91 1 0G91 066 60 1 A1 1 0 (0 (0 0 2611 FM 61 26 1 1 0 0 > 1 066 2611 1 1 1 0 12 0( 25 DM FG9 1 1 02G29 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ ? > . *46F11\0:8211\2 91\0:91\2£26 \0\2≥1 21\0:F1\2£61 68 TABLE 4.7 %DVLV Simplex Tableau: Optimal Solution \ 0 \ 2 6 0 6 2 EL 60 26 ( A1 1 0 (FG9 FG9 \2 8211 0 0 1 0G91 02G29 FM 6F11 8211 1 1 > 1 02G29 FG9 1 DM (2611 1 1 (066 * \041\2402G29+ ,> . 6F11¥1:8211¥02G294C569A E 3 & C02G29 & C569A ' 3 C02G29 & + , * ' 066 % $ 0L[WXUH$ 7 011& 21& F1& 21& OC1 & F1& OC02G29 &4C0B2G9 4C0B2G9 4C1 E C61 C0B2G9 C61(C0B2G94CFG9 * 0L[WXUH% C26 91& % >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ 91& OC1 4 C1 91& OC02G294C26 4C26 * * % 066 066¥9148211& ' E ' ' 8211( 6F1142611& ' 3 dŚĞD jǀĂůƵĞƐ;ŝŐŶŽƌŝŶŐƚŚĞŵŝŶƵƐƐŝŐŶƐͿĐŽƌƌĞƐƉŽŶĚŝŶŐƚŽƚŚĞƐůĂĐŬͬƐƵƌƉůƵƐǀĂƌŝĂďůĞƐŝŶĚŝĐĂƚĞƚŚĞŵĂƌŐŝŶĂů ƉƌŽĮƚĂďŝůŝƚǇŽƌƐŚĂĚŽǁƉƌŝĐĞƐŽĨƚŚĞƌĞƐŽƵƌĐĞƐͬǀĂƌŝĂďůĞƐƚŚĞǇƌĞƉƌĞƐĞŶƚ͘dŚĞƐŚĂĚŽǁƉƌŝĐĞŽĨĂŐŝǀĞŶĐŽŶƐƚƌĂŝŶƚ ŝƐƚŚĞĂŵŽƵŶƚďǇǁŚŝĐŚƚŚĞŽďũĞĐƟǀĞĨƵŶĐƟŽŶǁŽƵůĚďĞŝŵƉƌŽǀĞĚŝĨƚŚĞƌŝŐŚƚͲŚĂŶĚƐŝĚĞǀĂůƵĞŽĨƚŚĞĐŽŶƐƚƌĂŝŶƚ ŝƐŝŶĐƌĞĂƐĞĚďǇŽŶĞƵŶŝƚ͘ $3UREOHPZLWK8QXWLOLVHG5HVRXUFHV E * 752 ? > . =49[0:01[2:F[5 5[0:9[2:2[5£ A1 * 6[0:6[2:6[5£ 82 * 2[0:6[2:9[5£ 011 #& [0[2[5≥ 1 % 6F TABLE 4.8 Simplex Tableau: Optimal Solution %DVLV [0 [2 [5 60 62 65 EL [2 01 0G5 0 1 0G5 (0GA 1 F [5 F 2G5 1 0 (0G5 9G02 1 01 65 1 (FG5 1 1 0G5 (08G02 0 0F FM 9 01 F 1 1 1 > 1 F 01 1 1 0F DM (00G5 1 1 (2G5 (9G5 1 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ * % $ % & % C0A1 & D M C2G5 C9G5 % & / \0\2\5 % ? > . *4A1\0:82\2: 11\5 9\0:6\2:6\5≥01 2\0:6\2:9\5≥F \0\2\5≥1 TABLE 4.9 %DVLV 5\0:6\2:2\5≥9 Simplex Tableau: Optimal Solution \ 0 \ 2 \ 5 6 0 6 2 65 $ 0 $ 2 $ 5 EL \0 A1 0 1 (0G5 1 (0G5 0G5 (0GA 0G5 (0G5 2G5 60 1 1 1 FG5 0 (0G5 (2G5 (8GA 0G5 2G5 00G5 \2 82 1 0 08G02 1 0GA (9G02 0G02 (0GA 9G02 9G5 FM A1 82 011 1 1 1 0 0 0 > 2G5 9G5 1 00G5 1 1 1 1 1 DM 1 1 0F 1 F 01 0:6 0(F 0(01 / \0\2\5 & +, )RU3URGXFW$ - % C9 5 OC2G54 C 6 OC9G54 C 21G5 2 OC1 4 C 2 1 4 C 2AG5 > ' $ 2AG5 % C9 $ / $ 2AG5±9 C00G5 + D M [0, >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ +, )RU3URGXFW%DQG& 3URGXFW% C ' 3URGXFW& C 9 OC2G5 401G5 2 OC2G54 6G5 6 OC9G5 421G5 6 OC9G5421G5 OC 1 4 1 4C01 9 OC1 4 6 # % 4C01 1 4C F 4C F * ' 3 +, & 3 $ 6(16,7,9,7<$1$/<6,6 $ LO 6 Illustrate sensitivity "# FMDLMEL analysis & & - & * & C % C ' & 6HQVLWLYLW\DQDO\VLV SRVWRSWLPDOLW\DQDO\VLV ' - . = $$ - /% ' $ "# + , D % . + , D $$ + , D $$ % + , / % ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ *UDSKLF$SSURDFKWR6HQVLWLYLW\$QDO\VLV " 750 LO 7 ĂƌƌǇŽƵƚƐĞŶƐŝƟǀŝƚǇ ? > . = 461[0:59[2 # % 2[0:5[2 £A1 @ 6[0:5[2 £BA " [0[2 ≥1 ĂŶĂůǇƐŝƐƵƐŝŶŐŐƌĂƉŚŝĐ ĂƉƉƌŽĂĐŚ͕ƐŝŵƉůĞǆŵĞƚŚŽĚ ĂŶĚƐŽůǀĞƌ [0 $ [2 % $ % * 62 [04 0F[24F= 0111 % % [0[2 E ! . % +FM¢V, $ +EL¢V, &KDQJHVLQ2EMHFWLYH )XQFWLRQ&RHIÀFLHQWV E % $ % % % )LJXUH *UDSKLF3UHVHQWDWLRQ³6HQVLWLYLW\$QDO\VLV % % $ 30 % $ 30 / 30461 $ % 61[0:59[24D [24+( 61G59,[0: ( 61G59Q@ ' \4E[:D ' E* (2G5 (6G5R 30 $ % S + %, $ %30 $ % ±30G59 (2G5 ± 30G59 (2G5 $ % S +±30 59,;(2G5 30<81G5 > $ % & ( 6G5 >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ (30G59 < (6G5 30 ; 061G5 & 81G5£30£061G5 0F $F %= 81G5£30£061G5 % % %32 - (61G32;(2G5 32!A1 $ ±6132; ( 6G5 32<51 & 51£32£A1 &KDQJHVLQ5LJKWKDQGVLGH9DOXHV RI&RQVWUDLQWV 5LJKWKDQGVLGH5DQJLQJ $ & E0 ' / E04A1E0 E0 > + ' E0, * * 65 E0;BA E04BA )LJXUH *UDSKLF3UHVHQWDWLRQ5+65DQJLQJ . 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TABLE 4.10 5[0:9[2:2[5£A1 6[0:6[2:6[5£82 2[0:6[2:9[5£011 [0[2[5≥1 * * #& Simplex Tableau: Optimal Solution %DVLV [ 0 [2 [ 5 6 0 6 2 6 5 EL 0G5 0 1 0G5 (0GA 1 F [2 01 [5 F 2G5 1 0 (0G5 9G02 1 01 65 1 ( FG5 1 1 0G5 (08G02 0 0F FM 9 01 F 1 1 1 > 1 F 01 1 1 0F DM (00G5 1 1 (2G5 (9G5 1 D &KDQJHVLQ2EMHFWLYH)XQFWLRQ&RHIÀFLHQWVFM¢V % ' % . % S [0 N $ [2[5 L 9DULDEOHV,QFOXGHGLQWKH6ROXWLRQ D % + , = % D M + , ' * [2 D M (00G5 1 1 (2G5 (9G5 1 D % [2 0G5 0 1 0G5 (0GA 1 T (00 1 ( ( (2 01 ≠ ≠ " " ' + 01, % ' + (2, % C01 C2 % & CF C21+ 01(24F01:01421, % CF C21 % & [5* C6 C01 ' ( 6: 2 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ D M ( 00G5 1 1 (2G5 (9G5 1 D % [5 2G5 1 0 ( 0G5 9G02 1 T (00G2 ( 1 2 (6 ( ≠ ≠ " " LL 9DULDEOHVQRW,QFOXGHGLQWKH6ROXWLRQ / [0 D M ' (00G5 $ C00G5 % 00G5 % 00G5 % C2AG5+49:00G5, & % C00G5 % & ,QWURGXFLQJD1HZ3URGXFW - > % ' ' & % CF & % & > % U ' * ' ' C8 C 5 OC2G5 2 5 % OC9G5 9 2 & OC1 1 8 > % ' % E &KDQJHVLQWKHEL9DOXHV5LJKWKDQGVLGH5DQJLQJ / $ $ DM 606265 ' 2G59G51 % / %C2G5 % > % %C9G5 & % - + , + ,' * % . A1¥2G54C61 & % ' ULJKWKDQGVLGHUDQJLQJ 5+6UDQJLQJ >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ * % EL % DLM +&, EL DLM EL DLM F 0G5 26 ¨ " 01 (0G5 ( 51 ¨ " 0F 0G5 96 SRVLWLYH' ' GHFUHDVHG QHJDWLYH' ' LQFUHDVHG 26 51 A1 C2G5 +A1(26,45A +A1:51,B1 = ' C2G5 5AB1 % * EL DLM EL GDLM F (0GA (6F 01 9G02 26 ¨ " 0F (08G02 (20AG08 ¨ " - & 2620AG08 82(2646F82:20AG0840661G08 F9 & N 0F /0F DQ\ ' $$ & +011(0F,4F2 % ' VXEWUDFWLQJ @-> +EL, & @-> % + , VODFN $$ $' $ - $$ $' $ DGGLQJ @-> > @-> % dŽĚĞƚĞƌŵŝŶĞƚŚĞƌĂŶŐĞŽǀĞƌǁŚŝĐŚƐŚĂĚŽǁƉƌŝĐĞƐĂƌĞǀĂůŝĚ͕ǁĞĐŽŶƐŝĚĞƌbiͬaijƌĂƟŽƐĨŽƌĞĂĐŚƐůĂĐŬǀĂƌŝĂďůĞĂŶĚ ƚŚĞŶsubtractƚŚĞůĞĂƐƚƉŽƐŝƟǀĞĂŶĚůĞĂƐƚŶĞŐĂƟǀĞŽĨƚŚĞǀĂůƵĞƐĨƌŽŵƚŚĞŽƌŝŐŝŶĂůbiǀĂůƵĞƐƚŽǁŚŝĐŚƚŚĞƐůĂĐŬ ǀĂƌŝĂďůĞŝƐƌĞůĂƚĞĚ͘,ŽǁĞǀĞƌ͕ƚŚĞƐĞƋƵĂŶƟƟĞƐĂƌĞadded ŝĨƚŚĞǀĂƌŝĂďůĞŝŶǀŽůǀĞĚŝƐĂƐƵƌƉůƵƐǀĂƌŝĂďůĞ͘ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ F &KDQJHVLQWKH7HFKQRORJLFDO&RHIÀFLHQWVDLM·V ' % DLM!V DLM ) ' N $ [0 $ ' 562 % & > ' % W+ , $ % 600 TABLE 4.11 Simplex Tableau: Optimal Solution %DVLV [ 0 [ 2 [ 5 6 0 6 2 6 5 EL [2 01 1 1 + W 3 6 0 1 0G5 (0GA 1 F [5 F 2 5 - W 3 12 1 0 (0G5 9G02 1 01 65 1 8 17 - + W 3 12 1 1 0G5 (08G02 0 0F FM 9 01 F 1 1 1 > 1 F 01 1 1 0F DM 11 5 + W 3 3 1 1 (2G5 (9G5 1 - 601 [0 ' [0 W 62 + 62 & % , 1HZ[0FROXPQ ˘ È1 1 Í3 + 6 W ˙ ˙ Í Í2 - 5 W ˙ Í 3 12 ˙ ˙ Í Í -8 + 17 W ˙ Í 3 12 ˙ Í -11 5 ˙ Í + W˙ 3 ˚ Î 3 4 2OG[0FROXPQ PLQXVW 2OG62FROXPQ 4 È1 ˘ Í3 ˙ ˙ Í Í2 ˙ Í3 ˙ ˙ Í Í -8 ˙ Í3 ˙ Í -11˙ ˙ Í Î 3 ˚ È -1 ˘ Í6 W ˙ Í ˙ Í5W ˙ Í12 ˙ Í ˙ Í -17 W ˙ Í 12 ˙ Í -5 ˙ Í W ˙ Î3 ˚ E [0 ( >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ - 11 5 + W;1 3 3 W; 11 3 11 ¥ 4 3 5 5 $ % 00G9 + 6(00G94BG9 , $GGLQJRU'HOHWLQJ&RQVWUDLQWV "## = & & = & / &G ' 3 ' - $ D 65 + % , (a) Objective function coefficients Changes in objective function coefficients For basic variables in optimal solution For non-basic variables in optimal solution Calculate Dj /aij (aij values in the row of the variable) Subtract Dj value corresponding to the variable from the objective function coefficient, cj Obtain LN and LP values and add these to the objective function coefficient, cj Lower limit cj + LN Upper limit cj + LP Range Nature of objective function Lower limit Upper limit Max –• c j – Dj Min cj – Dj • ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ (b) RHS ranging Changes in the RHS values of constraints Calculate bi /aij from optimal solution table for slack/surplus variables (aij values in the column of relevant variable related to the constraint) For basic variables in optimal solution For non-basic variables in optimal solution Obtain LP values and find range Obtain LN and LP values and find range Nature of variable Lower limit Nature of variable Upper limit • bi – LP Slack –• bi + LP Surplus (The bi values here are from given LPP) Range Lower limit Upper limit bi – LN bi – LP Slack bi + LN bi + LP Surplus (The bi values here are from given LPP) (c) Changes in the constraints of the problem Change in constraint Addition of a constraint Deletion of a constraint Substitute solution values in the constraint Constraint not violated No change in solution Constraint may or may not be redundant Constraint is binding Constraint is not binding Need to re-solve the problem No change in solution Constraint violated Need to re-solve the problem &KDUW 6HQVLWLYLW\$QDO\VLVDWD*ODQFH 6ROYHUDQG6HQVLWLYLW\$QDO\VLV D 5 "## > 6HQVLWLYLW\5HSRUW/LPLWV5HSRUW= > @ . % +FM¶V, +EL¶V, >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ D > D 5* * 66 0LFURVRIW([FHO6HQVLWLYLW\5HSRUW :RUNVKHHW>%RRN@6KHHW 5HSRUW&UHDWHG0D\$0 $GMXVWDEOHFHOOV &HOO 1DPH )LQDO YDOXH 5HGXFHG FRVW 2EMHFWLYH FRHI¿FLHQW $OORZDEOH LQFUHDVH $OORZDEOH GHFUHDVH % 2XWSXWSURGXFW$ ± ( & 2XWSXWSURGXFW% ' 2XWSXWSURGXFW& &RQVWUDLQWV &HOO 1DPH )LQDO YDOXH 6KDGRZ SULFH &RQVWUDLQW 5+6 $OORZDEOH ,QFUHDVH $OORZDEOH 'HFUHDVH ( )DEULFDWLRQKRXUVXWLOLVHG ( )LQLVKLQJKRXUVXWLOLVHG ( 3DFNDJLQJKRXUVXWLOLVHG ( )LJXUH 6HQVLWLYLW\$QDO\VLV D M + 5HGXFHG &RVW, . % > E 6HQVLWLYLW\$QDO\VLVZLWK0XOWLSOH3DUDPHWHU&KDQJHV * @-> @-> LO 8 Execute 100 percent rule in sensitivity analysis with multiple parameter changes L &KDQJHVLQWKH2EMHFWLYH)XQFWLRQ&RHIÀFLHQWV % . % > 0 / % FM! $3 D M 2 = % FM! 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= 481[0:A9[2:F1[5:89[6 > . 6[06[25[58[6 £B1 A[05[29[56[6 £021 9[02[25[55[6 £A1 A[09[2 [5:2[6 £011 & "## [0[2[5[6 ≥1 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ ? > . =481[0:A9[2:F1[5:89[6:160:162:165:166 A[05[29[56[662 4021 9[02[25[55[665 A1 A[09[2 [5:2[6:66 4011 [0[2[5[660626566 ≥1 602(606* > 5 "## [041[2409[5401[641 =481 ¥1:A9¥09:F1¥01:89¥14C0889* 50/0 52 29&/2 09 6[06[25[58[660 B1 TABLE 4.12 Simplex Tableau 1: Non-optimal Solution %DVLV [0 [ 2 [ 5 [ 6 60 62 6 5 6 6 E L EL DLM 60 1 6 6 5 8 0 1 1 1 B1 51 62 1 A 5 9 6 1 0 1 1 021 26 65 1 925H 5 1 1 0 1 A1 20 ¨ 66 1 A 9 0 2 1 1 1 0 011 011 FM 81 A9 F1 89 1 1 1 1 > 1 1 1 1 B1 021 A1 011 DM 81 A9 F1 ≠ 89 1 1 1 1 TABLE 4.13 Simplex Tableau 2: Non-optimal Solution %DVLV [0 [ 2 [ 5 [6 6 0 62 6 5 6 6 E L EL DLM 09 ¨ (A1 60 62 1 1 (0 (8G5 2H (0G5 1 1 6 (0 0 1 1 0 (0 (9G5 1 1 51 21 [5 F1 9G5 2G5 0 0 1 1 0G5 1 21 51 66 1 05G5 05G5 1 0 1 1 (0G5 0 F1 261G05 FM 81 A9 F1 89 1 1 1 1 1 1 21 1 51 21 1 F1 DM > (0B1G5 59G5 1 (9 1 1 (F1G5 1 ≠ >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ TABLE 4.14 %DVLV Simplex Tableau 3: Optimal Solution [ 0 [2 [ 5 [ 6 6 0 62 6 5 6 6 EL [2 A9 (0G2 0 1 2 0G2 1 (0G2 1 09 62 1 (9G2 1 1 (0G5 0GA 0 (00GA 1 29 [5 F1 2 1 0 (0G5 (0G5 1 2G5 1 01 66 1 05G2 1 1 (25G5 (05GA 1 00GA 0 09 FM 81 A9 F1 89 1 1 1 1 > 1 09 01 1 1 29 1 09 DM (009G2 1 1 (F9G5 (59GA 1 (029GA 1 6HQVLWLYLW\$QDO\VLV L &KDQJHVLQ&RHIÀFLHQWVLQWKH2EMHFWLYH)XQFWLRQ * 3036 30 %C009G2 36 CF9G5 % 30 C009G2 % 36 CF9G5 * 3235& DM (009G2 1 1 (F9G5 (59GA 1 (029GA 1 DLMD % (0G2 0 1 2 0G2 1 (0G2 1 @ 009 1 ( (F9GA (81GA ( 029G5 ( / + , 029G5 60A8 + , 81GA 00A8 > 35 DM (009G2 1 1 (F9G5 (59GA 1 (029GA 1 DLMD % 2 1 0 (0G5 (0G5 1 2G5 1 @ (009G6 ( 1 F9 59G2 ( 029G6 ( * 59G240891 009G6 2F89 3URGXFW 9DULDEOH &XUUHQWFRHI¿FLHQW $OORZDEOHLQFUHDVH $OORZDEOHGHFUHDVH 5DQJH 30 32 35 [0 [2 [5 81 A9 F1 9891 60A8 0891 % 00A8 2F89 ( • 02891 9555 01AA8 9029 B891 36 [6 89 2F55 % ( • 01555 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ LL &KDQJHVLQWKH5LJKWKDQGVLGH9DOXHV & 9DULDEOH 6 0 62 0G2 0GA 5DWLRIRU 6 5 66 EL 6 0 1 ( 0G2 1 09 51 0 ( 00GA 1 29 091 (0G5 1 2G5 1 01 (51 ( 05GA 1 00GA 0 09 (B1G05 62 6 5 66 • (51 • 29 (091G00 • • 09 • • B1G00 09 * ' / &RQVWUDLQW &XUUHQW5+6YDOXH $OORZDEOHLQFUHDVH $OORZDEOHGHFUHDVH 5DQJH 0 B1 B1G05 51 A1 02A1G05 2 021 % 29 B9 • 5 A1 091G00 B1G00 981G00 F01G00 6 011 % 09 F9 • +, 3036 $3 D M 0* & . % 30±• 02891 36±• 01555 + , & + , 30 36 / $ +, + , - DF2 4F9(A9421 ,2460A8 U2 421G60A8416F / DF5 481(F14(01 '545029 U5 4(+(01,G 50294152 E U0U641 S UM41:16F:152:141F1 F1 > 011 ] 81¥1:F9¥09:81¥01:89¥14C0B89 >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ + , * D F0 4F1(81401 DF2 4889(A94029 DF5 4A1(F14(21 DF6 41 + , 52 - * 2011 @ & " * EL & / UL4DEL,L DEL!1 UL4±DEL'L DEL<1 DE0 4F1(B14(01 '0 451 U0 401G514155 DE5 4A8(A148 ,5 4091G00 U5 4+8¥00,G0914190 U04155U24111U54190U64111SUL41F6> 0 @ . = @-> D M 6065 ' ±59GA(029GA =$ ,0 49891 ,2 460A8 '5 45029 U0 401G98914108 U2 4029G60A84151 U5 421G502941A6 U6 41 - S UM4108:151:1A6:1114000> +, + , $$ - EL @-> L DEL @-> L D EM4EEL (X EL ,L @-> EL 'L @-> EL $ $ DEL41+LH EL ,UL41E UL£0 011 @ EL UL SUL!0 +, + , ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ &KDQJHLQSUR¿W ) 5001&O59GA ±9F55 /08 069F5 O029GA E F891 / %=4C0889:CF8914C0FA291+ % % [041[24A9[540F[641, + , - DE0 4B2(B142 DE2 4011(0214(21 DE5 4A9(A149 ,0 4B1G05 '2 429 ,5 4091G00 U0 42¥05GB1412B U2 421G2941F1 U5 49¥00G0914158 DE6 4021(011421 ,6 4• U6 429G•4111 > S UL;0 - =$ $ S UL406A 5(9,(:,//8675$7,216 ([DPSOH )LQGWKHGXDOSUREOHPIRUWKHIROORZLQJ 0LQLPLVH 6XEMHFWWR = [±[[ [[[ ≥ [[[ ≥ [±[±[ £ [±[[ ≥ [[±[ [[[ ≥ ? (0 (8[0 : 2[2 : [5 ≥ (01 > ' ' 2[0 : 9[2 ( 5[5 ≥ 5 2[0:9[2(5[5£5E 3ULPDO 'XDO ? > . = 49[0(A[2:F[5 ? > . *402\0:9\2(01\5:6\6:5\9(5\A 5[0:6[2:A[5 ≥02 5\0:\2(8\5:\6:2\9(2\A £9 [0:5[2:2[5 ≥9 6\0:5\2:2\5(2\6:9\9(9\A £(A (8[0:2[2:[5 ≥(01 A\0:2\2:\5:6\6(5\9:5\A £F [0(2[2:6[5 ≥6 \L≥1L402"A)Y >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ 2[0:9[2(5[5 ≥5 (2[0(9[2:5[5 ≥(5 [0[2[5 ≥1 \9(\A4\8 ? > . *402\0:9\2(01\5:6\6:5\8 ( 6\0(5\2(2\5:2\6(9\8≥ A A\0:2\2:\5:6\6(5\8£ F \0\2\5\6≥ 1\8 ([DPSOH 5\0:\2(8\5:\6:2\8£ 9 7KUHH SURGXFWV $ % DQG & DUH PDGH E\ PL[LQJ WKUHH PDWHULDOV7KH QXPEHU RI XQLWV RI GLIIHUHQWUDZPDWHULDOVUHTXLUHGWRSURGXFHWKHVHSURGXFWVWKHUDZPDWHULDODYDLODELOLW\DQGWKHXQLWSUR¿WRQ HDFKRIWKHSURGXFWVDUHJLYHQKHUH 3URGXFW $ 5DZPDWHULDO 8QLWSUR¿W 5 5 5 % & $YDLODELOLW\ L +RZPDQ\XQLWVRIHDFKSURGXFWVKRXOGEHSURGXFHGVRDVWRPD[LPLVHWKHWRWDOSUR¿W" LL :ULWHWKHGXDORIWKHSUREOHPDQGXVHLWWRFKHFNWKHRSWLPDOVROXWLRQ "[0[2[5 $%& * "## ? > . =461[0:51[2:21[5 2[09[21[5 £B11 2[09[25[5 £611 6[02[22[5 £A11 [0[2[5 ≥1 & ? =461[0:51[2:21[5:160:162:165 > . 2[09[21[5:60 4B11 2[09[25[5:62 4611 6[0:2[2:2[5:65 4A11 [0[2[5606265 ≥1 609(608 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 4.15 Simplex Tableau 1: Non-optimal Solution %DVLV [ 0 [ 2 [ 5 6 0 6 2 6 5 E L ELDLM 60 1 2 9 01 0 1 1 B11 691 62 1 2 9 5 1 0 1 611 211 6516H 2 2 1 1 0 A11 091 ¨ =41 FM 61 51 21 1 1 1 > 1 1 1 B11 611 A11 DM 61 51 21 1 1 1 ≠ TABLE 4.16 Simplex Tableau 2: Non-optimal Solution [ 0 [ 2 [5 6 0 6 2 6 5 EL %DVLV EL DLM 60 1 1 6 B 0 1 (0G2 A11 091 62 1 1 6H 2 1 0 (0G2 011 29 [0 61 0 0G2 0G2 1 1 0G6 091 511 FM 61 51 21 1 1 1 > 091 1 1 A11 011 1 DM 1 01 1 1 1 (01 ≠ TABLE 4.17 =4A111 Simplex Tableau 3: Optimal Solution %DVLV [0 [2 [ 5 6 0 6 2 6 5 60 [2 [0 1 51 61 1 1 0 1 0 1 8 0G2 0G6 0 1 1 (0 0G6 (0GF 1 (0GF 9G0A FM 61 51 21 1 1 1 > 0589 29 1 911 1 1 DM 1 1 (9 1 (9G2 (59G6 ¨ EL 911 29 0589 =4A291 * > 5 [040589[2429[541 =4A291 / X $ X % X & # % 0589 29 CA291 >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ "\0\2\5 "## ? *4B11\0:611\2:A11\5 > . 2\0:2\2:6\5 ≥61 9\:9\2:2\5 ≥51 1\0:5\2:2\5 ≥21 \0\2\5 ≥1 * 608 + D M & , \041\249G2\5459G6 > . . *4B11¥1:611 ¥9G2:A11¥59G64A291 ([DPSOH $FRPSDQ\PDQXIDFWXUHVDQGVHOOVWKUHHPRGHOVRIODUJHVL]HGSUHVVXUHFRRNHUVIRUFDQWHHQ XVH:KLOHPDUNHWGHPDQGVSRVHQRFRQVWUDLQWVVXSSOLHVRIDOXPLQXPOLPLWHGWRNJSHUZHHNDQGDYDLODELOLW\ RIPDFKLQHWLPHOLPLWHGWRKRXUVSHUZHHNUHVWULFWWKHSURGXFWPL[7KHUHVRXUFHXVDJHRIWKHWKUHHPRGHOV DQGWKHLUSUR¿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£891 5[0:6[2:9[5 £A11 [0[2[5 ≥1 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ & 6062 ' ? =4A1[0:21[2:F1[5:160:162 > . A[0:5[2:9[5:60 4891 5[0:6[2:9[5:62 4A11 [0[2[56062 ≥1 60F(621 TABLE 4.18 %DVLV Simplex Tableau 1: Non-optimal Solution [ 0 [ 2 [ 5 5 6 0 6 2 E L EL GDLM 6 0 1 A 9 0 1 891 091 62 1 569H 1 0 A11 021 ¨ FM A1 21 F1 1 1 > 1 1 1 891 A11 DM A1 21 F1 1 1 ≠ TABLE 4.19 Simplex Tableau 2: Non-optimal Solution [ 0 [2 [ 5 6 0 6 2 E L EL GDLM %DVLV 6 0 1 5H (0 1 0 (0 091 91 [5 F1 5G9 6G9 0 1 0G9 021 211 FM A1 21 F1 1 1 > 1 1 021 091 1 DM 02 (66 1 1 (0A ¨ =4BA11 ≠ TABLE 4.20 Simplex Tableau 3: Optimal Solution %DVLV [ 0 [2 [ 5 6 0 6 2 EL [ 0 [ 5 A1 F1 0 1 (0G5 0 1 0 0G5 (0G9 (0G5 2G9 91 B1 F M > D M A1 91 1 21 1 (61 F1 B1 1 1 1 (6 1 1 (02 =401211 0091 05B1 % C01211 >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ +, 5+65DQJLQJ * 60 * 62 E L DLM ELGDLM DLM ELGDLM 91 0G5 091 (0G5 (091 B1 (0G9 (691 2G9 229 @+891(091, +891:691, +A11(229, +A11:091, A11 0211 589 891 + , > 091& % 091¥64CA11 + , - % 091¥024C0F11 + , * 05 D M 1 (61 1 (6 (02 DLM 1 0 0 (0G9 2G9 @ ( (61 1 21 (51 \? 4C51 > % B1¥094C0591 + , 21 : 61 4 C A1 & 02 + , D %4C61 091 " % 45¥6:5¥024C6F E 4CFG ([DPSOH 7KHVLPSOH[WDEOHDXIRUDPD[LPLVDWLRQSUREOHPRIOLQHDUSURJUDPPLQJLVJLYHQDVIROORZV 3URGXFWPL[ F M [M [ [ 6 6 4XDQWLW\ EL [ 6 ± FM ] M FM±]M ± ± ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ $QVZHUWKHIROORZLQJTXHVWLRQVJLYLQJUHDVRQVLQEULHI D ,VWKLVVROXWLRQRSWLPDO" E $UHWKHUHPRUHWKDQRQHRSWLPDOVROXWLRQV" F ,VWKLVVROXWLRQGHJHQHUDWH" G ,VWKLVVROXWLRQIHDVLEOH" H ,I6LVVODFNLQPDFKLQH$ LQKRXUVZHHN DQG6LVVODFNLQPDFKLQH% LQKRXUVZHHN ZKLFKRIWKHVH PDFKLQHVLVEHLQJXVHGWRWKHIXOOFDSDFLW\ZKHQSURGXFLQJDFFRUGLQJWRWKLVVROXWLRQ" I $FXVWRPHUZRXOGOLNHWRKDYHRQHXQLWRISURGXFW[DQGLVZLOOLQJWRSD\LQH[FHVVRIWKHQRUPDOSULFH LQRUGHUWRJHWLW+RZPXFKVKRXOGWKHSULFHEHLQFUHDVHGLQRUGHUWRHQVXUHQRUHGXFWLRQRISUR¿WV" J +RZPDQ\XQLWVRI WKHWZRSURGXFWV[DQG[DUHEHLQJSURGXFHGDFFRUGLQJWRWKLVVROXWLRQDQGZKDW LVWKHWRWDOSUR¿W" K 0DFKLQH$ DVVRFLDWHGZLWKVODFN6LQKRXUVZHHN KDVWREHVKXWGRZQIRUUHSDLUVIRUWZRKRXUVQH[W ZHHN:KDWZLOOEHWKHHIIHFWRQSUR¿WV" L +RZPXFKZRXOG\RXEHSUHSDUHGWRSD\IRUDQRWKHUKRXU SHUZHHN RIFDSDFLW\HDFKRQPDFKLQH$ DQGPDFKLQH%" M $QHZSURGXFWLVSURSRVHGWREHLQWURGXFHGZKLFKZRXOGUHTXLUHSURFHVVLQJWLPHRIKRXURQPDFKLQH $DQGPLQXWHVRQPDFKLQH%,WZRXOG\LHOGDSUR¿WRICSHUXQLW'R\RXWKLQNLWLVDGYLVDEOHWR LQWURGXFHWKLVSURGXFW" 620 TABLE 4.21 %DVLV Simplex Tableau [0 [2 6 0 6 2 EL [2 9 0 0 0 1 01 62 1 0 1 (0 0 5 FM 6 9 1 1 > 1 01 1 5 (0 1 (9 1 D. +, Z D M! ' 3 +, E $ [060D M / +, $ 3 DM +, 3 +, Z % +, ? $ & 60 3 +, DM [0 (0 [0 C0+ , [0 C0 % >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ +, / [001 [2 % 6¥19¥014C91 +, $ D M C9 % 2$ 42¥94C01 + , $% C9C1 $ +., > $% C9C1# %+0G2,¥9+0G5,¥14C291> % C5 % 91 ([DPSOH 7KH FRPSDQ\ 3RUWODQG SOF KDV ¿YH SURGXFWV LQ LWV UDQJH DQG LV FXUUHQWO\ UXQQLQJ WKH IROORZLQJVDOHVSURGXFWLRQSURJUDPPH 'HWDLOV 6DOHVLQXQLWV 3HUXQLW 3URGXFWV $ % & ' ( 6DOHVSULFH C 9DULDEOHFRVW C /DERXUKRXUV 0DFKLQHKRXUV 7KLVSURJUDPPHIXOO\XWLOLVHVWKHDYDLODELOLW\RIODERXUDQGPDFKLQHWLPH $OLQHDUSURJUDPPHUHYHDOVWKDWWKHODERXUDQGPDFKLQHKRXUVKDYHDVKDGRZSULFHRIC0DQGCSHUKRXU 'HWHUPLQHWKHRSWLPDOSURGXFWLRQSURJUDPPHIURPWKLVLQIRUPDWLRQDQGFRPSDUHWKHFRQWULEXWLRQHDUQHGZLWK WKDWRIWKHH[LVWLQJSURJUDPPH ,&0$0D\$GDSWHG > " 491¥21:61¥09:81¥51:A1¥2F:21¥01 499F 99F111 ? 491¥01:61¥1F:81¥09:A1¥02:21¥16 42A8 2A8111 ? > . =4+591(091,$:+511(02A,%:+691(211,&:+911(256,':+211(1FF,( 21$:09%:51&:2F':01( £99F 01$:1F%:09&:02':16( £2A8 $%&'(≥1 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ ? > . 21;:01<≥211 09;:1F<≥086 51;:09<≥291 2F;:02<≥2AA 01;:16<≥002 *499F;:2A8< ;<≥1 ;401<415 > % 6041562416541B9664196941 % $3 %( 09%:01(4 99F 1F%:16(4 2A8 > ' %420B+ ,(422B9+ , = 086¥20B:002¥22B94A5F0+ , CA5F011 [ =491¥211:61¥086:81¥291:A1¥2AA:21¥002492AA+ ,4C92AA11 C000911 ([DPSOH $FRPSDQ\LVSURGXFLQJWKUHHYDULHWLHVRIDSURGXFW'HOX[HPRGHO(FRQRP\PRGHODQG ([SRUWPRGHO7KHSURGXFWVUHTXLUHZRUNLQWZRSURGXFWLRQIDFLOLWLHVQDPHO\PDFKLQLQJDQGDVVHPEO\HDFK RQHRIZKLFKLVOLPLWHGE\WKHQXPEHURIODERXUKRXUVDYDLODEOH7KHGHWDLOHGLQIRUPDWLRQLVJLYHQLQWKHIROORZLQJ WDEOH /DERXUKRXUVUHTXLUHGSHUXQLW 3URGXFWLRQIDFLOLW\ $YDLODEOHODERXU KRXUV LQPLOOLRQV 'HOX[HPRGHO (FRQRP\PRGHO ([SRUWPRGHO 0DFKLQLQJ $VVHPEO\ 'HPDQG IRUHFDVW µXQLWV 3UR¿WSHUµ >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ <RXDUHUHTXLUHGWR D )RUPXODWHWKHSUREOHPDV/33DQGREWDLQRSWLPDOVROXWLRQ E ,QWHUSUHWWKHD MYDOXHVLQWKH¿QDOVLPSOH[WDEOHDX'HWHUPLQHWKHUDQJHVRYHUZKLFKVKDGRZSULFHVDUH YDOLG F :ULWHWKHGXDORIWKHSUREOHPDQGREWDLQRSWLPDOYDOXHVRIWKHGXDOYDULDEOHV +, "[[2[5 + , ) 7 7 "## ? > . =40A11[0:0911[2:5111[5 [ £911 [2 £F11 [5 £511 09[:09[2:51[5 £6211 21[:21[2:61[5 £5A11 & 6062656669 > 0 622622 629 629 [0[2[5 ≥1 TABLE 4.22 %DVLV Simplex Tableau 1: Non-optimal Solution [0 [2 [5 60 62 65 66 69 EL EL DLM 60 1 0 1 1 0 1 1 1 1 911 ( 62 1 1 0 1 1 0 1 1 1 F11 ( 65 1 1 1 0H 1 1 0 1 1 511 511 66 1 09 09 5 1 1 1 0 1 6211 0611 69 1 2 2 6 1 1 1 1 0 5A11 B11 0A11 0911 5111 1 1 1 1 1 1 1 1 911 F11 511 6211 5A11 0A11 0911 5111 1 1 1 1 1 FM > D M =41 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 4.23 %DVLV Simplex Tableau 2: Non-optimal Solution [0 [2 [5 6 0 62 65 66 69 EL EL DLM 60 1 0H 1 1 0 1 1 1 1 911 911 62 1 1 0 1 1 0 1 1 1 F11 ( [5 5111 1 1 0 1 1 0 1 1 511 ( 66 1 09 09 1 1 1 (5 0 1 5511 2211 69 1 2 2 1 1 1 (6 1 0 2611 0211 0A11 0911 5111 1 1 1 1 1 1 511 911 F11 1 5511 2611 0A11 0911 1 1 1 (5111 1 1 FM > D M TABLE 4.24 %DVLV =4B11111 Simplex Tableau 3: Non-optimal Solution [0 [2 [5 6 0 62 6 5 6 6 6 9 EL EL DLM [0 0A11 0 1 1 0 1 1 1 1 911 ( 62 1 1 0 1 1 0 1 1 1 F11 F11 [5 5111 1 1 0 1 1 0 1 1 511 ( 66 1 1 09 1 (09 1 (5 0 1 2991 0811 69 1 1 2H 1 (2 1 (6 1 0 0611 811 0A11 0911 5111 1 1 1 1 1 911 1 511 1 F11 1 2991 0611 1 0911 1 (0A11 1 (5111 1 1 FM > D M TABLE 4.25 %DVLV =40811111 Simplex Tableau 4: Optimal Solution [0 [ 2 [ 5 60 62 65 66 6 9 EL [0 0A11 0 1 1 0 1 1 1 1 911 62 1 1 1 1 0 0 2 1 (19 011 [5 5111 1 1 0 1 1 0 1 1 511 66 1 1 1 1 1 1 1 0 (189 0911 [2 0911 1 0 1 (0 1 (2 1 19 811 FM 0A11 0911 5111 1 1 1 1 1 911 811 511 1 011 1 0911 1 1 1 1 (011 1 1 1 (891 > D M \42891111 >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ * 629 ) 7 7 911111C811111511111 % C2891111 ' $ 65D M' 3 +, ) 0111 %C011 > 0111 %C011* 0111 % C891 % % ' ELGDLM DLM EL EL DLM 60 62 65 66 6 9 6 0 62 6 5 66 69 911 0 1 1 1 1 911 ( ( ( ( 011 0 0 2 1 (19 011 ( ± 511 1 1 0 1 1 ( ( 511 ( ( 0911 1 1 1 0 (189 ( ( ( 0911 (2111 811 (0 1 (2 1 19 ± ( ± ( @ 9DULDEOH /HDVWSRVLWLYH EL DLM /HDVWQHJDWLYH EL DLM 2ULJLQDOEL YDOXH 5DQJH 60 011 (811 911 611(0211 62 011 F11 811(E 65 91 511 291(A91 66 0911 69 0611 (591 (211 6211 2811(E 5A11 2211(5F11 +, * \\2\5\6\9 ? > . *4911 \ :F11 \ 2:511 \ 5:6211\ 6:5A11\ 9 \:1\2:1\5:09\6:21\9 ≥0A11 1\:\2:1\5:09\6:21\9 ≥0911 1\:1\2:\5:51\6:61\9 ≥5111 = \04011\241\541\641\94891 \0\2\5\6\9 ≥1 ([DPSOH 7KHIROORZLQJLQIRUPDWLRQUHODWHVWRWKUHHSURGXFWVSURGXFHGE\DIDFWRU\7KHXQLWSUR¿W UHVRXUFHVUHTXLUHGDQGUHVRXUFHVDYDLODELOLW\DUHDVVKRZQKHUH ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 3URGXFW 5HVRXUFH 5HVRXUFH DYDLODELOLW\ $ % & 5DZPDWHULDO NJ )DEULFDWLRQKRXUV $VVHPEO\KRXUV 8QLWSUR¿W C )XUWKHUWKHRSWLPDOVROXWLRQWRWKH/3IRUPXODWLRQRIWKLVSUREOHPLVVKRZQLQ7DEOHZKLOHRSWLPDOVROXWLRQ WRWKHGXDORIWKLVOLQHDUSURJUDPPLQJSUREOHPLVJLYHQLQ7DEOH <RXDUHUHTXLUHGWR D )LQGWKHRSWLPDOVROXWLRQWRWKHSULPDODQGGXDOSUREOHPVDQGYHULI\WKDWWKHREMHFWLYHIXQFWLRQYDOXHV DUHHTXDOIRUWKHVH E 6WDWHVKDGRZSULFHVLQHDFKFDVH$OVRLQGLFDWHWKHHIIHFWRIDXQLWLQFUHDVHLQHDFKRIWKHUHVRXUFHV DQGLQWKHSUR¿WUDWHV F 'HWHUPLQHWKHUDQJHRYHURIYDOLGLW\RIWKH5+6YDOXHVDQGWKHUDQJHRIREMHFWLYHIXQFWLRQFRHI¿FLHQWV LQHDFKFDVH G &RPPHQWRQWKHUHVXOWVLQ E TABLE 4.26 Simplex Tableau: Optimal Solution %DVLV [0 [2 [5 6 6 2 6 5 EL [2 6 0G5 0 1 0G5 (0G5 1 21G5 [5 5 9GA 1 0 (0GA 2G5 1 91G5 65 1 (9G5 1 1 (2G5 (0G5 0 F1G5 FM 2 6 5 1 1 1 > 1 21G5 91G5 1 1 F1G5 (00GA 1 1 (9GA (2G5 1 DM TABLE 4.27 %DVLV Simplex Tableau: Optimal Solution \ 0 \ 2 \ 5 6 0 6 2 6 5 $ 0 $ 2 $ 5 EL \0 A1 0 1 2G5 1 (0G5 0GA 1 0G5 (0GA 9GA 60 1 1 1 9G5 0 (0G5 (9GA (0 0G5 9GA 00GA \2 61 1 0 0G5 1 0G5 (2G5 1 (0G5 2G5 2G5 FM A1 61 F1 1 1 1 ? ? ? > 9GA 2G5 1 00GA 1 1 1 1 1 1 1 F1G5 1 21G5 91G5 ? ?( 21G5 ?( 91G5 DM >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ +, = 3ULPDO 6ROXWLRQ 2EMHFWLYH IXQFWLRQ [041[24 2¥1:6¥ 'XDO 20 50 [54 3 3 5 2 \04 \24 \541 6 3 20 50 230 5 2 230 :5¥ 4 A1¥ :61¥ :F1¥14 3 3 3 6 3 3 +, > 3ULPDO 'XDO @ C9GA & # / C1 * C2G5 # C21G5 / C1 # D C91G5 [ ,QFUHDVHRI (IIHFWRQSUR¿W =& C9GA = C2G5 = E C0 % / E C0 % C21G5 C0 % D C91G5 +, 3ULPDOSUREOHP + , @->@ @ @-> VXEWUDFWLQJ EL GDL M % EL DLM )RU6 EL DLM DLM 21"# )RU62 )RU65 EL DLM DLM EL DLM (0G5 (21"E 1 • 21G5 0G5 91G5 (0GA (011 2G5 29"# 1 • F1G5 (2G5 (61"E (0G5 (F1 0 F1G5"# ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ E 5HVRXUFH EL @ 5DQJH /RZHUOLPLW 8SSHUOLPLW A1 A1(21461 A1(+(61,4011 * 61 61(29409 61(+(21,4A1 / F1 F1(F1G540A1G5 • + , @ . % D M GDLMDGGLQJ % 9DULDEOH D M (00GA 1 1 (9GA (2G5 1 [2 DLM 0G5 0 1 0G5 (0G5 1 @ (00G2 1 ( ± ( DLM 9GA 1 0 (0GA 2G5 1 @ (00G9 ( 1 ± ( [5 E 3URGXFW FM / 5DQJH /RZHUOLPLW 8SSHUOLPLW 2 (• 2:00GA425GA 6 6(9G245G2 6:24A D 5 5(042 5:94F 'XDOSUREOHP + , @->@ @ @-> DGGLQJ EL G DLM EL DLM )RU6 EL DLM DLM )RU62 EL DLM DLM 9GA 1 • (0G5 (9G2"E 0GA 00GA 0 00GA"# (0G5 (00G2 (9GA 2G5 1 • 0G5 2"# (2G5 )RU65 EL DLM 9"# (00G9 (0"E >ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ///͗ƵĂůŝƚǇĂŶĚ^ĞŶƐŝƟǀŝƚǇŶĂůǇƐŝƐഩഩ E 8QLWSUR¿W EL 5DQJH /RZHUOLPLW 8SSHUOLPLW # / 2 (• 2:00GA425GA # 6 6(9G245G2 6:24A # D 5 5(042 5:94F + , @ . % D M GDLMDGGLQJ % 9DULDEOH D M 1 1 F1G5 1 21G5 91G5 \ DLM 0 1 2G5 1 (0G5 0GA @ 1 ( ( ± 011 DLM 1 0 0G5 1 0G5 (2G5 @ ( 1 F1 ( ± \2 E 5HVRXUFH EL 5DQJH /RZHUOLPLW 8SSHUOLPLW @ A1 A1(21461 A1:614011 * 61 61(29409 61:214A1 / F1 F1(F1G540A1G5 • +, @-> . % > . % @-> ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 36KP2TCEVKEG 6JG$NGPFKPI2TQDNGO*GCNVJ[(QQFUCPF5PCEMU+PFKC.VF ? 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The transportation method is developed to deal with the transportation of goods from different sources to different destinations, given the relevant data like available quantities at various sources, demand at each of the destinations, the cost of shipping along each route and non-availability of certain routes, if any. It may be interesting to note that while such problems can also be handled as linear programming problems; the transportation method provides an efficient means to solve them. The method allows the manager to seek answers to the questions like the following: What is the optimal way of shipping goods from various sources (warehouses) to different markets so as to minimise the total cost involved in the shipping? How to handle a situation when some routes are not available or when some units have to be necessarily transported from a particular source to a particular market? How would the optimal shipping schedule change if some routes become cheaper/costlier? If it were possible to increase supply, which of the sources should be preferred? Instead of allowing shipping of goods only from listed sources to different destinations, if it were possible to ship goods from a particular source to another source or destination and then from there to a further destination, how much cost can be saved? This is what is called a transhipment problem. If an item can be produced at different locations at varying costs and sold in different markets at different prices, then what production and shipping plan will yield maximum profit? How can the production be scheduled, given the cost of production of an item and the cost of carrying stock, so as to meet the requirements in different periods? The use of transportation method is not limited to solution of the transportation problems alone. Problems of scheduling production, controlling inventory and management of funds over different time periods illustrate some other areas which lend themselves to handling by the transportation method. This chapter basically requires elementary arithmetic calculations. However, familiarity with summation notation, a basic knowledge of inequalities, matrices and their transpose is also necessary. While working through this chapter, master the skill of drawing of a closed path. ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ/͗dƌĂŶƐƉŽƌƚĂƟŽŶĂŶĚdƌĂŶƐŚŝƉŵĞŶƚWƌŽďůĞŵƐഩഩ Learning Objectives After reading this chapter, you should be able to: LO 1 LO 2 LO 3 /ĚĞŶƟĨǇƚƌĂŶƐƉŽƌƚĂƟŽŶƉƌŽďůĞŵ /ůůƵƐƚƌĂƚĞƚƌĂŶƐƉŽƌƚĂƟŽŶƉƌŽďůĞŵĂƐĂƐƉĞĐŝĂůĐĂƐĞŽĨ>WW ǆƉůĂŝŶŝŶŝƟĂůƐŽůƵƟŽŶƚŽĂƚƌĂŶƐƉŽƌƚĂƟŽŶƉƌŽďůĞŵďǇ EtƌƵůĞ͕>DĂŶĚsD LO 4 ^ŚŽǁŚŽǁƚŽŝŵƉƌŽǀĞĂŶŽŶͲŽƉƟŵĂůƐŽůƵƟŽŶƚŽĂƚƌĂŶƐƉŽƌƚĂƟŽŶ ƉƌŽďůĞŵďǇƐƚĞƉƉŝŶŐͲƐƚŽŶĞĂŶĚDK/ŵĞƚŚŽĚƐ LO 5 ĞƐĐƌŝďĞƵŶďĂůĂŶĐĞĚƚƌĂŶƐƉŽƌƚĂƟŽŶƉƌŽďůĞŵƐĂŶĚ ƉƌŽďůĞŵƐŽĨƉƌŽŚŝďŝƚĞĚƌŽƵƚĞƐ LO 6 ŝƐĐƵƐƐǁŚĞƚŚĞƌĂƐŽůƵƟŽŶŝƐƵŶŝƋƵĞŽƌŶŽƚ LO 7 ĞĂůǁŝƚŚĚĞŐĞŶĞƌĂƚĞƐŽůƵƟŽŶƐĂŶĚŵĂǆŝŵŝƐĂƟŽŶƉƌŽďůĞŵƐ LO 8 hŶĚĞƌƐƚĂŶĚĚƵĂůŽĨƚŚĞƚƌĂŶƐƉŽƌƚĂƟŽŶŵŽĚĞů LO 9 /ůůƵƐƚƌĂƚĞƐĞŶƐŝƟǀŝƚǇĂŶĂůǇƐŝƐĨŽƌƚƌĂŶƐƉŽƌƚĂƟŽŶƉƌŽďůĞŵƐ LO 10 /ŵƉůĞŵĞŶƚƉƌŽĚƵĐƟŽŶƐĐŚĞĚƵůŝŶŐĂŶĚŝŶǀĞŶƚŽƌǇĐŽŶƚƌŽů ƚŚƌŽƵŐŚƚƌĂŶƐƉŽƌƚĂƟŽŶŵŽĚĞů LO 11 ^ŽůǀĞƚƌĂŶƐŚŝƉŵĞŶƚƉƌŽďůĞŵƐ ,1752'8&7,21 :HKDYHGLVFXVVHGLQWKHSUHYLRXVFKDSWHUV WKHVROXWLRQWRWKHJHQHUDO OLQHDUSURJUDPPLQJSUREOHP1RZZHVKDOOFRQVLGHUDVSHFLDOEUDQFKRI LO 1 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LVDQGWKHFROXPQWRWDO LV6RDVVLJQXQLWVDQGPRYHYHUWLFDOO\GRZQWRWKHFHOO& 1RZWKHURZDQGFROXPQWRWDOVDUH DQG 7KXVDVVLJQLQWKHFHOODQGVKLIWWRWKHFHOO' :LWKFXUUHQW DQG DOORFDWHXQLWVKHUH)LQDOO\VKLIWWRWKHFHOO'FRUUHVSRQGLQJWRZKLFK DQGDUHERWKHTXDOWR6RDVVLJQWKLVTXDQWLW\WRWKHFHOO7KHVHDVVLJQPHQWVDUHVKRZQLQ7DEOH ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ/͗dƌĂŶƐƉŽƌƚĂƟŽŶĂŶĚdƌĂŶƐŚŝƉŵĞŶƚWƌŽďůĞŵƐഩഩ TABLE 5.3 To From Initial Feasible Solution: NWC Method Q P 180 A 12 R Supply 13 500 170 150 10 S 12 180 B 7 11 8 120 300 14 200 C 6 16 11 7 200 Demand 180 150 350 320 1,000 Total cost = 12 ¥ 180 + 10 ¥ 150 + 12 ¥ 170 + 8 ¥ 180 + 14 ¥ 120 + 7 ¥ 200 = ` 10,220 7KLVURXWLQJRIWKHXQLWVPHHWVDOOWKHULPUHTXLUHPHQWVDQGLQYROYHVDWRWDORI ± VKLSPHQWVDVWKHUH DUHVL[RFFXSLHGFHOOV,WLQYROYHVDWRWDOFRVWRIC,WPD\EHPHQWLRQHGWKDWWKLVPHWKRGRIREWDLQLQJ LQLWLDOVROXWLRQLVDUHODWLYHO\VLPSOHRQHEXWLWLVQRWFRQVLGHUHGYHU\HI¿FLHQWLQWHUPVRIFRVWPLQLPLVLQJ7KLV LVEHFDXVHLWWDNHVLQWRDFFRXQWRQO\WKHDYDLODEOHVXSSO\DQGGHPDQGUHTXLUHPHQWVLQPDNLQJDVVLJQPHQWVDQG 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ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ OHDVW LV FRUUHVSRQGLQJ WR WKH FHOO ' 7KH TXDQWLW\ DVVLJQHG WR WKLV FHOO LV EHLQJ ORZHU RI GHPDQG DQGVXSSO\ 'HOHWHWKHURZDQGDGMXVWWKHGHPDQGIRU'DW7KHOHDVWFRVWHOHPHQWLQ WKHUHPDLQLQJFHOOVLVIRU&:LWKWKHUHOHYDQWVXSSO\DQGGHPDQGYDOXHVHTXDOWRDQGXQLWV UHVSHFWLYHO\DVVLJQWRWKHFHOO&GHOHWHWKHURZDQGDGMXVWWKHGHPDQGIRU&DW1RZZLWKRQO\ RQHVXSSO\VRXUFHUHPDLQLQJWKHDPRXQWZRXOGEHWUDQVIHUUHGWRWKHFXUUHQWUHTXLUHPHQWVDWZDUHKRXVHV% &DQG'WR%WR&DQGWR' TABLE 5.4 To From A Obtaining Initial Feasible Solution: LC Method P Q 12 10 R 150 Supply S 50 12 13 300 500 300 B 7 11 8 14 C 6 16 11 7 200 20 Demand 180 150 350 50 320 300 1,000 180 300 20 7KHWUDQVSRUWDWLRQVFKHGXOHREWDLQHGLVUHSURGXFHGLQ7DEOH7KHVROXWLRQKDVVL[RFFXSLHGFHOOVDQGLWLV VHHQWRLQYROYHDWRWDOFRVWRIC TABLE 5.5 Initial Feasible Solution: Least Cost Method To P Q A 12 10 12 B 7 11 C 6 Demand 180 From R 50 150 Supply S 300 13 500 8 14 300 16 11 7 200 150 350 320 1,000 300 180 20 Total cost = 10 ¥ 150 + 12 ¥ 50 + 13 ¥ 300 + 8 ¥ 300 + 6 ¥ 180 + 7 ¥ 20 = ` 9,620 ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ/͗dƌĂŶƐƉŽƌƚĂƟŽŶĂŶĚdƌĂŶƐŚŝƉŵĞŶƚWƌŽďůĞŵƐഩഩ 9RJHO·V$SSUR[LPDWLRQ0HWKRG 9$0 7KH 9RJHO¶V$SSUR[LPDWLRQ 0HWKRG FRPPRQO\ NQRZQ DV 9$0 LV DQRWKHU 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ZKLFKFDQEHDVVLJQHGLQHDFKFDVHDQGVHOHFWWKHRQHZKHUHODUJHUTXDQWLW\FDQEH DVVLJQHG,QWKHRWKHUFDVHXQLWFRVWYDOXHVRIWKHWZROHDVWFRVWFHOOVLQWKHURZDQGFROXPQDQGFKRRVHWKH RQHZKLFKKDVDORZHUYDOXH 'HOHWHWKHURZRUFROXPQZKLFKKDVEHHQVDWLV¿HGE\WKHDOORFDWLRQDQGDGMXVWWKHTXDQWLW\RIGHPDQGVXSSO\ 5HFDOFXODWHWKHFRVWGLIIHUHQFHVIRUWKHUHGXFHGPDWUL[DQGSURFHHGLQWKHVDPHPDQQHUDVGLVFXVVHGHDUOLHU &RQWLQXHZLWKWKHSURFHVVXQWLODOOWKHXQLWVKDYHEHHQDVVLJQHG7KHLQLWLDOVROXWLRQWR([DPSOHXVLQJ 9$0LVREWDLQHGLQ7DEOH TABLE 5.6 Obtaining Initial Feasible Solution: VAM To Q From P R A 12 B 7 11 8 14 C 6 16 11 Demand 180 150 I 1 II III Iteration Supply S I II III 500 2 2 2 300 120 1 1 3 7 200 1 – – 350 230 320 120 1,000 1 3 6 5 1 4 1 – 1 4 1 10 150 230 12 180 13 120 120 200 7REHJLQZLWKWKHFRVWGLIIHUHQFHVEHWZHHQWKHSDLUVRIOHDVWFRVWFHOOVDUHWDNHQIRUHDFKURZDQGFROXPQ LWHUDWLRQ, 7KHODUJHVWRIWKHVHEHLQJ ± WKHFROXPQGHVLJQDWHG 'LVVHOHFWHG,QWKHORZHVWFRVW ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ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nitial Feasible Solution: VAM To P Q A 12 10 B 7 11 8 14 C 6 16 11 7 200 Demand 180 150 350 320 1,000 From R 230 150 12 180 Supply S 13 120 500 120 300 200 Total cost = 10 ¥ 150 + 12 ¥ 230 + 13 ¥ 120 + 7 ¥ 180 + 8 ¥ 120 + 7 ¥ 200 = ` 9,440 7KHIROORZLQJSRLQWVPD\EHQRWHG L 7KH9RJHO¶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LQ ZKLFK RSWLPDOLW\ WHVW LV DSSOLHG E\ FDOFXODWLQJ WKH RSSRUWXQLW\ FRVW RI HDFK HPSW\FHOO7KHRWKHUPHWKRGLVFDOOHGDVWKH 02', 7KLVRQHLVHDVLHUWR DSSO\DQGPRUHHI¿FLHQWWKDQWKHVWHSSLQJVWRQHPHWKRG,WLVEDVHGRQWKHFRQFHSWRIGXDOYDULDEOHVWKDW DUHXVHGWRHYDOXDWHWKHHPSW\FHOOV8VLQJWKHVHGXDOYDULDEOHVWKHRSSRUWXQLW\FRVWRIHDFKRIWKHHPSW\ FHOOVLVGHWHUPLQHG7KXVZKLOHERWKWKHPHWKRGVLQYROYHGHWHUPLQLQJRSSRUWXQLW\FRVWVRIHPSW\FHOOVWKH PHWKRGRORJ\HPSOR\HGE\WKHPGLIIHUV7KHRSSRUWXQLW\FRVWYDOXHVLQGLFDWHZKHWKHUWKHJLYHQVROXWLRQLV RSWLPDORUQRW %RWKWKHPHWKRGVFDQEHXVHGRQO\ZKHQWKHVROXWLRQLVDEDVLFIHDVLEOHVROXWLRQVRWKDWLWKDV ±EDVLF YDULDEOHV5HIHUWRVHFWLRQLIDVROXWLRQLQYROYHVVPDOOHUWKDQ ±QXPEHURIRFFXSLHGFHOOV EDVLF YDULDEOHV ,QFLGHQWO\QRQHRIWKHPHWKRGVXVHGWR¿QGLQLWLDOVROXWLRQZRXOG\LHOGDVROXWLRQZLWKJUHDWHU WKDQ ±QXPEHURIRFFXSLHGFHOOV START For each row and column, find the difference between the two least cost cells that have not been allocated Select the largest of the differences of rows and columns, and, in case of a tie, choose the one corresponding to which largest number of units can be assigned or the cost value is the lowest Assign the largest quantity permissible by the rim requirements to the cell in that row/ column with the smallest cost Eliminate the row/column that has been satisfied No Are all rim conditions satisfied? Yes STOP )LJXUH 6FKHPDWLF3UHVHQWDWLRQRI 9$0 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 6WHS,PSURYLQJWKH6ROXWLRQ %\DSSO\LQJHLWKHURIWKHPHWKRGVLIWKHVROXWLRQLVIRXQGWREHRSWLPDO WKHQWKHSURFHVVWHUPLQDWHVDVWKHSUREOHPLVVROYHG,IWKHVROXWLRQLVQRW LO 4 ^ŚŽǁŚŽǁƚŽŝŵƉƌŽǀĞ VHHQWREHRSWLPDOWKHQDUHYLVHGDQGLPSURYHGEDVLFIHDVLEOHVROXWLRQ ĂŶŽŶͲŽƉƟŵĂůƐŽůƵƟŽŶƚŽĂ LVREWDLQHG7KLVLVGRQHE\H[FKDQJLQJDQRQEDVLFYDULDEOHIRUDEDVLF ƚƌĂŶƐƉŽƌƚĂƟŽŶƉƌŽďůĞŵďǇ YDULDEOH,QVLPSOHWHUPVUHDUUDQJHPHQWLVPDGHE\WUDQVIHUULQJXQLWVIURP ƐƚĞƉƉŝŶŐͲƐƚŽŶĞĂŶĚDK/ DQRFFXSLHGFHOOWRDQHPSW\FHOOWKDWKDVWKHODUJHVWRSSRUWXQLW\FRVW ŵĞƚŚŽĚƐ DQGWKHQDGMXVWLQJWKHXQLWVLQRWKHUUHODWHGFHOOVLQDZD\WKDWDOOWKHULP UHTXLUHPHQWVDUHVDWLV¿HG7KLVLVGRQHE\¿UVWWUDFLQJD GLVFXVVHGLQGHWDLOODWHU 7KHVROXWLRQVRREWDLQHGLVDJDLQWHVWHGIRURSWLPDOLW\ VWHS DQGUHYLVHGLIQHFHVVDU\:HFRQWLQXHLQWKLV PDQQHUXQWLODQRSWLPDOVROXWLRQLV¿QDOO\REWDLQHG :HQRZGLVFXVVWKHVWHSSLQJVWRQHDQG02',PHWKRGVLQWXUQ 6WHSSLQJVWRQH0HWKRG 7KLVLVDSURFHGXUHIRUGHWHUPLQLQJWKHSRWHQWLDOLIDQ\RILPSURYLQJXSRQHDFKRIWKHQRQEDVLFYDULDEOHVLQ WHUPVRIWKHREMHFWLYHIXQFWLRQ7RGHWHUPLQHWKLVSRWHQWLDOHDFKRIWKHQRQEDVLFYDULDEOHV HPSW\FHOOV LV FRQVLGHUHGRQHE\RQH)RUHDFKVXFKFHOOZH¿QGZKDWHIIHFWRQWKHWRWDOFRVWZRXOGEHLIRQHXQLWLVDVVLJQHG WRWKLVFHOO:LWKWKLVLQIRUPDWLRQWKHQZHFRPHWRNQRZZKHWKHUWKHVROXWLRQLVRSWLPDORUQRW,IQRWZH LPSURYHWKDWVROXWLRQ7RXQGHUVWDQGLWZHUHIHUWR7DEOH ZKLFKLVDUHSURGXFWLRQRI7DEOH FRQWDLQLQJ WKHLQLWLDOEDVLFIHDVLEOHVROXWLRQREWDLQHGE\1:&RUQHU5XOH TABLE 5.8 To From Initial Feasible Solution: Testing for Optimality Q P 180 A 12 R Supply 13 500 170 150 10 S – 12 + 180 B 7 11 C 6 16 Demand 180 150 + 8 120 – 300 14 200 11 7 200 350 320 1,000 Total cost = 12 ¥ 180 + 10 ¥ 150 + 12 ¥ 170 + 8 ¥ 180 + 14 ¥ 120 + 7 ¥ 200 = ` 10,220 ,QWKLVWDEOHWKHUHDUHVL[HPSW\FHOOV' $ % $% DQG& ZLWKFRUUHVSRQGLQJQRQEDVLFYDULDEOHVDV DQG7RVWDUWZLWKOHWXVDVVHVVWKHSRWHQWLDOIRULPSURYHPHQWRIWKHQRQEDVLFYDULDEOH 8QIRUWXQDWHO\WKHHIIHFWRIVHQGLQJRQHXQLWRQWKLVURXWHFDQQRWEHGLUHFWO\LGHQWL¿HGEHFDXVHWKHVXSSOLHV RIWKHVRXUFHDUHDOUHDG\DVVLJQHGWR$%DQG&ZKLOHWKHGHPDQGDW' 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6KLIWLQJWKHVHXQLWVUHVXOWVLQWKHUHYLVHGVROXWLRQFRQWDLQHGLQ7DEOH 1RWLFHWKHYDULDEOHKDVUHSODFHG TABLE 5.9 To From Improved Solution: Non-Optimal Q P 180 R 50 150 Supply S 120 13 500 8 14 300 16 11 7 200 150 350 320 1,000 A 12 10 12 B 7 11 C 6 Demand 180 300 200 Total cost = 12 ¥ 180 + 10 ¥ 150 + 12 ¥ 50 + 13 ¥ 120 + 8 ¥ 300 + 7 ¥ 200 = ` 9,620 7KLVVROXWLRQLQYROYHVDWRWDOFRVWRICZKLFKLVORZHUE\C ¥ LQFRPSDULVRQWRWKHLQLWLDO VROXWLRQ+RZHYHULWLVQRWVXUHLIWKLVVROXWLRQLVRSWLPDO:HDJDLQDSSO\VWHSWRGHWHUPLQHRSWLPDOLW\ $SSOLFDWLRQRIWKLV\LHOGVWKHIROORZLQJUHVXOWV ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ/͗dƌĂŶƐƉŽƌƚĂƟŽŶĂŶĚdƌĂŶƐŚŝƉŵĞŶƚWƌŽďůĞŵƐഩഩ ) / $ $± & ± & ± $ ±± ± % % ± & ± & ± % ±± ± ' ' ± ' ± & ± & ±± ± $ $ ± ' ± '±$ ±± % % ± ' ± '±% ±± ± & &±'±'±& ±± ± 6LQFHDFHOOKDVDSRVLWLYHRSSRUWXQLW\FRVWWKHVROXWLRQLVQRWRSWLPDO)XUWKHUVLQFHWKHFHOO$KDVLWZH ZLOOLQFOXGHWKLVFHOOLQWKHWUDQVSRUWDWLRQVFKHGXOH:LWKWKHFHOOV& DQG$ ZLWKDµ±¶VLJQWKH TXDQWLW\WREHVKLIWHGDORQJWKHFORVHGORRSLV7KHUHYLVHGVROXWLRQZLWKWKLVFKDQJHLVJLYHQLQ7DEOH TABLE 5.10 To Improved Solution: Optimal P Q A 12 10 12 B 7 11 C 6 Demand 180 From R 230 150 Supply S 120 13 500 8 14 300 16 11 7 200 150 350 320 1,000 180 120 200 Total cost = 10 ¥ 150 + 12 ¥ 230 + 13 ¥ 120 + 7 ¥ 180 + 8 ¥ 120 + 7 ¥ 200 = ` 9,440 7KLVVROXWLRQDOVRLQYROYHVVL[RFFXSLHGFHOOVDQGLQYROYHVDWRWDOFRVWRIC7KHVROXWLRQZRXOGQRZEH WHVWHGIRURSWLPDOLW\7KLVLVGRQHKHUH ) / $ $ ± & ± & ± $ ±± ± % % ± & ± & ± % ±± ± ' ' ± ' ± & ± & ±± ± $ $ ± ' ± '±& 0 & 0 $ ±±± ± % % ± ' ± '±% ±± ± & &±'±'±& ±± ± ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 6LQFHWKHRSSRUWXQLW\FRVWVRIDOOWKHHPSW\FHOOVDUHQHJDWLYHWKHVROXWLRQREWDLQHGLVRSWLPDO$FDUHIXOORRN DWWKHVROXWLRQUHYHDOVWKDWLWLVLGHQWLFDOWRWKHLQLWLDOVROXWLRQREWDLQHGE\9$0DVJLYHQLQ7DEOH7KXV WKHLQLWLDOVROXWLRQE\DSSO\LQJ9$0LVRSWLPDODQGGRHVQRWUHTXLUHDQ\LPSURYHPHQW %HIRUHGLVFXVVLQJWKH02',PHWKRGRIWHVWLQJRSWLPDOLW\ZHFRQVLGHUWKHLQLWLDOVROXWLRQE\WKHOHDVWFRVW PHWKRGDQGWHVWLWVRSWLPDOLW\XVLQJWKHVWHSSLQJVWRQHPHWKRG7KHVROXWLRQLVUHSURGXFHGLQ7DEOH TABLE 5.11 Initial Feasible Solution (LCM): Testing for Optimality To P Q A 12 10 B 7 11 8 14 C 6 16 11 7 200 Demand 180 150 350 320 1,000 From R Supply S 50 150 13 12 300 500 300 300 20 180 Total cost = 10 ¥ 150 + 12 ¥ 50 + 13 ¥ 300 + 8 ¥ 300 + 6 ¥ 180 + 7 ¥ 20 = ` 9,620 7KHHPSW\FHOOVDUHFRQVLGHUHGRQHE\RQHDQGWKHUHVXOWVDUHJLYHQKHUH $ ) / $± ' ± ' ± $ ±± $ $ ± $ ± ' ± ' ±&±& ±±± ± % % ± & ± & ± % ±± ± ' ' ± ' ± & ± & ±± ± % % ± ' ± ' ± % ±± ± & &±'±'±& ±± ± 7KHVROXWLRQLVVHHQWREHQRQRSWLPDO&RQVLGHULQJWKHFHOO$WKHRQO\RQHZLWKDSRVLWLYHRSSRUWXQLW\FRVW ZHVKLIWXQLWVDORQJWKHFORVHGORRS$±$±'±'±&±&7KHUHYLVHGVROXWLRQLVSUHVHQWHGLQ 7DEOH ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ/͗dƌĂŶƐƉŽƌƚĂƟŽŶĂŶĚdƌĂŶƐŚŝƉŵĞŶƚWƌŽďůĞŵƐഩഩ TABLE 5.12 To From Improved Solution: Optimal Q P R 230 150 A 10 12 12 180 Supply S 120 13 500 300 120 B 7 11 8 14 C 6 16 11 7 200 Demand 180 150 350 320 1,000 200 Total cost = 10 ¥ 150 + 12 ¥ 230 + 13 ¥ 120 + 7 ¥ 180 + 8 ¥ 120 + 7 ¥ 200 = ` 9,440 ,WPD\EHQRWHGWKDWWKLVVROXWLRQPDWFKHVWKHRQHJLYHQLQ7DEOHDQGKHQFHLVRSWLPDO 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URZDUHDOORFFXSLHGFRQVLGHUWKHPRQHE\RQHDQGREWDLQWKHFRUUHVSRQGLQJFROXPQYDOXHVY$FFRUGLQJO\ 1 VLQFH 1 VLQFH DQG1 EHFDXVH 1H[W1FDQEHXVHGWR¿QG :LWK ± ±6LPLODUO\ZLWK 1 ± ± DQG ±1 ± ± )LJXUH 6FKHPDWLF3UHVHQWDWLRQRI 02', ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ/͗dƌĂŶƐƉŽƌƚĂƟŽŶĂŶĚdƌĂŶƐŚŝƉŵĞŶƚWƌŽďůĞŵƐഩഩ TABLE 5.13 Initial Solution: Testing for Optimality To Q P From R 150 180 A 12 10 B 7 11 – 12 170 +5 + S Supply ui 13 500 0 300 –4 – 11 180 +1 –5 + 8 120 – 14 200 C –5 6 – 17 16 – 10 11 7 200 1,000 Demand 180 150 350 320 vj 12 10 12 18 Total cost = 12 ¥ 180 + 10 ¥ 150 + 12 ¥ 170 + 8 ¥ 180 + 14 ¥ 120 + 7 ¥ 200 = ` 10,220 6WHS +DYLQJGHWHUPLQHGDOO DQGYYDOXHVFDOFXODWHIRUHDFKXQRFFXSLHGFHOOD Y± 7KHD ¶V UHSUHVHQWWKHRSSRUWXQLW\FRVWVRIYDULRXVFHOOV$IWHUREWDLQLQJWKHRSSRUWXQLW\FRVWVSURFHHGLQWKHVDPHZD\ DVLQWKHVWHSSLQJVWRQHPHWKRG,IDOOWKHHPSW\FHOOVKDYHQHJDWLYHRSSRUWXQLW\FRVWWKHVROXWLRQLVRSWLPDO DQGXQLTXH,IVRPHHPSW\FHOO V KDVD]HURRSSRUWXQLW\FRVWEXWLIQRQHRIWKHRWKHUHPSW\FHOOVKDYHSRVL WLYHRSSRUWXQLW\FRVWWKHQLWLPSOLHVWKDWWKHJLYHQVROXWLRQLVRSWLPDOEXWWKDWLWLVQRWXQLTXH²WKHUHH[LVWV RWKHUVROXWLRQ V WKDWZRXOGEHDVJRRGDVWKLVVROXWLRQ +RZHYHULIWKHVROXWLRQFRQWDLQVDSRVLWLYHRSSRUWXQLW\FRVWIRURQHRUPRUHRIWKHHPSW\FHOOVWKHVROXWLRQ LVQRWRSWLPDO,QVXFKDFDVHWKHFHOOZLWKWKHODUJHVWRSSRUWXQLW\FRVWYDOXHLVVHOHFWHGDFORVHGORRSWUDFHG DQGWUDQVIHUVRIXQLWVDORQJWKHURXWHDUHPDGHLQDFFRUGDQFHZLWKWKHPHWKRGGLVFXVVHGHDUOLHU7KHQWKH UHVXOWLQJVROXWLRQLVDJDLQWHVWHGIRURSWLPDOLW\DQGLPSURYHGLIQHFHVVDU\7KHSURFHVVLVUHSHDWHGXQWLODQ RSWLPDOVROXWLRQLVREWDLQHG /ĨĂůůѐijǀĂůƵĞƐ͕ƌĞƉƌĞƐĞŶƟŶŐŽƉƉŽƌƚƵŶŝƚǇĐŽƐƚƐŽĨǀĂƌŝŽƵƐƵŶŽĐĐƵƉŝĞĚĐĞůůƐ͕ŝŶĂŐŝǀĞŶƐŽůƵƟŽŶĂƌĞŶĞŐĂƟǀĞƚŚĞŶ ƚŚĞƐŽůƵƟŽŶŝƐŽƉƟŵĂůĂŶĚƵŶŝƋƵĞ͘&ƵƌƚŚĞƌ͕ŝĨŶŽŶĞŽĨƚŚĞѐijǀĂůƵĞƐŝƐƉŽƐŝƟǀĞĂŶĚŽŶĞŽƌŵŽƌĞŽĨƚŚĞŵĂƌĞ ĞƋƵĂůƚŽǌĞƌŽ͕ƚŚĞŶƚŚĞƉƌŽďůĞŵŚĂƐŵƵůƟƉůĞŽƉƟŵĂůƐŽůƵƟŽŶƐ͘ )RUWKHLOOXVWUDWLRQXQGHUFRQVLGHUDWLRQWKHD YDOXHVDUHVKRZQFDOFXODWHGEHORZDQGJLYHQLQFLUFOHVLQWKH HPSW\FHOOV7KH\DUH D ± D ±± ± D ±± D ±± ± D ±± ± DQG D ±± ± +HUHWKHFHOO'KDVWKHODUJHVWRSSRUWXQLW\FRVW D DQGWKHUHIRUHZHVHOHFWDIRULQFOXVLRQDVD EDVLFYDULDEOH7KHFORVHGORRSVWDUWLQJZLWKWKLVFHOOLVVKRZQLQ7DEOH7KHUHYLVHGVROXWLRQLVJLYHQLQ 7DEOH ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 5.14 Improved Solution: Non-optimal To Q P From – B + +1 12 10 7 11 Supply ui 13 500 0 14 300 –4 –6 S 50 150 180 A R + – 8 –5 120 12 300 –5 200 C 0 6 – 12 16 –5 11 7 200 1,000 Demand 180 150 350 320 vj 12 10 12 13 Total cost = 12 ¥ 180 + 10 ¥ 150 + 12 ¥ 50 + 13 ¥ 120 + 8 ¥ 300 + 7 ¥ 200 = ` 10,220 7KHVROXWLRQLQWKLVWDEOHLVWHVWHGIRURSWLPDOLW\DQGLVIRXQGWREHQRQRSWLPDO+HUHWKHFHOO$KDVDSRVLWLYH RSSRUWXQLW\FRVWDQGWKHUHIRUHDFORVHGORRSLVWUDFHGVWDUWLQJZLWKLW7KLVHQDEOHVXQLWVWREHWUDQVIHUUHG WRWKLVURXWH7KHUHVXOWLQJVROXWLRQLVJLYHQLQ7DEOH TABLE 5.15 Improved Solution: Optimal To Q P From R –1 10 12 B 7 C 6 180 ui 13 500 0 14 300 –4 –6 230 150 A Supply S 120 12 120 11 8 –5 –5 200 –1 – 12 16 –5 11 7 200 1,000 Demand 180 150 350 320 vj 11 10 12 13 Total cost = 10 ´ 150 + 12 ´ 230 + 13 ´ 120 + 7 ´ 180 + 8 ´ 120 + 7 ´ 200 = ` 9,440 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LVLQWURGXFHGZLWK VXSSO\RI± XQLWVDQG]HURWUDQVSRUWDWLRQFRVWVLQFHQRSHQDOWLHVDUHSURYLGHGIRUQRWVDWLVI\LQJ WKHGHPDQGDWHDFKPDUNHW)XUWKHUVLQFHURXWHV DQGDUHJLYHQDVSURKLELWHGWKHFRVWHOHPHQWIRU HDFKRIWKHVHLVUHSODFHGE\- 7KHLQIRUPDWLRQLVSUHVHQWHGLQ7DEOH TABLE 5.16 Transportation Problem: Unbalanced, with Prohibited Routes -2 * 'HPDQG ' 7KHLQLWLDOVROXWLRQLVFRQWDLQHGLQ7DEOH ,WLVREWDLQHGXVLQJ9$0 TABLE 5.17 Initial Basic Feasible Solution by VAM B A Iteration Market Warehouse Supply C I II III IV V 180 5 5 – – – 100 60 5 5 5 3 3 M 160 4 4 4 4 – 9 120 80 2 2 2 4 4 0 0 100 0 – – – – 660 1 M 2 14 3 9 4 11 5 0 Demand 240 140 200 40 220 40 I 9 5 6 II 2 2 1 III 2 2 3 IV 2 2 - V 3 4 - 60 12 7 11 6 180 5 80 100 7 160 40 40 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ ,QWKH¿UVWVWHSFRVWGLIIHUHQFHVDUHFDOFXODWHGDQGWKHODUJHVWRQHLVVHOHFWHGDQGXQLWVDUHSODFHGLQ WKHFHOO! 7KHURZ LVGHOHWHGDQGGLIIHUHQFHVDUHUHFDOFXODWHG1RZDWLHLVREVHUYHGLQWKHODUJHVWYDOXH RIWKHGLIIHUHQFHVWKDWLV ,IURZLVVHOHFWHGWKHQWKHDSSURSULDWHFHOOZRXOGEHZKHUHFRQVLGHULQJ GHPDQGDQGVXSSO\XQLWVFDQEHVHQWZKLOHLIURZLVFKRVHQWKHQLQWKHOHDVWFRVWFHOO!+ XQLWVFDQ EHSODFHG7KXVZHPD\SUHIHUDEO\FKRRVH$IWHUWKLVZHSURFHHGLQWKHXVXDOPDQQHU²FRVWGLIIHUHQFHV DUHUHFDOFXODWHGDQGTXDQWLWLHVDVVLJQHGWRWKHOHDVWFRVWFHOOVRIWKHURZVFROXPQVVHOHFWHG7KHVROXWLRQLV UHSURGXFHGLQ7DEOHDQGWHVWHGIRURSWLPDOLW\6LQFHDOOD £WKHVROXWLRQLVVHHQWREHRSWLPDO1RWH WKDWD LVQRWFDOFXODWHGIRUWKHSURKLELWHGFHOOV! DQG(YHQLID LVFDOFXODWHGIRUDSURKLELWHGURXWH LWLVERXQGWREHH[WUHPHO\QHJDWLYHVLQFHLWZRXOGLQYROYH0- 7KHVROXWLRQJLYHQLQ7DEOHLVQRWXQLTXHEHFDXVHLQFHOOD 7RREWDLQDQDOWHUQDWLYHRSWLPDO VROXWLRQZHGUDZDFORVHGORRSDVVKRZQLQWKLVWDEOH7KHUHYLVHGVROXWLRQLVJLYHQLQ7DEOH %RWKWKH VROXWLRQVLQYROYHDWRWDOFRVWRIC TABLE 5.18 Initial Feasible Solution: Optimal 'HJHQHUDF\ :HKDYHDOUHDG\VHHQWKDWDEDVLFIHDVLEOHVROXWLRQRIDWUDQVSRUWDWLRQ SUREOHPKDV ±EDVLFYDULDEOHVZKLFKPHDQVWKDWWKHQXPEHURI RFFXSLHG FHOOV LQ VXFK D VROXWLRQ LV OHVV WKDQ WKH QXPEHU RI URZV SOXVWKHQXPEHURIFROXPQV,WPD\KDSSHQVRPHWLPHVWKDWWKHQXPEHU RIRFFXSLHGFHOOVLVVPDOOHUWKDQ ±6XFKDVROXWLRQLVFDOOHGD LO 7 Deal with degenerate solutions and maximisation problems 'HJHQHUDF\LQDWUDQVSRUWDWLRQSUREOHPFDQ¿JXUHLQWZRZD\V'HJHQHUDF\FDQDULVHLQWKH¿UVWLQVWDQFH ZKHQDQLQLWLDOIHDVLEOHVROXWLRQLVREWDLQHG VHH([DPSOH 6HFRQGO\WKHSUREOHPPD\EHFRPHGHJHQHUDWH ZKHQWZRRUPRUHFHOOVDUHYDFDWHGVLPXOWDQHRXVO\LQWKHSURFHVVRIWUDQVIHUULQJXQLWVDORQJWKHFORVHGSDWK ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ/͗dƌĂŶƐƉŽƌƚĂƟŽŶĂŶĚdƌĂŶƐŚŝƉŵĞŶƚWƌŽďůĞŵƐഩഩ TABLE 5.19 Alternate Optimal Solution 7KHGLI¿FXOW\ZKHQDVROXWLRQLVGHJHQHUDWHLVWKDWLWFDQQRWEHWHVWHGIRURSWLPDOLW\%RWKWKHVWHSSLQJVWRQH PHWKRGDQGWKHPRGL¿HGGLVWULEXWLRQPHWKRG 02', DUHLQRSHUDWLYHLQVXFKDFDVH7KHIRUPHUFDQQRWEH DSSOLHGEHFDXVHIRUVRPHRIWKHHPSW\FHOOVWKHFORVHGORRSVFDQQRWEHWUDFHGZKLOHWKHODWWHUIDLOVEHFDXVH DOO DQG1YDOXHVFDQQRWEHGHWHUPLQHG 7RRYHUFRPHGHJHQHUDF\ZHSURFHHGE\DVVLJQLQJDQLQ¿QLWHVLPDOO\VPDOODPRXQWFORVHWR]HURWRRQH RU PRUHLIWKHQHHGEH HPSW\FHOODQGWUHDWWKHFHOODVDQRFFXSLHGFHOO7KLVDPRXQWLVUHSUHVHQWHGE\D*UHHN OHWWHUe HSVLORQ DQGLVWDNHQWREHVXFKDQLQVLJQL¿FDQWYDOXHWKDWZRXOGQRWDIIHFWWKHWRWDOFRVW7KXVLWLV ELJHQRXJKWRFDXVHWKHSDUWLFXODUURXWHWRZKLFKLWLVDVVLJQHGWREHFRQVLGHUHGDVDEDVLFYDULDEOHEXWQRW ODUJHHQRXJKWRFDXVHDFKDQJHLQWKHWRWDOFRVWDQGRWKHUQRQ]HURDPRXQWV$OWKRXJKeLVWKHRUHWLFDO\QRQ ]HURWKHRSHUDWLRQVZLWKLWLQWKHFRQWH[WRISUREOHPDWKDQGDUHJLYHQKHUH 2e 2 2±e 2 e e ee e e±e DQG 2¥e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ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ ƐŽůƵƟŽŶĐĂŶďĞƚĞƐƚĞĚĨŽƌŽƉƟŵĂůŝƚǇŽŶůǇǁŚĞŶƚŚĞŶƵŵďĞƌŽĨŽĐĐƵƉŝĞĚĐĞůůƐŝƐĞƋƵĂůƚŽmнnʹϭ͘/ĨƚŚĞ ŶƵŵďĞƌŝƐƐŵĂůůĞƌƚŚĂŶƚŚŝƐ͕ƚŚĞƐŽůƵƟŽŶŝƐƐĂŝĚƚŽďĞdegenerate͘ĞŐĞŶĞƌĂĐǇŝŶƚŚĞƐŽůƵƟŽŶŝƐƌĞƐŽůǀĞĚďǇ ƉůĂĐŝŶŐĂŶŝŶĮŶŝƚĞƐŝŵĂůůǇƐŵĂůůƋƵĂŶƟƚǇeŝŶĞĂĐŚŽĨƚŚĞƌĞƋƵŝƌĞĚŶƵŵďĞƌŽĨƵŶŽĐĐƵƉŝĞĚĐĞůůƐ͘dŚĞĐŚŽƐĞŶĐĞůůƐ ŵƵƐƚďĞŝŶĚĞƉĞŶĚĞŶƚ͘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± UHTXLUHGIRUDQRQGHJHQHUDWHVROXWLRQLQWKLVFDVH+HUH HPSW\FHOOV$-$-$-$-$-$-DQG$-DUHLQGHSHQGHQWZKLOHRWKHUVDUHQRW)RUUHPRYLQJ GHJHQHUDF\OHWXVSODFHe LQWKHFHOO$-DQGWKHQWHVWLWIRURSWLPDOLW\7KLVLVVKRZQLQ7DEOH ,WPD\ EHLQWHUHVWLQJIRUWKHUHDGHUWRWU\E\LQVHUWLQJeLQDQ\RIWKHFHOOVZKLFKDUHQRWLQGHSHQGHQWDQGVHHWKH UHVXOW 7KHVROXWLRQLVIRXQGWREHQRQRSWLPDO7KHLPSURYHGVROXWLRQLVJLYHQLQ7DEOH TABLE 5.20 Initial Feasible Solution: Non-optimal To Æ M1 M2 M3 M4 P1 30 6 – 4 9 +1 P2 20 From Ø –4 –11 M5 Supply ui 0 40 0 40 2 0 50 2 0 90 1 220 10 –2 20 P3 –12 1 7 –4 6 –11 11 – 3 50 + 1 0 1 –11 14 20 + – 0 e 30 60 P4 7 1 Demand 90 30 50 30 20 vj 6 0 –2 1 –2 –13 12 –4 6 –1 ` ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ/͗dƌĂŶƐƉŽƌƚĂƟŽŶĂŶĚdƌĂŶƐŚŝƉŵĞŶƚWƌŽďůĞŵƐഩഩ TABLE 5.21 Revised Solution: Optimal To Æ From Ø M1 P1 6 P2 20 M2 M3 M4 4 9 1 11 3 M5 Supply ui 0 40 0 0 40 2 0 50 1 0 90 1 220 10 30 –4 –10 –2 20 20 – 12 –4 6 –10 e 50 P3 7 P4 7 1 –12 12 –4 6 Demand 90 30 50 30 20 vj 6 0 –1 1 –2 0 0 1 –12 14 –1 30 60 –1 Total cost = 6 ¥ 30 + 1 ¥ 10 + 3 ¥ 20 + 0 ¥ 50 + 7 ¥ 60 + 1 ¥ 30 = ` 700 1RWLFHWKDWLQLPSURYLQJWKHVROXWLRQRQO\ePRYHVDVWKLVLVWKHPLQLPXPTXDQWLW\LQWKHFHOOVEHDULQJµ±¶ VLJQRQWKHFORVHGORRS7KHRSWLPDOLW\WHVW02',VXJJHVWVWKDWWKHVROXWLRQJLYHQLQ7DEOHLVDQRSWLPDO RQH7KHVROXWLRQFDQEHVWDWHGDV + DOORWKHU WRWDOFRVW C :KHQWKHSUREOHPEHFRPHVGHJHQHUDWHDWWKHVROXWLRQUHYLVLRQVWDJHHSVLORQ e LVSODFHGLQRQH RUPRUH LIQHFHVVDU\ RIWKHUHFHQWO\YDFDWHGFHOOVZLWKWKHPLQLPXPFRVW$QGWKHQZHSURFHHGZLWKWKHSUREOHPLQ WKHXVXDOPDQQHU ([DPSOH 3ODQW 6ROYHWKHIROORZLQJWUDQVSRUWDWLRQSUREOHPWDNLQJLQLWLDOVROXWLRQE\9$0 0DUNHW 6XSSO\ 0 0 0 0 3 3 3 'HPDQG 7KHLQLWLDOEDVLFIHDVLEOHVROXWLRQLVVKRZQLQ7DEOH7KHVROXWLRQREWDLQHGXVLQJ9$0LVIRXQGQRWWR EHRSWLPDO ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 5.22 Initial basic feasible solution: Non-optimal Market Supply ui 8 150 0 5 125 –1 3 Plant M1 M3 M2 65 P1 85 7 5 10 – –3 + 110 P2 M4 –4 15 6 8 4 – + –2 –2 65 P3 +1 + 9 120 12 – 10 7 185 460 –4 Demand 110 80 150 120 vj 7 5 7 4 Total cost = 65 ¥ 5 + 85 ¥ 7 + 110 ¥ 6 + 15 ¥ 4 + 65 ¥ 10 + 120 ¥ 7 = ` 3,130 TABLE 5.23 Revised Solution: Optimal Market Supply ui 8 150 0 5 125 0 3 Plant M1 M2 M3 10 5 7 M4 150 P1 –4 –4 –1 45 P2 80 6 8 4 –1 –1 e 65 P3 9 –5 120 12 10 7 185 460 Demand 110 80 150 120 vj 6 4 7 4 Total cost = 150 ¥ 7 + 45 ¥ 6 + 80 ¥ 4 + 65 ¥ 9 + 120 ¥ 7 = ` 3,065 7KHUHYLVHGVROXWLRQLVVKRZQLQ7DEOH,WLVIRXQGWREHGHJHQHUDWHVLQFHWZRFHOOV$ -DQG$ -DUH YDFDWHGLQWKHSURFHVVRILPSURYLQJWKHQRQRSWLPDOVROXWLRQ$FFRUGLQJO\WKHUHDUHQRZDWRWDORI¿YHRFFXSLHG ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ/͗dƌĂŶƐƉŽƌƚĂƟŽŶĂŶĚdƌĂŶƐŚŝƉŵĞŶƚWƌŽďůĞŵƐഩഩ FHOOVLQVWHDGRIVL[7RFRSHXSZLWKWKHGHJHQHUDF\ZHDVVLJQDQHSVLORQWRRQHRIWKHVHUHFHQWO\YDFDWHG FHOOVDQGSURFHHG7KHVROXWLRQLVWKHQWHVWHGIRURSWLPDOLW\7KHRSWLPDOLW\WHVWLVIRXQGWREHRSWLPDO7KH RSWLPDOWUDQVSRUWDWLRQSODQWKHQLV 7RWDOFRVW C 0D[LPLVDWLRQ3UREOHP $VSUHYLRXVO\PHQWLRQHGWKHFODVVLFDOWUDQVSRUWDWLRQSUREOHPLVRQHRIWKHPLQLPLVDWLRQW\SHV+RZHYHUD WUDQVSRUWDWLRQWDEOHDXPD\FRQWDLQXQLWSUR¿WVLQVWHDGRIXQLWFRVWVDQGWKHREMHFWLYHIXQFWLRQEHPD[LPLVDWLRQ RIWRWDOSUR¿WV7RVROYHDPD[LPLVDWLRQW\SHRISUREOHPWKHWUDQVSRUWDWLRQPHWKRGLVQRWDSSOLHGGLUHFWO\ 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LQWRDPLQLPLVDWLRQSUREOHP6LPLODUO\LIVXFKDSUREOHPKDVDSURKLELWHGURXWHWKHQWKHSD\RIIHOHPHQWIRU VXFKDURXWHVKRXOGEHVXEVWLWXWHGE\0- EHIRUHSURFHHGLQJWRFRQYHUWWRPLQLPLVDWLRQW\SH /ĨĂƚƌĂŶƐƉŽƌƚĂƟŽŶƉƌŽďůĞŵĐĂůůƐĨŽƌŵĂǆŝŵŝƐĂƟŽŶŽĨƚŚĞŽďũĞĐƟǀĞĨƵŶĐƟŽŶ͕ǁĞŶĞĞĚƚŽƚƌĂŶƐĨŽƌŵŝƚŝŶƚŽĂŶ ĞƋƵŝǀĂůĞŶƚŵŝŶŝŵŝƐĂƟŽŶƉƌŽďůĞŵĨŽƌƐŽůƵƟŽŶ͘dŚŝƐŝƐĚŽŶĞďǇƐƵďƚƌĂĐƟŶŐĞĂĐŚŽĨƚŚĞŐŝǀĞŶĐĞůůǀĂůƵĞƐ;ĞǆĐůƵĚŝŶŐ ƚŚŽƐĞĐŽŶƚĂŝŶŝŶŐƐƵƉƉůǇͬĚĞŵĂŶĚƋƵĂŶƟƟĞƐͿĨƌŽŵĂĐŽŶƐƚĂŶƚ͕ǁŚŝĐŚŝƐƵƐƵĂůůǇƚĂŬĞŶĂƐƚŚĞůĂƌŐĞƐƚŽĨƚŚŽƐĞǀĂůƵĞƐ͘ ([DPSOH 6ROYHWKHIROORZLQJWUDQVSRUWDWLRQSUREOHPIRUPD[LPXPSUR¿W 3HU8QLW3UR¿W $ ; :DUHKRXVH < = 0DUNHW % & ' $YDLODELOLW\DWZDUHKRXVHV 'HPDQGLQWKHPDUNHWV ;XQLWV $XQLWV <XQLWV %XQLWV =XQLWV &XQLWV 'XQLWV 7RVROYHWKLVSUREOHPZH¿UVWFRQYHUWWKHXQLWSUR¿WPDWUL[LQWRRSSRUWXQLW\ORVVPDWUL[E\VXEWUDFWLQJHDFK RIWKHYDOXHVLQLWIURPWKHODUJHVWYDOXHHTXDOWR7KHLQIRUPDWLRQLVSUHVHQWHGLQ7DEOH ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 5.24 Opportunity Loss Matrix: Initial Solution Market Warehouse A B 13 7 17 + 18 C Availability ui 200 0 500 11 + 5 300 15 1,000 D 200 X – 17 Y – 10 180 22 19 – 20 100 –4 – 7 15 –5 20 14 0 100 Z 11 Demand 180 320 100 400 vj –4 7 –1 –4 – +6 400 Total profit = 18 ¥ 200 + 7 ¥ 100 + 18 ¥ 400 + 14 ¥ 180 + 3 ¥ 20 + 11 ¥ 100 = ` 15,180 $OVRJLYHQLQWKHWDEOHLVWKHLQLWLDOIHDVLEOHVROXWLRQREWDLQHGXVLQJ9$07KHVROXWLRQLVWHVWHGIRURSWLPDOLW\ XVLQJWKH02',PHWKRGDQGLVIRXQGWREHQRQRSWLPDO$FORVHGORRSLVWUDFHGEHJLQQLQJZLWKWKHFHOO=' ZKLFKKDVWKHODUJHVWSRVLWLYHRSSRUWXQLW\FRVW7KHUHYLVHGVROXWLRQLVJLYHQLQ7DEOH TABLE 5.25 Revised Solution: Non-optimal Market Warehouse A B 13 7 17 18 C Availability ui 200 0 500 11 9 D 200 X – 11 19 – 14 –4 0 380 120 Y –4 Z 11 Demand 180 vj 2 180 + 1+ – 7 15 100 –6 20 14 + 5 300 320 100 400 1,000 7 5 –4 22 – Total profit = 18 ¥ 200 + 7 ¥ 120 + 18 ¥ 380 + 14 ¥ 180 + 11 ¥ 100 + 20 ¥ 20 = ` 15,300 7KLVVROXWLRQLVWHVWHGIRURSWLPDOLW\DQGLVDOVRIRXQGWREHQRQRSWLPDO,WLVIXUWKHUUHYLVHGE\PDNLQJ DEDVLFYDULDEOH*LYHQLQ7DEOHWKLVLVWHVWHGWREHRSWLPDO ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ/͗dƌĂŶƐƉŽƌƚĂƟŽŶĂŶĚdƌĂŶƐŚŝƉŵĞŶƚWƌŽďůĞŵƐഩഩ TABLE 5.26 Revised Solution: Optimal Market Warehouse A B 13 7 17 18 C Availability ui 200 0 500 11 9 D 200 X Y – 11 –4 –15 19 0 –4 280 100 120 15 7 14 5 300 1,000 120 180 Z 11 Demand 180 320 100 400 vj 2 7 4 –4 –6 22 –1 Total profit = 18 ¥ 200 + 7 ¥ 120 + 10 ¥ 100 + 18 ¥ 280 + 14 ¥ 180 + 20 ¥ 120 = ` 15,400 START Write the problem in tabular form Is it balanced? No Balance the table using dummy row/column Yes Is it a maximisation problem? Convert it into a minimisation problem: subtract each element of the profit matrix from its highest value Yes No Find an initial basic feasible solution using NWC rule, VAM etc. Is it degenerate Yes Eliminate degeneracy by assigning e to requisite number of cells No Generate an improved solution No Is it optimal? 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ZRXOGHTXDOWR]HUREHFDXVHQRXQLWVZRXOGEHVKLSSHGIURPDWHUPLQDOWRLWVHOI1RZ DVVXPHWKDWDW WHUPLQDOV # # « # WKHWRWDORXWVKLSPHQWH[FHHGVWKHWRWDOLQVKLSPHQWE\DPRXQWVHTXDO WR « UHVSHFWLYHO\DQGDWWKHUHPDLQLQJ WHUPLQDOV # " # "«# WKHWRWDOLQVKLSPHQW H[FHHGVWKHWRWDORXWVKLSPHQWE\DPRXQWV « UHVSHFWLYHO\,IWKHWRWDOLQVKLSPHQWDWWHUPLQDOV ## « # EH « UHVSHFWLYHO\DQGWKHWRWDORXWVKLSPHQWDWWKHWHUPLQDOV# # « # "EH « "UHVSHFWLYHO\ZHFDQZULWHWKHWUDQVKLSPHQWSUREOHPDV 0LQLPLVH m+n m+n Â i = Â j = 6XEMHFWWR ± ± ± ± ± + ± m m+n i = j = m+ Â ai = Â b j « « « « $VFDQEHHDVLO\REVHUYHGWKHVHFRQVWUDLQWVDUHVLPLODUWRWKHFRQVWUDLQWVRIDWUDQVSRUWDWLRQSUREOHPZLWK " VRXUFHVDQG GHVWLQDWLRQVZLWKWKHGLIIHUHQFHVWKDWKHUHWKHUHDUHQR DQG WHUPVDQGWKDW IRU « DQG IRU « 7KHWHUPV DQGLQWKHVHFRQVWUDLQWVPD\EHVHHQ DVWKHDOJHEUDLFHTXLYDOHQWVRI DQG1RZZHFDQYLHZWKLVSUREOHPDVDQHQODUJHGSUREOHPDQGVROYH LWE\XVLQJWKHWUDQVSRUWDWLRQPHWKRG ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 7KHWUDQVKLSPHQWSUREOHPFDQEHGHSLFWHGLQWDEXODUIRUPDVVKRZQLQ7DEOH TABLE 5.33 # Transhipment Problem > ± ± " ± ± " + ± S S # Ø # Æ "= ' = " " 'HPDQG 7KH¿UVW URZVUHSUHVHQWWKH FRQVWUDLQWVJLYHQLQ ZKLOHWKHUHPDLQLQJ URZVVKRZWKHFRQVWUDLQWV JLYHQLQ 7KHFRQVWUDLQWVLQ DQG DUHUHSUHVHQWHGE\WKH¿UVW FROXPQVDQGWKHUHPDLQLQJ FROXPQV UHVSHFWLYHO\$OOWKHYDOXHVDUHSODFHGRQWKHGLDJRQDOIURPOHIWWRSWRULJKWERWWRP(DFKRIWKHPEHDUVQHJDWLYH VLJQZKLFKPXVWEHFRQVLGHUHGFDUHIXOO\ZKHQDLVLQYROYHGLQWKHUHDGMXVWPHQW GXULQJWKHVROXWLRQSURFHVV ƚƌĂŶƐƉŽƌƚĂƟŽŶƉƌŽďůĞŵŝƐĐĂůůĞĚĂƐĂtranshipment problemǁŚĞŶƐŚŝƉŵĞŶƚƐŽĨŐŽŽĚƐĂƌĞĂůƐŽĂůůŽǁĞĚĨƌŽŵ ŽŶĞƐŽƵƌĐĞƚŽĂŶŽƚŚĞƌĂŶĚĨƌŽŵŽŶĞĚĞƐƟŶĂƟŽŶƚŽĂŶŽƚŚĞƌƐŽƚŚĂƚĂƚƌĂŶƐƉŽƌƚĂƟŽŶƉƌŽďůĞŵǁŝƚŚmͲŽƌŝŐŝŶƐ ĂŶĚnͲĚĞƐƟŶĂƟŽŶƐďĞĐŽŵĞƐĂƚƌĂŶƐŚŝƉŵĞŶƚƉƌŽďůĞŵǁŝƚŚmнnƐŽƵƌĐĞƐĂŶĚĂŶĞƋƵĂůŶƵŵďĞƌŽĨĚĞƐƟŶĂƟŽŶƐ͘ tŝƚŚƐŽŵĞŵŝŶŽƌŵŽĚŝĮĐĂƟŽŶƐ͕ŝƚŝƐƐŽůǀĞĚĂƐĂƚƌĂŶƐƉŽƌƚĂƟŽŶƉƌŽďůĞŵ͘ ([DPSOH 5HFRQVLGHU([DPSOH1RZDVVXPHWKDWLWLVSRVVLEOHIRUWKHLWHPLQTXHVWLRQWREHVKLSSHG EHWZHHQVRXUFHVDQGEHWZHHQGHVWLQDWLRQVDVZHOO7KHFRVWVLQYROYHGDUHVKRZQKHUH )URP SODQW $ 7RSODQW )URP ZDUHKRXVH $ % & 3 7RZDUHKRXVH 3 4 5 6 % 4 & 5 6 ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ/͗dƌĂŶƐƉŽƌƚĂƟŽŶĂŶĚdƌĂŶƐŚŝƉŵĞŶƚWƌŽďůĞŵƐഩഩ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nitial Feasible Solution: Non-optimal B A –0 C 2 P 8 R Q 12 10 150 – 12 Supply ui 500 0 300 –4 7 200 –6 8 0 – 11 2 0 – 10 10 0 – 12 – 13 S 230 13 120 A 0 B 5 – 0 10 9 0 7 6 0 16 8 0 5 4 0 4 8 0 0 1,000 –9 +2 + –2 –0 –1 7 7 180 –5 11 +8 16 11 –5 120 –5 14 200 –0 C – 16 – 11 6 –1 – 12 –5 –0 P – 23 Q – 20 R – 24 12 10 12 – 14 – 17 – 16 11 8 – 11 – 20 – 17 11 –7 –6 –7 –6 6 –6 –0 –1 5 –6 3 +1 –0 –9 –0 S – 26 13 – 23 14 – 14 7 – 11 9 –7 –9 Demand 0 0 0 180 150 350 320 vj 0 4 6 11 10 12 13 Total cost = 10 ¥ 150 + 12 ¥ 230 + 13 ¥ 120 + 7 ¥ 180 + 8 ¥ 120 + 7 ¥ 200 = ` 9,440 )URP7DEOHWKHRSWLPDOVROXWLRQLV 6HQGIURPSODQW 6HQGIURPSODQW 6HQGIURPSODQW 6HQGIURPZDUHKRXVH% XQLWVWRSODQWDQGXQLWVWRZDUHKRXVH% XQLWVWRZDUHKRXVH$DQGXQLWVWRZDUHKRXVH& XQLWVWRZDUHKRXVH' XQLWVWRZDUHKRXVH' 7KLVWUDQVSRUWDWLRQSDWWHUQZRXOGLQYROYHDWRWDOFRVWRICUHVXOWLQJLQDVDYLQJRIC±C CRQDFFRXQWRIWKHSRVVLELOLW\RIWUDQVKLSPHQW 0 vj 13 12 10 0 – 26 – 22 – 20 Demand S R Q 2 0 14 8 – 14 – 15 – 20 –9 6 0 7 11 16 –0 – 13 –6 –9 –3 –3 9 180 9 5 8 0 6 7 12 P –0 180 –7 –4 –7 – 12 –3 10 150 4 4 0 6 16 11 10 Q –0 150 – 11 –3 –4 –7 –2 10 350 8 0 3 5 11 8 12 R –0 350 –7 +1 –4 –3 13 320 0 –0 1,000 0 – 13 – 10 0 2 0 – 10 0 8 10 –9 200 7 –6 –2 300 14 200 0 ui 500 120 Supply 13 S Total cost = 2 ¥ 230 + 10 ¥ 150 + 13 ¥ 120 + 7 ¥ 180 + 8 ¥ 350 + 7 ¥ 200 = ` 8,980 – 25 – 16 – 19 11 6 – 21 – 14 7 P 12 0 – 13 9 – 16 –3 7 10 C –7 – 230 0 5 8 B –2 2 230 C 0 –0 B A A Revised Solution: Non-optimal + + TABLE 5.35 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 0 vj 13 12 10 12 0 – 25 – 22 – 20 – 21 – 15 10 Demand S R Q P C 5 B –7 0 A –0 2 0 14 8 – 14 – 16 5 0 7 11 – 12 –6 –9 9 180 9 5 8 – 21 16 11 0 –2 6 6 – 10 –0 7 7 7 –3 12 –0 180 –6 –4 –7 – 11 –3 10 270 – 120 150 4 4 0 6 16 11 10 Q – 10 –3 –4 –6 –2 10 350 8 0 3 5 11 8 12 R –0 350 –8 –5 –4 –1 12 320 0 10 2 8 7 14 13 S Total cost = 2 ¥ 230 + 10 ¥ 270 + 7 ¥ 180 + 8 ¥ 350 + 7 ¥ 200 + 2 ¥ 120 = ` 8,860 – 24 – 16 – 19 – 14 – 12 –4 –3 8 P 0 – 230 230 C To 9 0 2 B Revised Solution: Optimal A TABLE 5.36 –0 120 200 1,000 0 0 0 0 200 300 500 Supply – 12 – 10 – 10 –9 –5 –2 0 ui ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ/͗dƌĂŶƐƉŽƌƚĂƟŽŶĂŶĚdƌĂŶƐŚŝƉŵĞŶƚWƌŽďůĞŵƐഩഩ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 7KH6ROYHUDQG7UDQVKLSPHQW3UREOHP /LNHIRUWUDQVSRUWDWLRQSUREOHPVZHFDQXVH6ROYHUWRREWDLQVROXWLRQWRWKHWUDQVKLSPHQWSUREOHPV:HQRZ FRQVLGHUWKLV,WLVLOOXVWUDWHGE\UHFRQVLGHULQJ([DPSOH7KHUHOHYDQWFRVWPDWULFHVDUHVKRZQLQ)LJXUH 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DQGWKHQ/9)LQDOO\ZHFOLFNDW' 1DQGJHWWKH VROXWLRQ7KHVROXWLRQLVSUHVHQWHGLQ)LJXUH )URPWKHWDEOHLWLVHYLGHQWWKDWWKHPRYHPHQWRILWHPVZRXOGEHDVIROORZV ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ ( # ) 6 5 3ODQW$ 3ODQW% 3ODQW$ :DUHKRXVH4 3ODQW% :DUHKRXVH3 3ODQW% :DUHKRXVH5 3ODQW& :DUHKRXVH6 :DUHKRXVH4 :DUHKRXVH6 – – – – – – )LJXUH 2SWLPDO6ROXWLRQWRWKH7UDQVKLSPHQW3UREOHP 7KLVSDWWHUQRIVKLSPHQWZRXOGLQYROYHDWRWDOFRVWRICDVDJDLQVWCLQFDVHRIVLPSOHWUDQVSRUWDWLRQ ZLWKQRIDFLOLW\IRUWUDQVKLSPHQWWKXVUHVXOWLQJLQDVDYLQJRIC1RWLFHWKDWWKLVVROXWLRQPDWFKHVZLWK WKHVROXWLRQREWDLQHGHDUOLHU ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ/͗dƌĂŶƐƉŽƌƚĂƟŽŶĂŶĚdƌĂŶƐŚŝƉŵĞŶƚWƌŽďůĞŵƐഩഩ 5(9,(:,//8675$7,216 ([DPSOH 6ROYHWKHIROORZLQJSUREOHPXVLQJWKHWUDQVSRUWDWLRQPHWKRGREWDLQLQJWKHLQLWLDOIHDVLEOH VROXWLRQE\9$07KHFHOOHQWULHVLQWKHWDEOHDUHXQLWFRVWV LQC )URP 7R 6XSSO\ 'HPDQG 7KLVLQLWLDOIHDVLEOHVROXWLRQXVLQJ9$0LVGLVSOD\HGLQ7DEOH6LQFHWKHQXPEHURIRFFXSLHGFHOOV LVOHVVWKDQ ± DQeLVLQWURGXFHGLQDFHOOWRWHVWZKHWKHUWKHVROXWLRQLVRSWLPDO7KLVVROXWLRQLV IRXQGQRQRSWLPDO,WLVLPSURYHGDQGSUHVHQWHGLQ7DEOH7KLVRQHLVDOVRIRXQGQRQRSWLPDO7KHQH[W UHYLVHGVROXWLRQSUHVHQWHGLQ7DEOHLVIRXQGWREHRSWLPDO TABLE 5.37 From To 1 2 – 24 – 22 Initial Feasible Solution: Non-optimal 1 2 3 80 69 103 3 47 – 59 4 12 12 + 100 72 6 + 64 10 16 4 86 – 87 Supply ui 61 12 0 40 16 – 31 94 20 – 40 25 8 – 57 19 8 – 52 64 10 65 –20 16 3 5 4 – 71 + 103 15 – 24 6 87 36 2 57 19 – 11 8 – 63 e 20 Demand 16 14 18 6 10 vj 56 72 103 76 71 – 23 – 21 94 + 27 5 72 – 11 – 70 Total cost = 103 ¥ 12 + 72 ¥ 6 + 40 ¥ 10 + 16 ¥ 16 + 36 ¥ 4 + 15 ¥ 6 + 19 ¥ 2 + 20 ¥ 8 = ` 2,756 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 5.38 To From 1 2 – 36 – 34 Improved Solution: Non-optimal 1 2 3 80 69 103 3 47 – 59 10 64 + 72 Supply ui + 61 12 0 40 16 – 31 94 20 – 28 25 8 – 57 – 52 2 10 100 5 4 8 8 65 – 32 16 4 3 16 4 86 15 27 20 Demand 16 14 vj 44 72 – 59 103 – 12 87 36 – 51 8 – 99 – 11 57 – 12 19 – 11 2 6 5 – 35 – 21 72 94 19 8 18 6 10 64 103 64 71 – 82 Total cost = 103 ¥ 10 + 64 ¥ 2 + 72 ¥ 8 + 40 ¥ 8 + 16 ¥ 16 + 36 ¥ 4 + 15 ¥ 8 + 20 ¥ 6 + 19 ¥ 2 = ` 2,732 TABLE 5.39 From To 1 – 36 Improved Solution: Optimal 1 2 3 80 69 103 100 72 103 87 –7 2 64 Supply ui 61 12 0 40 16 – 31 94 20 – 28 25 8 – 47 – 42 5 4 8 2 16 2 – 34 47 – 69 – 32 65 16 – 10 4 3 16 4 86 15 27 20 Demand 16 14 vj 44 62 – 69 – 12 36 – 61 8 – 89 –1 57 –2 19 – 11 2 6 5 – 25 – 11 72 94 19 8 18 6 10 64 103 64 61 – 72 Total cost = 103 ¥ 2 + 64 ¥ 2 + 61 ¥ 8 + 72 ¥ 16 + 16 ¥ 16 + 36 ¥ 4 + 15 ¥ 8 + 20 ¥ 6 + 19 ¥ 2 = ` 2,652 ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ/͗dƌĂŶƐƉŽƌƚĂƟŽŶĂŶĚdƌĂŶƐŚŝƉŵĞŶƚWƌŽďůĞŵƐഩഩ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¿UVWEDODQFHGE\DGGLQJDGXPP\SODQW #ZLWKVXSSO\YDOXHRIXQLWVDQGFRVWHOHPHQWVHTXDOWRWKHJLYHQSHQDOW\YDOXHVDQGLQFROXPQV KHDGHG78DQG(UHVSHFWLYHO\7KHLQIRUPDWLRQLVVKRZQLQ7DEOH7KHLQLWLDOIHDVLEOHVROXWLRQWRWKH SUREOHPXVLQJ9$0LVDOVRJLYHQLQWKHWDEOH8SRQWHVWLQJWKHVROXWLRQLVIRXQGWREHRSWLPDO,WLQYROYHVD WRWDOFRVWRIWUDQVSRUWDWLRQHTXDOWRCDQGDSHQDOW\FRVWRIC TABLE 5.40 Initial Solution–VAM Buyer Plant A –2 D E 8 4 B 9 C 6 T –1 4 600 7 150 –1 –2 5 F 100 –4 100 10 9 –2 100 8 3 3 Demand 750 200 500 vj 6 4 6 400 Supply ui 100 0 800 3 150 0 400 –3 1,450 Total cost = 4 ¥ 100 + 9 ¥ 600 + 7 ¥ 100 + 9 ¥ 100 + 6 ¥ 150 = ` 8,300 and Penalty = 3 ¥ 400 = ` 1,200 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ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eLVSODFHGLQWKHFHOO&DVLWLVDQLQGHSHQGHQWFHOO7KHVROXWLRQLVWHVWHGIRURSWLPDOLW\ DQGLVIRXQGQRWWREHRSWLPDO$FORVHGSDWKLVGUDZQDVVKRZQLQWKHWDEOHDQGDUHYLVHGVROXWLRQLV REWDLQHGDVJLYHQLQ7DEOH,WLVWHVWHGDQGIRXQGWREHRSWLPDO7KHWRWDOFRVWRIWUDQVSRUWDWLRQ LQYROYHGLVC TABLE 5.41 Initial Solution-VAM Destination Source P A 6 – B 13 –8 C 9+ D 12 –5 200 Q R 11 9 e –5 –5 150 9 8 250 14 –2 –3 + +1 10 10 –2 Supply ui 8 200 0 15 300 –1 500 3 100 1 S 300 –7 – 12 12 10 Demand 350 250 300 200 vj 6 6 9 9 100 100 1,100 Total cost = 6 ¥ 200 + 8 ¥ 300 + 9 ¥ 150 + 9 ¥ 250 + 12 ¥ 100 + 10 ¥ 100 = ` 9,400 ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ/͗dƌĂŶƐƉŽƌƚĂƟŽŶĂŶĚdƌĂŶƐŚŝƉŵĞŶƚWƌŽďůĞŵƐഩഩ TABLE 5.42 Revised Solution–Optimal Destination Source P A 6 B 13 –8 C 9 D 12 –4 Q R 11 9 10 8 Supply ui 200 0 15 300 –1 12 500 3 100 2 S e 100 –5 –5 250 9 –2 10 250 –2 –1 14 8 300 –8 –1 12 10 Demand 350 250 300 200 vj 6 6 9 8 100 100 1,100 Total cost = 6 ¥ 100 + 8 ¥ 100 + 8 ¥ 300 + 9 ¥ 250 + 9 ¥ 250 + 10 ¥ 100 = ` 9,300 E 7KHRSWLPDOVROXWLRQREWDLQHGLQ7DEOHLVDXQLTXHRQHVLQFHQRD LVHTXDOWR]HUR F $ SHUFHQW UHGXFWLRQ RQ WKH URXWH ' LPSOLHV WKDW WKH XQLW FRVW RQ WKLV URXWH ZLOO EH UHYLVHG WR C,QWKHRSWLPDOVROXWLRQVKRZQLQ7DEOHWKHD ±ZLWKDFRVWYDOXHRICIRUWKLVURXWH ZRXOGEHUHYLVHGWR±ZLWKWKHXQLWFRVWRIC+RZHYHUVLQFHD FRQWLQXHVWREHQHJDWLYHQRFRVW VDYLQJLVSRVVLEOHE\VXSSO\LQJJRRGVRQWKLVURXWH+HQFHWKHRSWLPDOVROXWLRQZLOOQRWEHDIIHFWHG E\WKLVFKDQJH G 7KHRSWLPDOVROXWLRQFDOOVIRUWUDQVIHUULQJXQLWVRQWKHURXWH$1RZLILWLVGHVLUHGWRVHQG XQLWVRQWKLVURXWHZHVKDOOUHGXFHWKHVXSSO\DWVRXUFHDQGWKHGHPDQGDWGHVWLQDWLRQ$E\ XQLWVDQGDJDLQVROYHWKHSUREOHP)RUWKLVVROXWLRQZHPDNHWKHURXWH$SURKLELWHGVLQFHQR PRUHXQLWVFDQEHVHQWRQLW 7KHVROXWLRQLVJLYHQLQ7DEOH,WLVWHVWHGIRURSWLPDOLW\DQGIRXQGWREHRSWLPDO$FFRUGLQJO\ WKHWRWDOFRVWRIWUDQVSRUWDWLRQZRUNVRXWWREH¥¥¥¥¥ ¥¥ C H 7RWUDQVIHUXQLWVRQWKHURXWH'ZHGUDZDFORVHGORRSEHJLQQLQJZLWKWKLVFHOOLQ7DEOHWR VHHLILWFDQLQGHHGEHGRQH7KHFORVHGSDWKLV' Æ'Æ$Æ$Æ':LWKJLYHQTXDQWLWLHV LWLVHYLGHQWWKDWLWLVSRVVLEOHWRGRVR7KHTXDQWLWLHVRIWKHVHFHOOVZRXOGEHUHYLVHGWR±'$ $DQG')XUWKHUVLQFHD ±IRUWKHFHOO'WKLVWUDQVIHURIXQLWVZLOOFDXVH WKHWRWDOFRVWWRLQFUHDVHE\¥ C ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 5.43 Revised Solution–2 Destination Source P Q R A M 11 9 B 13 10 8 S e –9 –6 200 C 9 D 12 –5 –6 9 –3 10 250 –1 –1 8 300 –8 50 15 14 12 12 10 Demand 200 250 300 200 vj 5 5 9 8 50 100 Supply ui 50 0 300 –1 500 4 100 2 950 Total cost = 6 ´ 150 + 8 ´ 50 + 8 ´ 300 + 9 ´ 200 + 9 ´ 250 + 12 ´ 50 + 10 ¥ 100 = ` 9,350 I 6LQFH$LVDQRFFXSLHGFHOOWKHUHTXLUHGDQVZHUFDQEHREWDLQHGDVIROORZV/HWWKHFRVWHOHPHQW RIWKLVFHOOFKDQJHWR27R¿QG2ZHVROYHWKHIROORZLQJHTXDWLRQV Y Y Y Y Y 2Y DQGY 7KLVJLYHV ± 2 Y Y ±2Y DQGY :LWKWKHVHYDOXHV D ±2± ±±2£ RU 2≥± D ±± ± D ±±2± ±±2£ RU 2≥± D ±± ± D 2± 2±£ RU 2£ D 2± 2±£ RU 2£ D ± ± D ±2± ±2±£ RU 2≥± D ± ± 7KHOHDVWSRVLWLYHYDOXHRI2LVZKLOHWKHOHDVWQHJDWLYHYDOXHLV±$FFRUGLQJO\±£2£ZLOOFDXVHQR FKDQJHLQWKHFXUUHQWRSWLPDOVROXWLRQ7KXV 2RU± DQG ,WPHDQVWKDWQRFKDQJH LQWKHFXUUHQWRSWLPDOVROXWLRQZLOORFFXULIWKHXQLWFRVWRIWKHURXWH$YDULHVEHWZHHQCDQGC ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ/͗dƌĂŶƐƉŽƌƚĂƟŽŶĂŶĚdƌĂŶƐŚŝƉŵĞŶƚWƌŽďůĞŵƐഩഩ ([DPSOH $FRQVWUXFWLRQ¿UPKDVFXUUHQWO\XQGHUWDNHQWKUHHSURMHFWVRQHHDFKLQ1RLGD*UHDWHU1RLGD DQG*XUXJUDP(DFKSURMHFWUHTXLUHVDVSHFL¿FVXSSO\RIJUDYHOZKLFKWKH¿UPVRXUFHVIURP'HKUDGXQ$OLJDUK %KRSDODQG.DQSXU6KLSSLQJFRVWV LQUXSHHV GLIIHUIURPORFDWLRQWRORFDWLRQ7KHIROORZLQJWDEOHVXPPDULVHV WKHSUREOHPWKHPDQDJHURIWKHFRQVWUXFWLRQ¿UPIDFHV )URP 7R $YDLODEOH VXSSO\ 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Advertising Campaign Schedule - =?0=B =C0>D >E0?D 79 5DGLR 0DJD]LQHV 0LQ([SRVXUHV LQPLOOLRQV 7 - 1 7KHLQLWLDOVROXWLRQXVLQJ9$0LVSUHVHQWHGLQ7DEOH7KHVROXWLRQLVGHJHQHUDWHEHFDXVHWKHUH DUHVL[RFFXSLHGFHOOVDJDLQVWWKHUHTXLUHGQXPEHURIVHYHQ ± 'HJHQHUDF\LVUHVROYHGE\ SODFLQJDQeLQWKHFHOOUHSUHVHQWHGE\ODVWFROXPQRIWKHVHFRQGURZ 7KHVROXWLRQLVWHVWHGIRURSWLPDOLW\DQGLVIRXQGWREHQRQRSWLPDO7KXVDFORVHGSDWKLVGUDZQ DVVKRZQ 7KHLPSURYHGVROXWLRQLVJLYHQLQ7DEOH ,WLVWHVWHGDQGIRXQGWREHRSWLPDO,WPD\EHQRWHG KRZHYHULWLVQRWXQLTXHVLQFHWKHFHOO D 7KHVROXWLRQLV 7KURXJK79 7KURXJK5DGLR 7KURXJK0DJD]LQHV PSHRSOHWRUHDFKLQWKHDJHJURXS±PSHRSOHWRUHDFKLQWKHDJH JURXSDQGRYHU PSHRSOHWRUHDFKLQWKHDJHJURXS± PSHRSOHWRUHDFKLQWKHDJHJURXS± 7RWDOPLQLPXPFRVW CODNK ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 5.46 Initial Solution: Non-optimal Age-groups Media 13–18 19–25 12 7 26–35 25 TV –1 5 – 10 36 & over 10 10 1 Dummy Max. Exp. ui + 0 40 0 30 –1 –1 e 30 10 Magazines 14 –4 –3 –6 9 –3 12 12 + 9 –1 10 10 0 10 12 – Radio 0 20 90 –3 Min. Exp 30 25 15 10 10 vj 11 7 10 10 1 Dummy 10 36 & over 10 10 12 10 0 12 TABLE 5.47 Improved Solution: Optimal Age-groups Media 13–18 19–25 12 7 26–35 25 TV –2 –1 Max. Exp. ui 40 0 30 0 0 20 0 90 5 0 e 30 Radio 10 Magazines 14 –2 9 –3 0 15 –4 –5 12 9 5 –2 Min. Exp 30 25 15 10 10 vj 10 7 9 10 0 LL 6LQFHWKHRSWLPDOVROXWLRQUHTXLUHVUHDFKLQJPSHRSOHLQWKHDJHJURXS±WKURXJK79 LWHQWLUHO\PHHWVWKHUHTXLUHPHQWRIDWOHDVWPVXFKH[SRVXUHV1RZWRHQVXUHPH[SRVXUHV WKURXJK79LQWKHDJHJURXS±ZHSODFHDLQWKHFHOO DQGGUDZDFORVHGSDWKDQG PDNHDGMXVWPHQWVLQWKHFHOOVZKLFKOLHRQLW7KHFHOOVLQFOXGH DQG 7KHUH VXOWLQJVROXWLRQLVSURYLGHGLQ7DEOH ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ/͗dƌĂŶƐƉŽƌƚĂƟŽŶĂŶĚdƌĂŶƐŚŝƉŵĞŶƚWƌŽďůĞŵƐഩഩ TABLE 5.48 Revised Solution Age-groups Media 19–25 Dummy Max. Exp. 7 10 36 & over 10 10 10 9 12 10 0 Magazines 14 12 9 12 0 20 Min. Exp 30 25 15 10 10 90 13–18 25 4 TV 12 Radio 26–35 1 0 40 26 4 30 5 15 7KHUHYLVHGSODQLV 7KURXJK79 PSHRSOHWRUHDFKLQWKHDJHJURXS±PSHRSOHWRUHDFKLQWKHDJH JURXS±PSHRSOHWRUHDFKLQWKHDJHJURXSDQGDERYH 7KURXJK5DGLR PSHRSOHWRUHDFKLQWKHDJHJURXS± 7KURXJK0DJD]LQHV PSHRSOHWRUHDFKLQWKHDJHJURXS±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ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ LLL ,IWKHFRPSDQ\ZDQWVDWOHDVWTXLQWDOVRIFHPHQWDUHWUDQVSRUWHGIURPIDFWRU\&WRGLVWULEXWLRQ FHQWUH<ZLOOWKHWUDQVSRUWDWLRQVFKHGXOHEHDQ\GLIIHUHQW",IVRZKDWZLOOEHWKHQHZRSWLPDOVFKHGXOH DQGWKHHIIHFWRQFRVW" LY 6XSSRVHWKHFRPSDQ\GHVLUHVWRVHQGDWPRVWTXLQWDOVRIFHPHQWIURPIDFWRU\&WRGLVWULEXWLRQ FHQWUH<ZKDWZLOOWKHRSWLPDOVFKHGXOHEH"$OVRREWDLQWKHWRWDOWUDQVSRUWDWLRQFRVWLQVXFKDFDVH Y 7KHPRQWKO\SURGXFWLRQDWIDFWRU\&DQGWKHPRQWKO\GHPDQGDWGLVWULEXWLRQFHQWUH;LQFUHDVHE\ TXLQWDOVHDFK:KDWHIIHFWZLOOLWKDYHRQWKHRSWLPDOVROXWLRQDQGRQWKHWRWDOFRVW" L 7KHLQLWLDOVROXWLRQWRWKHJLYHQSUREOHPXVLQJ9$0LVVKRZQLQ7DEOH TABLE 5.49 Initial Solution: VAM Distribution centres Factory Z W X Y 10 8 5 Supply ui 7,000 0 8,000 10 10 7,000 A – 14 –9 6,000 B –1 7 9 1,000 – 4 –6 + 8 14 – 8 10,000 25,000 4,000 6,000 1,000 15 C 6 Demand 6,000 6,000 8,000 5,000 vj –4 –1 5 –2 –1 10 1 + 7KLVVROXWLRQLVWHVWHGIRURSWLPDOLW\DQGIRXQGWREHQRQRSWLPDO$QLPSURYHGVROXWLRQLVJLYHQLQ 7DEOH7KLVVROXWLRQLVIRXQGWREHRSWLPDO 7KHRSWLPDOVROXWLRQLVLQGLFDWHGEHORZ ( Ø# Æ * 7RWDOFRVW ¥¥¥¥¥O¥ C ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ/͗dƌĂŶƐƉŽƌƚĂƟŽŶĂŶĚdƌĂŶƐŚŝƉŵĞŶƚWƌŽďůĞŵƐഩഩ TABLE 5.50 Improved Solution: Optimal Distribution centres Factory W X Y 10 8 5 Z Supply ui 7,000 0 8,000 9 9 7,000 A – 13 –8 –5 4 6,000 B –1 7 9 2,000 –1 8 15 1,000 6,000 3,000 10 14 8 10,000 6,000 6,000 8,000 5,000 25,000 –3 0 5 –1 C 6 Demand vj –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evised Problem I: Initial Solution Distribution centres Factory Supply ui 7,000 0 8 8,000 4 8 5,000 3 20,000 Z W X Y 10 8 5 4 15 3,000 A –7 –3 4,000 6,000 1,000 1,000 B 7 C 6 Demand 6,000 6,000 3,000 5,000 vj 3 5 5 4 9 –6 5,000 –2 10 –6 14 –1 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 7KHVROXWLRQLVVHHQWREHRSWLPDO7KHWRWDOWUDQVSRUWDWLRQFRVWLV¥¥¥ ¥¥¥ CSOXV¥ URXWH0F C7KH VFKHGXOHLV ( Ø# Æ * LY 7KHRSWLPDOVROXWLRQREWDLQHGLQ L LQYROYHVGLVSDWFKLQJTXLQWDOVIURPIDFWRU\ WRFHQWUH +RZHYHULIQRWPRUHWKDQTXLQWDOVLVWREHVHQWRQWKLVURXWHWKHQZHZRXOGUHSODFHWKHRULJLQDO SUREOHPE\RQHZKHUHVXSSO\DW DQGGHPDQGDWZRXOGEHUHGXFHGE\ VHQGLQJXQLWVRQ URXWH0 DQGUHSODFLQJLWVFRVWHOHPHQWE\- VRWKDWQRPRUHXQLWVPD\EHVHQWRQWKLVURXWH 7KHUHYLVHGSUREOHPLVVKRZQLQ7DEOH+ ZKHUHLQLQLWLDOVROXWLRQLVDOVRSURYLGHG7KHVROXWLRQ REWDLQHGWKURXJK9$0LVIRXQGWREHRSWLPDO 7KHWRWDOFRVWLQYROYHGLVC TABLE 5.52 Revised Problem II: Initial Solution Distribution centres Factory W X Y 10 8 5 Supply ui 7,000 0 8,000 10 10 Z 7,000 A – 14 –9 6,000 B –1 7 –6 4 1,500 500 9 15 8 10 M 8 9,500 24,500 3,500 6,000 C 6 Demand 6,000 6,000 7,500 5,000 vj –4 –1 5 –2 –1 Y 5HFRQVLGHUWKHVROXWLRQJLYHQLQ7DEOH7KHFHOOLVDQHPSW\FHOOKHUH6RZHGUDZD FORVHGORRSEHJLQQLQJZLWKWKLVFHOODVÆ ÆÆÆ7KXVZHDGGWRWKH TXDQWLWLHVRIWKHFHOOV 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7KHWHDPFRQVWLWXWHGIRUWKLVSXUSRVHZRUNHGH[WHQVLYHO\IRUGD\VWUDYHOOLQJDOORYHUFROOHFWLQJUHOHYDQW LQIRUPDWLRQ ,W LGHQWL¿HG ¿YH ORFDWLRQV IRU HVWDEOLVKLQJ ZDUHKRXVHV DW$KPHGDEDG ,QGRUH 1DJSXU 1HZ'HOKLDQG.DQSXU)RUHDFKRIWKHVHLWHVWLPDWHGWKHFDSDFLW\LQWHUPVRIWKHQXPEHURIW\UHVWKH RYHUKHDGFRVWVDVVRFLDWHGDVDOVRWKHFRVWRIVHQGLQJW\UHVWRWKHPIURPWKHSODQWLQ0DKDUDVKWUD7KLV LQIRUPDWLRQLVVKRZQLQ7DEOH TABLE 1 Cost of Transportation, Capacity and Overhead Cost Information * $KPHGDEDG # G F 6 GHHH 5 F /1 GHHH F ,QGRUH 1HZ'HOKL 1DJSXU .DQSXU 7KHWHDPDOVRFROOHFWHGLQIRUPDWLRQRQWKHFRVWRIWUDQVSRUWLQJW\UHVIURPYDULRXVZDUHKRXVHVWRGLIIHUHQW FXVWRPHUORFDWLRQVDQGWKHWRWDOGHPDQGLQWKRXVDQGVRIW\UHVDWYDULRXVFXVWRPHUSRLQWV7KLVLQIRUPDWLRQ LVSUHVHQWHGLQ7DEOH ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 2 Transportation Cost (per tyre) from Various Regional Warehouses < ): 7 ) 9 7 $KPHGDEDG $JUD ,QGRUH 5RKWDN 1HZ'HOKL .DQSXU 6XUDW :DUGKD 1DJSXU 9DGRGDUD &RQVLGHULQJ WKH LQIRUPDWLRQ SUHVHQWHG KHUH SUHSDUH D UHSRUW SURYLGLQJ DQVZHUV WR WKH 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But everyone is not equally efficient at every job. Someone may be more efficient on one and less efficient on the other job, while it might be otherwise for someone else. The relative efficiency is reflected in terms of the time taken for, or the cost associated with, performance of different jobs by different people. An obvious problem for a manager to handle is to assign jobs to various workers in a manner that they can be done in the most efficient way. Interestingly, such problems can be formulated as linear programming problems or as transportation problems and solved as such, but a method, called Hungarian Assignment Method provides an easy route. It allows a manager to obtain answers to questions like the following: How to assign the given jobs to some workers on a one-to-one basis when completion time of performance is given for each combination and it is desired that the jobs are completed in the least time or at the least cost. Further, in the assignment pattern so obtained, is only one option available or there are other equally attractive ones also? How to deal with situations when the number of jobs do not match with the number of job performers, when some job (s) cannot be performed by, or is not be given to, a particular performer, or when a certain job has to be given to a particular individual? How should the salesmen of a company be assigned to different sales zones so that the total expected sales are maximised? Given the order of preferences that different managers have for different rooms in a hotel on one of its floors, what pattern of assignment of rooms to the managers will satisfy their requirements the most? How to schedule the flights or the bus routes between two cities so that the layover times for the crew can be minimised? In fact, the assignment method works for any problems in which one-to-one matching is called for in the light of the given payoffs, where the total pay-off is sought to be minimised or maximised. Only ordinary arithmetical skills are required for working through this chapter. Also needed for some part of it is familiarity with the summation notation, algebraic inequalities, matrices and their transpose. ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ Learning Objectives After reading this chapter, you should be able to: LO 1 LO 2 LO 3 LO 4 LO 5 LO 6 LO 7 <ŶŽǁĂŶĂƐƐŝŐŶŵĞŶƚƉƌŽďůĞŵ /ůůƵƐƚƌĂƚĞŚŽǁƚŽƐŽůǀĞĂŶĂƐƐŝŐŶŵĞŶƚƉƌŽďůĞŵďǇĐŽŵƉůĞƚĞĞŶƵŵĞƌĂƟŽŶĂŶĚ ƚƌĂŶƐƉŽƌƚĂƟŽŶŵĞƚŚŽĚ ĞƚĞƌŵŝŶĞŚŽǁƚŽƐŽůǀĞĂƐƐŝŐŶŵĞŶƚƉƌŽďůĞŵĂƐĂŶ>WWĂŶĚƵƐŝŶŐ,D ŝƐĐƵƐƐƵŶďĂůĂŶĐĞĚĂŶĚĐŽŶƐƚƌĂŝŶĞĚĂƐƐŝŐŶŵĞŶƚƉƌŽďůĞŵƐ ŽŵƉĂƌĞƵŶŝƋƵĞĂŶĚŵƵůƟƉůĞŽƉƟŵĂůƐŽůƵƟŽŶƐƚŽĂŶĂƐƐŝŐŶŵĞŶƚƉƌŽďůĞŵ ^ŽůǀĞƉƌŽďůĞŵƐĐĂůůŝŶŐĨŽƌŵĂǆŝŵŝƐĂƟŽŶŽĨŽďũĞĐƟǀĞĨƵŶĐƟŽŶ ĞƚĞƌŵŝŶĞƚŚĞĚƵĂůƚŽĂŶĂƐƐŝŐŶŵĞŶƚƉƌŽďůĞŵ ,1752'8&7,21 ,QWKHSUHYLRXVFKDSWHUZHGLVFXVVHGDERXWVROXWLRQWRWKHWUDQVSRUWDWLRQ SUREOHP1RZZHFRQVLGHUDQRWKHUW\SHRIVSHFLDOOLQHDUSURJUDPPLQJ SUREOHPFDOOHGWKH LO 1 Know an assignment problem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¿WV $W\SLFDODVVLJQPHQWSUREOHPIROORZV ([DPSOH $SURGXFWLRQVXSHUYLVRULVFRQVLGHULQJKRZKHVKRXOGDVVLJQWKHIRXUMREVWKDWDUHWREH SHUIRUPHGE\IRXURIWKHZRUNHUV+HZDQWVWRDVVLJQWKHMREVWRWKHZRUNHUVVXFKWKDWWKHDJJUHJDWHWLPHWR SHUIRUPWKHMREVLVOHDVW%DVHGRQSUHYLRXVH[SHULHQFHKHKDVWKHLQIRUPDWLRQRQWKHWLPHWDNHQE\WKHIRXU ZRUNHUVLQSHUIRUPLQJWKHVHMREVDVJLYHQLQ7DEOH TABLE 6.1 Time Taken by Workers on Various Jobs (in minutes) ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ :HVKDOOQRZFRQVLGHUDQDO\WLFDOO\WKHVROXWLRQWRWKLVSUREOHPRIWKHVXSHUYLVRU 7KHUHDUHIRXUPHWKRGVRIVROYLQJDQDVVLJQPHQWSUREOHP D FRPSOHWHHQXPHUDWLRQPHWKRG E WUDQVSRUWDWLRQ PHWKRG F VLPSOH[PHWKRGDQG G +XQJDULDQDVVLJQPHQWPHWKRG:HVKDOOGLVFXVVWKHVHPHWKRGVQRZ &RPSOHWH(QXPHUDWLRQ0HWKRG ,QWKLVPHWKRGDOOSRVVLEOHDVVLJQPHQWVDUHOLVWHGRXWDQGWKHDVVLJQPHQW LQYROYLQJWKHPLQLPXPFRVW RUPD[LPXPSUR¿WLIWKHSUREOHPLVRIWKH PD[LPLVDWLRQW\SH LVVHOHFWHG7KLVUHSUHVHQWVWKHRSWLPDOVROXWLRQ,Q FDVH WKHUH DUH PRUH WKDQ RQH DVVLJQPHQW SDWWHUQV LQYROYLQJ WKH VDPH OHDVWFRVWWKH\DOOVKDOOUHSUHVHQWRSWLPDOVROXWLRQV²WKHSUREOHPKDV PXOWLSOHRSWLPDWKHQ LO 2 Illustrate how to solve an assignment problem by complete enumeration and transportation method )RU WKH SUREOHP JLYHQ LQ ([DPSOH WKHUH DUH D WRWDO RI DVVLJQPHQWV SRVVLEOH 7KHVH DUH OLVWHG LQ 7DEOHZKHUHLQWKHWLPHLQYROYHGZLWKHDFKDVVLJQPHQWSDWWHUQLVDOVRJLYHQ ,WLVFOHDUWKDWDVVLJQPHQWQXPEHUOLVWHGLQWKHWDEOHWKDWLQYROYHVZRUNHU±MREDVVLJQPHQWV DQG LVWKHRSWLPDORQHDVLWLQYROYHVWKHOHDVWWRWDOWLPHHTXDOWRPLQXWHV TABLE 6.2 Worker–Job Assignments O ,WPD\EHQRWHGWKDWLQJHQHUDOIRUDQZRUNHUMRESUREOHPWKHUHDUH QXPEHURIZD\VLQZKLFKWKHMREV FDQEHDVVLJQHGWRWKHZRUNHUV7KXVWKHOLVWLQJDQGHYDOXDWLRQRIDOOWKHSRVVLEOHDVVLJQPHQWVLVDVLPSOH PDWWHUZKHQLVDVPDOOQXPEHU%XWZKHQ LVDQHYHQPRGHUDWHO\ODUJHQXPEHUWKLVPHWKRGLVQRWYHU\ SUDFWLFDO)RUH[DPSOHIRUD¿YHSHUVRQDQG¿YHMRESUREOHPDWRWDORI DVVLJQPHQWVZLOOQHHGWREH HYDOXDWHGZKLOHDSUREOHPLQYROYLQJHLJKWSHUVRQVDQGHLJKWMREVZLOOUHTXLUHHQXPHUDWLRQDQGHYDOXDWLRQRI DVPDQ\DV DVVLJQPHQWVDQGDSUREOHPZLWKZRUNHUVDQGDVPDQ\MREVWKHQXPEHURISRVVLEOH DVVLJQPHQWVZRUNVRXWWREH7KHUHIRUHWKHPHWKRGLVQRWVXLWDEOHIRUUHDOZRUOGVLWXDWLRQV ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ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¿FDQWSRLQW WRQRWHKHUH7KHVROXWLRQREWDLQHGE\DSSO\LQJWKLVPHWKRGZRXOGEHVHYHUHO\GHJHQHUDWH7KLVLVEHFDXVH WKHRSWLPDOLW\WHVWLQWKHWUDQVSRUWDWLRQPHWKRGUHTXLUHVWKDWWKHUHPXVWEH ± ±ZKHQ EDVLFYDULDEOHV)RUDQDVVLJQPHQWSUREOHPRIWKHRUGHU ¥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¥DVVLJQPHQW SUREOHP+HUHWKH¶V DUHWKHFRVWFRHI¿FLHQWVDQGDSDUWLFXODUYDOXHRI LQGLFDWHVWKHFRVW DOWKRXJKWLPHV WDNHQE\GLIIHUHQWZRUNHUVDUHJLYHQWKH\DUHWDNHQWREHµFRVW¶VLQFHLWLVVRXJKWWRPLQLPLVHWKHWRWDOWLPH DQGKHQFHWKHFRVW RIDVVLJQLQJWKMREWRWKHWKZRUNHU7KHVROXWLRQWRWKLVSUREOHPFRQVLVWVRIDVVLJQLQJ ¶VRQHRIWKHWZRYDOXHV²RU²LQVXFKDPDQQHUWKDWWKHWRWDODVVLJQPHQWFRVWLVWKHPLQLPXP TABLE 6.3 The Assignment Problem as a Transportation Problem 'HPDQG ([DPSOH 6ROYHWKHDVVLJQPHQWSUREOHPJLYHQLQ([DPSOHXVLQJWKHWUDQVSRUWDWLRQPHWKRG 7KHSUREOHPLVUHSURGXFHGLQ7DEOHZKHUHLQWKHLQLWLDOVROXWLRQXVLQJ9$0LVFRQWDLQHG7KHUHDUHRQO\ IRXURFFXSLHGFHOOVDJDLQVWWKHUHTXLUHGQXPEHURI± DQGKHQFHWKHVROXWLRQLVGHJHQHUDWH ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ TABLE 6.4 Initial Solution: VAM Job Supply Worker B A C D 51 67 1 55 1 e e 1 1 45 40 2 57 42 63 3 49 52 48 64 1 4 41 45 60 55 1 Demand 1 1 1 1 4 1 e 1 1 7RUHPRYHGHJHQHUDF\DQeLVSODFHGLQWKUHHRIWKHLQGHSHQGHQWFHOOVQDPHO\DQG7KHVROXWLRQ LVUHSURGXFHGLQ7DEOHDQGWHVWHGIRURSWLPDOLW\:LWKDWRWDOFRVWHTXDOWR WKLV VROXWLRQLVIRXQGWREHQRQRSWLPDO$FFRUGLQJO\DFORVHGSDWKLVGUDZQVWDUWLQJZLWKWKHFHOOZKLFK KDVWKHODUJHVW DYDOXHHTXDOWR TABLE 6.5 Initial Solution: Non-optimal Job Supply ui 67 1 0 55 1 2 64 1 4 55 1 –4 4 Worker B A e 1 45 2 57 – 10 + – C D e 40 –7 – 51 1 42 – 17 + 14 63 e 1 1 3 49 4 41 Demand 1 1 1 1 vj 45 40 44 67 –8 52 48 7 1 –9 45 – 20 60 8 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 6.6 Revised Solution: Optimal Job Worker B A e C Supply ui 1 0 55 1 2 64 1 4 55 1 –4 4 D 1 1 45 40 2 57 42 –7 51 – 14 67 1 – 10 – 17 63 e 1 3 49 4 41 Demand 1 1 1 1 vj 45 40 44 53 52 –8 48 –7 1 45 –9 – 20 60 –6 7KHUHYLVHGVROXWLRQLVJLYHQLQ7DEOH,WLVWHVWHGIRURSWLPDOLW\DQGWKHD YDOXHVLQGLFDWHWKDWLWLVLQGHHG RSWLPDO 7KHVROXWLRQLV 7RWDO ,WLVLGHQWLFDOWRWKHRQHREWDLQHGHDUOLHU 6LPSOH[0HWKRG :H KDYH VHHQ WKDW DQ DVVLJQPHQW SUREOHP FDQ EH IRUPXODWHG DV D WUDQVSRUWDWLRQ SUREOHP ZKLFK LQ WXUQ LV D VSHFLDO W\SH RI OLQHDU SURJUDPPLQJ SUREOHP $FFRUGLQJO\ DQ DVVLJQPHQW SUREOHP FDQ EH IRUPXODWHGDVDOLQHDUSURJUDPPLQJSUREOHP ZLWKLQWHJHUYDULDEOHV DQG VROYHGXVLQJDPRGL¿HGVLPSOH[PHWKRGRURWKHUZLVH7KLVLVGLVFXVVHGLQ LO 3 Determine how to solve assignment problem as an LPP and using HAM ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ GHWDLOLQ&KDSWHU,WPD\EHQRWHGWKDWLWVIRUPXODWLRQUHTXLUHVWKDWWKHGHFLVLRQYDULDEOHVWDNHRQO\RQHRIWKH WZRYDOXHVRUDFFRUGLQJO\DVDQDVVLJQPHQW RIDZRUNHUWRDMRE LVPDGHRUQRW ,QJHQHUDOOHW Ï1 if ith person is assigned the jth job xij = Ì Ó0 if ith person is nott assigned the jth job 7KHSUREOHPZRXOGEH n n ! Â Â cij xij 0LQLPLVH i = j = 6XEMHFWWR n Â xij = IRU « j = n Â xij = IRU « i = RUO IRUDOODQG ([DPSOHGDWDFDQEHIRUPXODWHGDVDQ/33DVIROORZV 0LQLPLVH 6XEMHFWWR DOO¶V RU ,WPD\EHHDVLO\YLVXDOLVHGWKDWLWLVUHODWLYHO\WHGLRXVWRVROYHWKHSUREOHPLQWKLVIRUPDW7KXVWKLVDSSURDFK WRWKHVROXWLRQLVQRWFRQVLGHUHG ŶĂƐƐŝŐŶŵĞŶƚƉƌŽďůĞŵĐĂŶďĞƐŽůǀĞĚŝŶŵĂŶǇǁĂǇƐ͘/ƚĐĂŶďĞƐŽůǀĞĚďǇĐŽŵƉůĞƚĞĞŶƵŵĞƌĂƟŽŶŽĨĂůůƚŚĞƉŽƐƐŝďůĞ ĂƐƐŝŐŶŵĞŶƚƉĂƩĞƌŶƐďƵƚƚŚŝƐŵĞƚŚŽĚŝƐǀĞƌǇƟŵĞͲĐŽŶƐƵŵŝŶŐ͘ŶĂƐƐŝŐŶŵĞŶƚƉƌŽďůĞŵĐĂŶĂůƐŽďĞĨŽƌŵƵůĂƚĞĚĂŶĚ ƐŽůǀĞĚĂƐĂƚƌĂŶƐƉŽƌƚĂƟŽŶƉƌŽďůĞŵďƵƚƚŚĞƐŽůƵƟŽŶŽďƚĂŝŶĞĚďǇƚŚŝƐŵĞƚŚŽĚŝƐŐĞŶĞƌĂůůǇƐĞǀĞƌĞůǇĚĞŐĞŶĞƌĂƚĞ ĂŶĚƚŚĞƌĞĨŽƌĞƌĞƋƵŝƌĞƐŚĂŶĚůŝŶŐĚĞŐĞŶĞƌĂĐǇ͘,ĞŶĐĞ͕ŝƚŝƐĂůƐŽŝŶĞĸĐŝĞŶƚ͘dŚĞĨŽƌŵƵůĂƟŽŶĂŶĚƐŽůƵƟŽŶŽĨĂŶ ĂƐƐŝŐŶŵĞŶƚƉƌŽďůĞŵĂƐĂůŝŶĞĂƌƉƌŽŐƌĂŵŵŝŶŐƉƌŽďůĞŵƚŽŽŝƐĐƵŵďĞƌƐŽŵĞ͘ +XQJDULDQ$VVLJQPHQW0HWKRG +$0 " # $ % ! " # $ %#$& ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ " ' ( ) " * ) " + " " + START Write the problem in tabular form Is it a balanced problem? No Add dummy row(s)/column(s) Yes Is it a maximisation problem? Yes Convert it into a minimisation problem, either (i) by changing the signs of the elements of the table or by subtracting all the values from the largest value. No Obtain reduced cost tables. (i) Subtract from all entries in each row the least value in the row. (ii) From this table, subtract from all entries in each column the least value in the column. Can all zeros be covered by less than n lines? No Yes Improve the solution. For this: (i) Select the minimum of the uncovered (by lines) cell values. (ii) Subtract this value from all uncovered cell values. (iii) Add this value to the cells lying on the intersection of any pair of lines. (iv) Leave the cell values covered by only one line undisturbed. Make assignments on one-to-one match basis considering zeros in rows/columns. STOP )LJXUH 6FKHPDWLF3UHVHQWDWLRQRI +XQJDULDQ$VVLJQPHQW0HWKRG ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ ' , , 6WHS - . + &%% 6WHS / ' + 6WHS 2 + % & ' 3+ 4 ' % 5 & 6 6WHS + + " + . + % & ' # . 7 + % & " / ' % % & %% " 6WHS ( 86 6WHS 9 , 3+ : 4 " %& - 3+ 4 , ) + , %& ( %& + / 3+ 4 + % & %& 3+ 4 5 , % & . + %& ( %& %& % & 2 dŚĞ,ƵŶŐĂƌŝĂŶƐƐŝŐŶŵĞŶƚDĞƚŚŽĚŝƐĂŶĞĸĐŝĞŶƚŵĞƚŚŽĚŽĨƐŽůǀŝŶŐĂƐƐŝŐŶŵĞŶƚƉƌŽďůĞŵƐ͘/ƚŝŶǀŽůǀĞƐĚĞǀĞůŽƉŝŶŐ ĂƐĞƌŝĞƐŽĨ͚ZĞĚƵĐĞĚŽƐƚ͛ƚĂďůĞƐ͘dŚĞŵĞƚŚŽĚŝŶǀŽůǀĞƐƚŚƌĞĞƐƚĞƉƐŽĨ;ŝͿŽďƚĂŝŶŝŶŐǌĞƌŽƐŝŶĞǀĞƌǇƌŽǁĂŶĚĞǀĞƌǇ ĐŽůƵŵŶďǇƌŽǁƌĞĚƵĐƟŽŶƐĂŶĚĐŽůƵŵŶƌĞĚƵĐƟŽŶƐ͖;ŝŝͿĐŽǀĞƌŝŶŐĂůůǌĞƌŽƐǁŝƚŚĂŵŝŶŝŵƵŵŶƵŵďĞƌŽĨŚŽƌŝǌŽŶƚĂůͬ ǀĞƌƟĐĂůůŝŶĞƐƚŽĐŚĞĐŬǁŚĞƚŚĞƌĂƐƐŝŐŶŵĞŶƚƐĐĂŶďĞŵĂĚĞ͖ĂŶĚ;ŝŝŝͿŵĂŬŝŶŐĂƐƐŝŐŶŵĞŶƚƐĂƚǌĞƌŽƐŽŶůǇƐƵĐŚƚŚĂƚ ĞĂĐŚƌŽǁĂŶĚĞĂĐŚĐŽůƵŵŶŚĂƐĂƐŝŶŐůĞĂƐƐŝŐŶŵĞŶƚŵĂĚĞŝŶŝƚ͘DŽĚŝĮĐĂƟŽŶƐĂƌĞŶĞĞĚĞĚŝĨƚŚĞŶƵŵďĞƌŽĨůŝŶĞƐ ĐŽǀĞƌŝŶŐĂůůǌĞƌŽƐŝƐƐŵĂůůĞƌƚŚĂŶn͕ŽƌĚĞƌŽĨƚŚĞƐƋƵĂƌĞŵĂƚƌŝǆ͘ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ ([DPSOH 6ROYHWKHDVVLJQPHQWSUREOHPJLYHQLQ([DPSOHIRURSWLPDOVROXWLRQXVLQJ+$07KH LQIRUPDWLRQLVUHSURGXFHGLQ7DEOH TABLE 6.7 ! Time Taken (in minutes) by four Workers 6; 6< ;! = > ;= 6> 8 ;; 8 6? ;> 6@ 6 6 6! 6; < ;; " A 6WHS " ( ) " @ TABLE 6.8 Reduced Cost Table 1 ! ; < !! >= > !; < >! !8 8 ! 6 < ! 6 < 6 !? !6 6WHS * + # " ? TABLE 6.9 Reduced Cost Table 2 ! ; < !! !6 > !; < >! < 8 ! 6 < 8 6 < 6 !? ! ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ 6WHS 2 + 5 + " ? " !< TABLE 6.10 Reduced Cost Table 3 ! ; < !! !6 > !; < >! < 8 ! 6 < 8 6 < 6 !? ! 6WHS / ' 6%B& " + ' " !! TABLE 6.11 Assignment of Jobs ! ; 0 !! !6 > !; 0 >! 0 8 ! 6 0 8 6 0 6 !? ! ( !86 + / !* 8* 6* / ! , + >, " !A>A8* 6* 6<+ ;; + 6@C6!B!@6 " D ([DPSOH 8VLQJWKHIROORZLQJFRVWPDWUL[GHWHUPLQH D RSWLPDOMREDVVLJQPHQWDQG E WKHFRVWRI DVVLJQPHQWV 0DFKLQLVW -RE $ % & ' ( ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ ,WHUDWLRQ * TABLE 6.12 #$ Reduced Cost Table 1 , - . / 0 @ ! ! < = ; < ; ; 8 6 < > ! 8 < > 1 8 6 8 < 6 ,WHUDWLRQ * TABLE 6.13 #$ + Reduced Cost Table 2 , - . / 0 = < < < 6 6 ; < 8 6 > 8 < < < > ; < < 1 > 8 > < > / + 5 " !8 " >" % A6 A6 A; A;& " !6 ,WHUDWLRQ " + " !6 ' ; ) : %& / , 6) + *6 1*6 %& ) 6 , !) + 1*! %& / + 1 , 8 + *8 % & " + , > , ; " A, A> *6 *; *! 1*8 8C>C6C8C?B>! ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ TABLE 6.14 Reduced Cost Table 3 #$ , - . / 0 = 0 0 > 6 > 8 0 8 > 0 ! 0 0 0 > ; > > 1 0 ! 0 0 > 620(63(&,$/&$6(6 8QEDODQFHG$VVLJQPHQW3UREOHPV " # ' ' E LO 4 Discuss unbalanced ' % and constrained assignment % " problems F A A F F % & F D F % &5 % & + F F 6¥; 3+ 4 5 &RQVWUDLQHG$VVLJQPHQW3UREOHPV / , , " , F % # & " " G, ([DPSOH <RXDUHJLYHQWKHLQIRUPDWLRQDERXWWKHFRVWRISHUIRUPLQJGLIIHUHQWMREVE\GLIIHUHQWSHUVRQV 7KHMRE±SHUVRQPDUNLQJ¥LQGLFDWHVWKDWWKHLQGLYLGXDOLQYROYHGFDQQRWSHUIRUPWKHSDUWLFXODUMRE8VLQJWKLV LQIRUPDWLRQVWDWH L WKHRSWLPDODVVLJQPHQWRIMREVDQG LL WKHFRVWRIVXFKDVVLJQPHQW 3HUVRQ 3 3 3 3 -RE - - - - - ¥ ¥ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ H 2! 3 828 3 6 " !; . " !. # # F " + / + ' 6E " != TABLE 6.15 2 Cost Table ! > 8 6 ; 2! 2> 28 26 2;%2& >= 8! >< >> < !@ >6 != >@ < # >! >< >< < >< !> # ! < >! != ! >= < TABLE 6.16 Reduced Cost Table 1 2 ! > 8 6 ; 2! 2> 28 26 2;%2& ? !? 6 < < !> ! !> < # ? 6 6 < > < # < < 8 ; < !! < TABLE 6.17 Reduced Cost Table 2 2 2! 2> 28 26 2;%2& ! > 8 6 ; ? !; 6 > 0 0 @ ! @ 0 # ; 6 0 0 0 # 0 6 8 ! 0 = 0 " + ' 0%! & " 2!3> 2>3 6 283 ; 26 3 8 , ! " !@C!>C!C><B ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ dŚĞ,ƵŶŐĂƌŝĂŶƐƐŝŐŶŵĞŶƚDĞƚŚŽĚŝŶǀŽůǀĞƐŵĂŬŝŶŐĂƐƐŝŐŶŵĞŶƚƐŽŶĂŽŶĞͲƚŽͲŽŶĞďĂƐŝƐ͘,ĞŶĐĞ͕ĂŶĂƐƐŝŐŶŵĞŶƚ ƉƌŽďůĞŵƐŚŽƵůĚďĞĂďĂůĂŶĐĞĚŽŶĞǁŝƚŚĂŶĞƋƵĂůŶƵŵďĞƌŽĨƌŽǁƐĂŶĚĐŽůƵŵŶƐ͘/ŶĐĂƐĞƚŚĞǇĂƌĞƵŶĞƋƵĂů͕ǁĞŶĞĞĚ ƚŽďĂůĂŶĐĞŝƚďǇĂĚĚŝŶŐĂƐŵĂŶǇƌŽǁƐŽƌĐŽůƵŵŶƐĂƐǁŝůůŵĂŬĞƚŚĞŶƵŵďĞƌŽĨƌŽǁƐĂŶĚĐŽůƵŵŶƐĞƋƵĂů͘&ƵƌƚŚĞƌ͕ ŝĨƐŽŵĞĂƐƐŝŐŶŵĞŶƚŝƐƉƌŽŚŝďŝƚĞĚ͕ƚŚĞĐŽƐƚĞůĞŵĞŶƚĨŽƌƚŚĂƚƐŚŽƵůĚďĞƚĂŬĞŶƚŽďĞǀĞƌǇŚŝŐŚǀĂůƵĞ;ĞƋƵĂůƚŽMͿ͘ 8QLTXHYV0XOWLSOH2SWLPDO6ROXWLRQV 5 + # LO 5 Compare unique and multiple optimal solutions + to an assignment problem F # ' E ' F , % & * F ' ) F ([DPSOH 6ROYHWKHIROORZLQJDVVLJQPHQWSUREOHPDQGREWDLQWKHPLQLPXPFRVWDWZKLFKDOOWKHMREV FDQEHSHUIRUPHG -RE &RVWLQ :RUNHU $ % & ' " , ; 1 " !@ TABLE 6.18 6E Balancing the Assignment Problem , - . / 0 >; !@ 8> >< >! 86 >; >! !> != >< != >< 8> ! >< >@ >< ! >= 1 < < < < < 6WHS * " " !? ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 6.19 Reduced Cost Table 1 , - . / 0 = < !6 > 8 >> !8 ? < ; 6 ! 6 ! < 6 !> 6 < !! 1 < < < < < / + + ; % F&# " 6/ " >< " " >< + FE " + " + " *> + 1*> + E *6 *; + *6 1*; . + " F " + TABLE 6.20 Reduced Cost Table 2 , - . / 0 = 0 !6 8 !@ ? ; 0 ! 6 ! 6 >< 0 0 @ 0 0 = 1 0 0 0 6 0 %& / + A! + *8 1A!% , ! &. F 1 , 8 + 7 + 1*8 %& / + *8) + A!1*8 . F + 1*!* + 1*! " ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ , - 4566 7 4566 7 > !@ > !@ 6 !> 6 !> ; ! ; ! ! >< 8 >< I 8 I ! " " /ŶƚŚĞĐŽƵƌƐĞŽĨŵĂŬŝŶŐĂƐƐŝŐŶŵĞŶƚƐ͕ŝĨǁĞůĂŶĚŝŶĂƐŝƚƵĂƟŽŶǁŚĞƌĞĞĂĐŚŽĨƚŚĞƌŽǁƐĂŶĚĐŽůƵŵŶƐůĞŌŚĂƐ ŵƵůƟƉůĞǌĞƌŽƐ͕ƚŚĞŶƚŚĞƉƌŽďůĞŵǁŝůůŚĂǀĞŵƵůƟƉůĞŽƉƟŵĂůƐŽůƵƟŽŶƐ͘KƚŚĞƌǁŝƐĞƚŚĞŽƉƟŵĂůƐŽůƵƟŽŶŽďƚĂŝŶĞĚ ƐŚĂůůďĞƵŶŝƋƵĞ͘ 0D[LPLVDWLRQ&DVH F , F ' % J F &* F F LO 6 Solve problems calling for maximisation of objective function F ' " A F % &8 " A % ' + &K F % &" ([DPSOH $FRPSDQ\SODQVWRDVVLJQ¿YHVDOHVPHQWR¿YHGLVWULFWVLQZKLFKLWRSHUDWHV(VWLPDWHVRI VDOHVUHYHQXHLQWKRXVDQGVRIUXSHHVIRUHDFKVDOHVPDQLQGLIIHUHQWGLVWULFWVDUHJLYHQLQWKHIROORZLQJWDEOH ,Q\RXURSLQLRQZKDWVKRXOGEHWKHSODFHPHQWRIWKHVDOHVPHQLIWKHREMHFWLYHLVWRPD[LPLVHWKHH[SHFWHG VDOHVUHYHQXH" ([SHFWHG6DOHV'DWD 6DOHVPDQ 'LVWULFW ' ' ' ' ' 6 6 6 6 6 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ / F ' 6? " " >! TABLE 6.21 ! > 8 6 ; Opportunity Loss Matrix ! > 8 6 ; ? ! < !? !> 8 != !6 8 @ ! !8 @ < ! !8 >< !! ; ! ; 6 ; > . 6WHS / " >> TABLE 6.22 ! > 8 6 ; " Reduced Cost Table 1 ! > 8 6 ; @ < < !? !! > ! !6 8 = < !> @ < < !> !? !! ; ; < 6 6 ; ! 6WHS / ( ) " ! " + " " >8 TABLE 6.23 ! > 8 6 ; Reduced Cost Table 2 ! > 8 6 ; @ < < !? !! < !6 !> ! ; < !> @ < < = !6 < < < 6 6 ; ! / + ' 6E " >6 6WHS " ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ TABLE 6.24 Reduced Cost Table 3 District Salesman D1 D2 D3 D4 D5 S1 12 0 0 7 0 S2 0 10 8 10 0 S3 0 8 4 2 0 S4 23 1 0 0 5 S5 15 5 0 0 1 " F + " !G>>G!8G;6G8;G6L !G>>G;8G!6G8;G6L !G>>G;8G!6G6;G8L !G>>G!8G;6G6;G8 7 F ' C>8!<<< /ŶĐĂƐĞŽĨĂŶĂƐƐŝŐŶŵĞŶƚƉƌŽďůĞŵŝŶǀŽůǀŝŶŐŵĂǆŝŵŝƐĂƟŽŶŽĨŽďũĞĐƟǀĞĨƵŶĐƟŽŶ͕ƚŚĞƐŽůƵƟŽŶŝƐŽďƚĂŝŶĞĚďǇĮƌƐƚ ĐŽŶǀĞƌƟŶŐŝƚŝŶƚŽĂŶĞƋƵŝǀĂůĞŶƚŵŝŶŝŵŝƐĂƟŽŶƉƌŽďůĞŵ͘dŚŝƐŝƐĚŽŶĞďǇƐƵďƚƌĂĐƟŶŐĞĂĐŚĞůĞŵĞŶƚŽĨƚŚĞŵĂƚƌŝǆ ĨƌŽŵĂĐŽŶƐƚĂŶƚ͘,ŽǁĞǀĞƌ͕ǁĞƐŚŽƵůĚďĂůĂŶĐĞŝƚŝĨŝƚŝƐƵŶďĂůĂŶĐĞĚ͕ĂŶĚƌĞƉůĂĐĞƚŚĞŐŝǀĞŶĐĞůůǀĂůƵĞďǇʹMŝŶ ƌĞƐƉĞĐƚŽĨĂƉƌŽŚŝďŝƚĞĚĂƐƐŝŐŶŵĞŶƚ͘ '8$/2)7+($66,*10(17352%/(0 E F BMM 99 ¥ ≥ < B!>:) L F;BMM 99 LO 7 Determine the dual to an assignment problem B Â cij xij B ij " B > B>:) n n i =! j =! Â xij B Â xij B! ' , %& %& ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 1 if worker i is assigned to job j ! ÏÌ Ó0 if worker i is not assigned to job j F; ' + . Y4 : ≥ C Y B< B CY >< ≥ Â %¥ Â &C Â %Y¥ Â & Â B Â B! / ≥ Â C Â Y (#/ F : Â < ij < B Â C Â Y $F /, ([DPSOH ≥CYB!>N Y :ULWHWKHGXDORIWKHSUREOHPJLYHQLQ([DPSOHDQGREWDLQWKHRSWLPDOYDOXHVRIWKHGXDO YDULDEOHV < ! +Y Y Y Y ! + Y£" + Y£# + Y£$ + Y£" + Y£" + Y£% + Y£&' + Y£& + Y£"' + Y£ + Y£ + Y£" + Y£& + Y£&$ + Y£"" + Y£"" ( Y ≥ $()(((* ! ((( + ( Y,( &-& &-" ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ TABLE 6.25 Obtaining Optimal Values of Dual Variables Job Worker B A 1 45 2 57 3 49 4 41 vj 45 e 1 40 e 42 e 1 C D 51 67 63 55 52 48 45 40 1 ui 0 1 2 64 4 60 55 –4 44 53 )$()( )( ).(Y )"(Y)$(Y )Y )" %( / -2/ // dŚĞŽƉƟŵĂůǀĂůƵĞƐŽĨƚŚĞĚƵĂůǀĂƌŝĂďůĞƐĨŽƌĂŶĂƐƐŝŐŶŵĞŶƚƉƌŽďůĞŵĂƌĞŐŝǀĞŶďǇuiĂŶĚvjǀĂůƵĞƐŽĨƚŚĞŽƉƟŵĂů ƐŽůƵƟŽŶƚŽƚŚĞƚƌĂŶƐĨŽƌŵĞĚƚƌĂŶƐƉŽƌƚĂƟŽŶƉƌŽďůĞŵ͘ĐĐŽƌĚŝŶŐůǇ͕ƚŚĞƐƵŵŵĂƟŽŶŽĨƚŚĞƐĞǀĂůƵĞƐŝƐĞƋƵĂůƚŽ ƚŚĞŽďũĞĐƟǀĞĨƵŶĐƟŽŶǀĂůƵĞŽďƚĂŝŶĞĚĨŽƌƚŚĞŽƉƟŵĂůƐŽůƵƟŽŶƚŽƚŚĞŐŝǀĞŶĂƐƐŝŐŶŵĞŶƚƉƌŽďůĞŵ͘ 7+(62/9(5$1'$66,*10(17352%/(0 - ([DPSOH - ( $PDQDJHUKDVWRDVVLJQ¿YHMREVWRKLVWHDPRI¿YHZRUNHUVHDFKRIZKRPLVWREHDVVLJQHG RQHMRE7KHFRVW C RISHUIRUPLQJWKHMREVE\HDFKRIWKHZRUNHUVLVHVWLPDWHGDVVKRZQEHORZ :RUNHU -RE : : : : : 7KHPDQDJHUZDQWVWRDVVLJQRQHMREWRHDFKRIWKHZRUNHUV'HWHUPLQHWKHRSWLPDODVVLJQPHQWSDWWHUQ$OVR REWDLQWKHPLQLPXPWRWDOFRVWLQYROYHG ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 6HWWLQJXSWKH3UREOHP (3 4 5 - ( 4- 67' 4 7 &-- )LJXUH $VVLJQPHQW3UREOHP 8( 7 &- 4 7 &-- / 4 ! 4 -7 ( %- - 9 /, 4 4 ( /(9:, ! 4 -6 /(9, 4 ! ; - 6$<%-2( 4 - = > 7- (= 9 ?@>7A,(= 9 ?@>7A,(- ( >7" 4( ==& B ; - /( '-< (>' 9) ?@>>&A,* C'9) ?@CC&A,( - ==&>'7' / / - )LJXUH 6HWWLQJ8SRI $VVLJQPHQW3UREOHPIRU6ROXWLRQ ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ 6 3 - ! - /3 / 4 .! $ / - <%( 9) ?DEF:?C@>7'(>7&A,2EWDLQLQJWKH6ROXWLRQ 8 ( / -7 ( 4 4 ' - ' 2 -+ / 6HW7DUJHW&HOO7 ( 4 (4 <%( ( 1 -6 (G<G% (TXDO7R ( # %\&KDQJLQJ&HOOV B -+ >7&( 4 -( 4 $ ( ( G>GG7G& -+ 1 6XEMHFWWRWKH&RQVWUDLQWV7 4 ( / -6 ( >7(=( B - / =(=(=" =&-H4( >'@>>&A(C'(---( B - ( %% ( 4 &= ==&-6 ( 9), -+ 4 <<&- G=GG=G& )G<GG<G&-6 4 %% -+ G>G'G7G')G>G%G7%-7 / / $( / >7&9 , - G>GG7G&) /-6 (8 ( ' 2 - 2SWLRQV ( -2 F ( 4 > # % * '-6 (8( 4 D 6ROYH7 /( 4 ' ' & -29 ' = % % % ?%,5 7 &- 7 &-- )LJXUH 2SWLPDO6ROXWLRQWR$VVLJQPHQW3UREOHP ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 7 3 ( 4 7 ("$ (&$ " (&$ ('$ " ("'" '("#" + 8 ' -+/ @ & 4(8B - -7 /(/ 6RPH6SHFLDO6LWXDWLRQV H ( I - 7 < 6 ( B / 4 I! 3 -2/ -2(/ <6 ( /(/ -2( / / - ( ( J 9), J 9K, - 7 ( / I- 2 < 6 ( / 4 J#@.# A /-< ( #-6 ( / -F ( 4+ ! / < 6 (3 B -< ( ( - / / D ( # 1A -2 I( /J @ 4.# < 6 A([DPSOH 7KHUHDUHIRXUPDFKLQHVWREHLQVWDOOHGLQDIDFWRU\IRUZKLFKVL[ORFDWLRQVDUHDYDLODEOH7KH FRVWRILQVWDOOLQJHDFKPDFKLQHLQGLIIHUHQWORFDWLRQVLVJLYHQEHORZ ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ 0DFKLQH /RFDWLRQ $ % & ' ( ) 0 0 ± 0 ± 0 )RUVRPHWHFKQLFDOUHDVRQVLWLVQRWSRVVLEOHWRLQVWDOOPDFKLQH0 DWORFDWLRQ&DQGPDFKLQH0DWORFDWLRQ $2EWDLQWKHRSWLPDOLQVWDOODWLRQRIPDFKLQHVDQGWKHFRVWDVVRFLDWHGZLWKLW 63 ( 4 7 &-"-8 . ( @ AB "($$$- )LJXUH 0DFKLQH/RFDWLRQ3UREOHP ( 4 7 &-&- . 4- (( > = "-( ( >" 9) ?@>>A,- /( < -(< 9 ?@>=A,*<9 ?@>=A,* -7 /( 2&9 ?DEF:?C@>=&(>=A,- )LJXUH 6HWWLQJXSIRU6ROXWLRQWR0DFKLQH/RFDWLRQ3UREOHP ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 8 ( ( 5B >/ ! -+ ' 2 ( G2G& G>GG=G G>G"G=G"L)G>G&G=G& G<GG<G)G2GG2G G>GG=G) / 7 / ( > # % * ' '- 7 &-'- )LJXUH 2SWLPDO6ROXWLRQWR0DFKLQH/RFDWLRQ3UREOHP . #$ > # %#$ # %#$ # 1 %'$ # %'$ ("$ 5(9,(:,//8675$7,216 ([DPSOH $VROLFLWRU¶V¿UPHPSOR\VW\SLVWVRQKRXUO\SLHFHUDWHEDVLVIRUWKHLUGDLO\ZRUN7KHUHDUH¿YH W\SLVWVDQGWKHLUFKDUJHVDQGVSHHGDUHGLIIHUHQW$FFRUGLQJWRDQHDUOLHUXQGHUVWDQGLQJRQO\RQHMRELVJLYHQ WRRQHW\SLVWDQGWKHW\SLVWLVSDLGIRUDIXOOKRXUHYHQZKHQKHZRUNVIRUDIUDFWLRQRIDQKRXU)LQGWKHOHDVW FRVWDOORFDWLRQIRUWKHIROORZLQJGDWD ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ 7\SLVW 5DWHKRXU 1XPEHURISDJHVW\SHGKRXU -RE 1RRISDJHV $ 3 % 4 & 5 ' 6 ( 7 0%$'HOKL$SULO 8VLQJWKHJLYHQLQIRUPDWLRQZH¿UVWREWDLQWKHFRVWPDWUL[ZKHQGLIIHUHQWMREVDUHSHUIRUPHGE\GLIIHUHQW W\SLVWV7KLVLVVKRZQLQ7DEOH7KHHOHPHQWVRIWKHPDWUL[DUHREWDLQHGDVIROORZV7RLOOXVWUDWHLIW\SLVW LVJLYHQMRE2 KHZRXOGUHTXLUH KRXUVDQGKHQFHEHSDLGIRUKRXUV#CSHUKRXU 7KLVUHVXOWVLQDFRVWRICIRUWKLVFRPELQDWLRQ TABLE 6.26 Total Cost Matrix 2 B & 1 6XEWUDFWLQJ WKH PLQLPXP HOHPHQW RI HDFK URZ IURP DOO LWV HOHPHQWV ZH REWDLQ 5&7 DV VKRZQ LQ 7DEOH TABLE 6.27 Reduced Cost Table 1 2 B & 1 1RZVXEWUDFWLQJWKHPLQLPXPHOHPHQWRIHDFKFROXPQIURPDOOWKHHOHPHQWVZHJHW5&7JLYHQLQ7DEOH ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 6.28 Reduced Cost Table 2 2 B & 1 +HUHWKHPLQLPXPQXPEHURIOLQHVWRFRYHUDOO]HURVLVHTXDOWRZKLFKLVVPDOOHUWKDQWKHRUGHU RIWKHJLYHQ PDWUL[$FFRUGLQJO\WKHUHYLVHGWDEOHLVSUHSDUHGE\FRQVLGHULQJWKHOHDVWXQFRYHUHGYDOXHHTXDOWRDQG DGMXVWLQJLWZLWKXQFRYHUHGFHOOYDOXHVDQGWKRVHO\LQJDWWKHLQWHUVHFWLRQRIOLQHV7DEOHFRQWDLQV5&7 TABLE 6.29 Reduced Cost Table 3 2 B & 1 ,Q7DEOHDVZHOOIRXUOLQHVFDQFRYHUDOO]HURV$FFRUGLQJO\5&7LVREWDLQHGGUDZQXSDVWKHUHYLVHG WDEOH7KLVLVJLYHQLQ7DEOH TABLE 6.30 Reduced Cost Table 4 Job Typist P Q R S T A 2 1 2 3 0 B 4 1 0 7 0 C 0 0 2 0 2 D 0 1 0 1 0 E 3 0 3 0 3 ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ ,QWKLVFDVHWKHPLQLPXPQXPEHURIOLQHVWRFRYHUDOO]HURVHTXDOV ZKLFKPDWFKHVZLWKWKHRUGHURIWKH PDWUL[$FFRUGLQJO\DVVLJQPHQWVKDYHEHHQPDGHDVGHVFULEHGEHORZ & B 2 1 7RWDO 7KLVRSWLPDOVROXWLRQKRZHYHULVQRWXQLTXH ([DPSOH :HOOGRQH&RPSDQ\KDVWDNHQWKHWKLUGÀRRURIDPXOWLVWRUH\HGEXLOGLQJIRUUHQWZLWKDYLHZ WRORFDWHRQHRIWKHLU]RQDORI¿FHV7KHUHDUH¿YHPDLQURRPVLQWKLVWREHDVVLJQHGWR¿YHPDQDJHUV(DFK URRPKDVLWVRZQDGYDQWDJHVDQGGLVDGYDQWDJHV 6RPHKDYHZLQGRZVVRPHDUHFORVHUWRWKHZDVKURRPVRUWRWKHFDQWHHQRUVHFUHWDULDOSRRO7KHURRPVDUHRI DOOGLIIHUHQWVL]HVDQGVKDSHV(DFKRIWKH¿YHPDQDJHUVZHUHDVNHGWRUDQNWKHLUURRPSUHIHUHQFHVDPRQJVWWKH URRPVDQG7KHLUSUHIHUHQFHVZHUHUHFRUGHGLQDWDEOHDVLQGLFDWHGEHORZ 0$1$*(5 0 0 0 0 0 0RVWRIWKHPDQDJHUVGLGQRWOLVWDOOWKH¿YHURRPVVLQFHWKH\ZHUHQRWVDWLV¿HGZLWKVRPHRIWKHVHURRPVDQG WKH\KDYHOHIWWKHVHIURPWKHOLVW$VVXPLQJWKDWWKHLUSUHIHUHQFHVFDQEHTXDQWL¿HGE\QXPEHUV¿QGRXWDV WRZKLFKPDQDJHUVKRXOGEHDVVLJQHGWRZKLFKURRPVRWKDWWKHLUWRWDOSUHIHUHQFHUDQNLQJLVDPLQLPXP &$1RYHPEHU ,QWKH¿UVWVWHSZHIRUPXODWHWKHDVVLJQPHQWSUREOHPXVLQJSUHIHUHQFHUDQNV7KLVLVVKRZQLQ7DEOH 1RWLFHWKDWWKHURRPVQRWUDQNHGE\DPDQDJHUDUHUHSUHVHQWHGE\#²DVSURKLELWHGDVVLJQPHQWV ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 6.31 & Assignment Problem # # # # # # # # # # # # 6LQFHJUHDWHUSUHIHUHQFHVDUHVKRZQE\ORZHUQXPEHUVWKHRSWLPDOVROXWLRQFDOOVIRUPLQLPLVLQJWKHWRWDO SUHIHUHQFHUDQNLQJ7RVROYHZHREWDLQ5&7OE\VXEWUDFWLQJWKHOHDVWYDOXHIURPHDFKYDOXHRIWKHURZIRU HDFKRIWKHURZV7KLVLVJLYHQLQ7DEOH TABLE 6.32 & Reduced Cost Table 1 # # # # # # # # # # # # 6LQFHHDFKRIWKHFROXPQVDVZHOODVURZVKDVD]HUROLQHVDUHGUDZQWRFRYHUDOO]HURV)XUWKHUWKHQXPEHU RIOLQHVEHLQJHTXDOWRWKHRUGHURIWKHJLYHQPDWUL[DVVLJQPHQWVFDQEHPDGHDVVKRZQLQWKH7DEOH TABLE 6.33 Assignment of Rooms to Managers Manager Room M1 M2 M3 M4 M5 301 M 3 1 M 0 302 0 0 4 0 1 303 1 M 0 3 M 304 1 0 1 1 1 305 M 1 2 0 M ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ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¿UVWLQWRPLQLPLVDWLRQW\SH7KLV LVVKRZQLQ7DEOHZKHUHLQWKHYDULRXVHOHPHQWVDUHREWDLQHGE\VXEWUDFWLQJHDFKRIWKHHOHPHQWVLQWKH JLYHQWDEOHIURP TABLE 6.34 Opportunity Loss Matrix C CC CCC CD 6XEWUDFWLQJ PLQLPXP YDOXH LQ HDFK URZ IURP HDFK RI WKH URZ HOHPHQWV ZH GHULYH 5&7 VKRZQ LQ 7DEOH ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 6.35 Reduced Cost Table 1 C CC CCC CD 1RZVXEWUDFWLQJPLQLPXPYDOXHLQHDFKFROXPQIURPHDFKRIWKHFROXPQHOHPHQWVRI5&7ZHJHW5&7 DVVKRZQLQ7DEOH TABLE 6.36 Reduced Cost Table 2 C CC CCC CD 6LQFHWKHPLQLPXPQXPEHURIOLQHVWRFRYHUDOO]HURV LVVPDOOHUWKDQWKHPDWUL[RUGHU 5&7LVGHULYHG DVDUHYLVHGWDEOH7KLVLVJLYHQLQ7DEOH TABLE 6.37 Reduced Cost Table 3 I II III IV A 10 55 30 0 B 0 35 0 30 C 10 0 0 10 D 0 0 10 0 :LWKWKHQXPEHURIOLQHVWRFRYHUDOO]HURVEHLQJHTXDOWRWKHRUGHURIWKHJLYHQPDWUL[DVVLJQPHQWVFDQEH PDGHDVVKRZQLQ7DEOH+RZHYHUWKHSUREOHPKDVDQDOWHUQDWLYHRSWLPDOVROXWLRQDVZHOO%RWKRIWKHVH DUHJLYHQEHORZ ' , ,9 , ,,, ,, 7RWDO ' ,9 ,,, ,, , 7RWDO ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ ,IVDOHVPDQ FDQQRWEHDVVLJQHGWRWHUULWRU\,,, DOWHUQDWLYH WKHQDOWHUQDWLYHDERYHPD\EHDGRSWHG ZLWKRXWDGYHUVHHIIHFWRQVDOHLQFUHDVH ([DPSOH $¿UPSURGXFHVIRXUSURGXFWV7KHUHDUHIRXURSHUDWRUVZKRDUHFDSDEOHRISURGXFLQJDQ\ RIWKHVHIRXUSURGXFWV7KH¿UPUHFRUGVKRXUVDGD\DQGDOORZVPLQXWHVIRUOXQFK7KHSURFHVVLQJWLPH LQPLQXWHVDQGWKHSUR¿WIRUHDFKRIWKHSURGXFWVDUHJLYHQEHORZ 2SHUDWRU 3UR¿W C SHUXQLW 3URGXFWV $ % & ' )LQGWKHRSWLPDODVVLJQPHQWRISURGXFWVWRRSHUDWRUV &$1RYHPEHU $QKRXUZRUNLQJGD\ZLWK DPLQXWHOXQFKWLPHDOORZHGLPSOLHVWKDWQHWZRUNLQJWLPHDYDLODEOHSHUGD\LV KRXUVDQGPLQXWHVWKDWLVPLQXWHV7KHQXPEHURIXQLWVRIGLIIHUHQWSURGXFWVZKLFKFRXOGEHSURGXFHG E\WKHIRXURSHUDWRUVFDQEHFDOFXODWHGE\GLYLGLQJE\WKHJLYHQSURFHVVLQJWLPHV:LWKWKHSUR¿WSHUXQLW RIHDFKSURGXFWEHLQJJLYHQZHPD\FDOFXODWHWKHSUR¿WUHVXOWLQJIURPHDFKSRVVLEOHDVVLJQPHQW7KHSUR¿W PDWUL[LVJLYHQLQ7DEOH7KHYDOXHVLQWKLVPDWUL[DUHGHULYHGDVIROORZV)RUH[DPSOHRSHUDWRUFDQ SURGXFH XQLWVRISURGXFW ZKLFKDWDSUR¿WUDWHRICSHUXQLWLPSOLHVDWRWDOSUR¿WRIC TABLE 6.38 ( Profit Matrix 2 % 7RVROYHWKHSUREOHPLWLV¿UVWFRQYHUWHGLQWRDPLQLPLVDWLRQSUREOHPE\REWDLQLQJRSSRUWXQLW\ORVVPDWUL[ E\VXEWUDFWLQJHDFKYDOXHIURP±WKHKLJKHVWSUR¿WYDOXHLQWKHWDEOH,WLVJLYHQLQ7DEOH TABLE 6.39 ( Opportunity Loss Matrix 2 % ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 8VLQJWKH+XQJDULDQPHWKRGZH¿UVWREWDLQ]HURVLQHYHU\URZDQGFROXPQDVJLYHQLQ7DEOHVDQG TABLE 6.40 ( Reduced Cost Table 1 2 % TABLE 6.41 ( Reduced Cost Table 2 2 % ,WLVREVHUYHGWKDWDOO]HURVDUHFRYHUHGE\WKUHHOLQHVZKLFKLVRQHOHVVWKDQWKHRUGHURIWKHPDWUL[7KH LPSURYHGPDWUL[LVREWDLQHGDVVKRZQLQ7DEOHZLWKDGMXVWPHQWZLWKWKHOHDVWXQFRYHUHGYDOXHRI TABLE 6.42 Reduced Cost Table 3 Product Operator A B C D 1 0 0 21 0 2 30 0 146 150 3 0 24 0 4 4 0 0 21 120 6LQFH WKH PLQLPXP QXPEHU RI OLQHV WR FRYHU DOO ]HURV LV ZKLFK PDWFKHV ZLWK WKH RUGHU RI WKH PDWUL[ DVVLJQPHQWVFDQEHPDGHDVVKRZQLQ7DEOH7KXVWKHRSWLPDODVVLJQPHQWLV ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ ( 2 % 2 ? 7RWDO ([DPSOH $QDLUOLQHRSHUDWLQJVHYHQGD\VDZHHNKDVJLYHQWKHIROORZLQJVFKHGXOHRILWVÀLJKWVEHWZHHQ 1HZ'HOKLDQG0XPEDL7KHFUHZVVKRXOGKDYHDPLQLPXPRI¿YHKRXUVEHWZHHQWKHÀLJKWV2EWDLQWKHSDLULQJ RIÀLJKWVWKDWPLQLPLVHVOD\RYHUWLPHDZD\IURPKRPH)RUDQ\JLYHQSDLULQJWKHFUHZZLOOEHEDVHGDWWKHFLW\ WKDWUHVXOWVLQWKHVPDOOHVWOD\RYHU 1HZ'HOKL²0XPEDL 0XPEDL²1HZ'HOKL )OLJKW1R 'HSDUWXUH $UULYDO )OLJKW1R 'HSDUWXUH $UULYDO DP DP DP DP DP DP DP DP SP SP SP SP SP SP SP SP 0&RP'HOKL 7REHJLQZLWKZH¿UVWDVVXPHWKDWDOOWKHFUHZLVEDVHGDW1HZ'HOKL8VLQJWKLVDVVXPSWLRQZHFDQREWDLQ WKHOD\RYHUWLPHVRIYDULRXVFRPELQDWLRQVRIÀLJKWVDVVKRZQLQ7DEOH7RLOOXVWUDWHWKHÀLJKWZKLFK VWDUWVIURP1HZ'HOKLDWDPUHDFKHV0XPEDLDWDP,ILWLVWRUHWXUQDVWKHVFKHGXOHGWLPHIRU ZKLFKLVDPWKHQLWFDQGRVRRQO\DIWHUKRXUVVLQFHDPLQLPXPOD\RYHUWLPHRI¿YHKRXUVLVUHTXLUHG 6LPLODUO\OD\RYHUWLPHVIRURWKHUÀLJKWFRPELQDWLRQVFDQEHREWDLQHGDVVKRZQLQWKHWDEOH TABLE 6.43 Layover Time—Crew at New Delhi E$ -6, -6- -6. -6/ 7KHOD\RYHUWLPHVIRUYDULRXVÀLJKWFRPELQDWLRQVZKHQFUHZLVDVVXPHGWREHEDVHGDW0XPEDLDUHVLPLODUO\ FDOFXODWHGDQGVKRZQLQ7DEOH ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 6.44 Layover Time—Crew at Mumbai E$ -6, -6- -6. -6/ 1RZVLQFHWKHFUHZFDQEHEDVHGDWHLWKHURIWKHVWDWLRQVPLQLPXPOD\RYHUWLPHVFDQEHREWDLQHGIRUGLIIHUHQW ÀLJKWFRPELQDWLRQVE\VHOHFWLQJWKHFRUUHVSRQGLQJORZHUYDOXHRXWRIWKHDERYHWZRWDEOHV)RULQVWDQFHLQ FRPELQLQJ±ZHVHOHFWZKLFKLVORZHURIWKHWZRYDOXHVDQG 7DEOHVDQG 7KH YDOXHVDUHVKRZQLQ7DEOH TABLE 6.45 Layover Time—Crew at New Delhi/Mumbai E$ -6, -6- -6. -6/ ZKHQFUHZLVEDVHGDW0XPEDL 7RFRQWLQXHZHFDQREWDLQWKHRSWLPDOSDLULQJRIÀLJKWVVRDVWRPLQLPLVHWKHWRWDOOD\RYHUWLPHXVLQJWKH +XQJDULDQ$VVLJQPHQW0HWKRG7DEOHJLYHVWKH5&7O TABLE 6.46 Reduced Cost Table 1 Flight 201 202 203 204 101 14 13 0 3 102 13 12 7 0 103 0 1 6 1 104 1 0 7 12 :LWKD]HURLQHDFKFROXPQDVZHOOWKHUHLVQRQHHGWRSHUIRUPFROXPQUHGXFWLRQV)XUWKHUVLQFHWKHQXPEHU RIOLQHVFRYHULQJDOO]HURVLVHTXDOWRWKHRUGHURIWKHJLYHQPDWUL[ZHFDQREWDLQWKHRSWLPDODVVLJQPHQWDV VKRZQLQWKHWDEOH 7KHRSWLPDOSDLULQJRIÀLJKWVLVJLYHQRQQH[WSDJH ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ E$ E$ > ' @ % 1HZ'HOKL 1HZ'HOKL 1HZ'HOKL 0XPEDL 7RWDO ([DPSOH $FRPSDQ\KDVMXVWGHYHORSHGDQHZLWHPIRUZKLFKLWSURSRVHVWRXQGHUWDNHDQDWLRQDO WHOHYLVLRQSURPRWLRQDOFDPSDLJQ,WKDVGHFLGHGWRVFKHGXOHDVHULHVRIRQHPLQXWHFRPPHUFLDOVGXULQJSHDN DXGLHQFHYLHZLQJKRXUVRI±SP7RUHDFKWKHZLGHVWSRVVLEOHDXGLHQFHWKHFRPSDQ\ZDQWVWRVFKHGXOH RQHFRPPHUFLDORQHDFKRIWKHQHWZRUNVDQGWRKDYHRQO\RQHFRPPHUFLDODSSHDUGXULQJHDFKRIWKHIRXU RQHKRXUWLPHEORFNV7KHH[SRVXUHUDWLQJVIRUHDFKKRXUZKLFKUHSUHVHQWWKHQXPEHURIYLHZHUVSHUC VSHQWDUHJLYHQEHORZ 1HWZRUN 9LHZLQJ KRXUV $ % & ' ±SP ±SP ±SP ±SP D :KLFKQHWZRUNVKRXOGEHVFKHGXOHGHDFKKRXUWRSURYLGHWKHPD[LPXPDXGLHQFHH[SRVXUH" E +RZZRXOGWKHVFKHGXOHFKDQJHLILWLVGHFLGHGQRWWRXVHQHWZRUN$EHWZHHQDQGSP" 0&RP'HOKL D 7RVROYHWKLVSUREOHPZH¿UVWPXOWLSO\HDFKYDOXHLQWKHPDWUL[E\WRH[SUHVVH[SRVXUHUDWLQJVSHU C ODNK ,W VLPSOL¿HV WKH FDOFXODWLRQ ZRUN VRPHZKDW )XUWKHU EHLQJ D PD[LPLVDWLRQ SUREOHP ZH VXEWUDFWHDFKYDOXHIURPWKHODUJHVWYDOXHWRJHWWKHRSSRUWXQLW\ORVVPDWUL[7KHUHVXOWRIWKHVHVWHSV LVJLYHQLQ7DEOH TABLE 6.47 Opportunity Loss Matrix Network Viewing hours A B C D 1–2 p.m. 0 90 158 176 2–3 p.m. 82 116 100 165 3–4 p.m. 79 86 172 194 4–5 p.m. 156 57 103 143 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 1RZZHDSSO\+$0WRVROYHWKHSUREOHP5RZUHGXFWLRQVDQGFROXPQUHGXFWLRQVDUHVKRZQLQ7DEOHV DQGUHVSHFWLYHO\ TABLE 6.48 Reduced Cost Table 1 Network Viewing hours A B C D 1–2 p.m. 0 90 158 176 2–3 p.m. 0 34 18 83 3–4 p.m. 0 7 93 115 4–5 p.m. 99 0 46 86 TABLE 6.49 Reduced Cost Table 2 Network Viewing hours A B C D 1–2 p.m. 0 90 140 93 2–3 p.m. 0 34 0 0 3–4 p.m. 0 7 75 32 4–5 p.m. 99 0 28 3 6LQFH WKH QXPEHU RI OLQHV FRYHULQJ DOO ]HURV LV ZH LPSURYH WKH VROXWLRQ DV VKRZQ LQ 7DEOHV DQG TABLE 6.50 Viewing hours Reduced Cost Table 3 Network A B C D 1–2 p.m. 0 90 137 90 2–3 p.m. 3 37 0 0 3–4 p.m. 0 7 72 29 4–5 p.m. 99 0 25 0 ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ TABLE 6.51 Reduced Cost Table 4 Network Viewing hours A B C D 1–2 p.m. 0 83 130 83 2–3 p.m. 10 37 0 0 3–4 p.m. 0 0 65 22 4–5 p.m. 106 0 25 0 $VVLJPHQWVPDGHDUHVKRZQLQ7DEOH7KHRSWLPDOVROXWLRQLV D@ $ @ ±SP ±SP ±SP ±SP )RUWKHJLYHQUHVWULFWLRQWKHSURKLELWHGWLPHVORWVDUHUHSODFHGE\#LQ7DEOHDQGWKHSUREOHPLVVROYHG 7DEOHVKRZVWKHHIIHFWRIURZDQGFROXPQUHGXFWLRQVERWK6LQFHWKHQXPEHURIOLQHVFRYHULQJDOO]HURV LVIXUWKHUFDOFXODWLRQVOHDGWRYDOXHVJLYHQLQ7DEOH TABLE 6.52 Reduced Cost Table 5 Network Viewing hours A B C D 1–2 p.m. M 0 68 21 2–3 p.m. M 16 0 0 3–4 p.m. 0 7 93 50 4–5 p.m. 99 0 46 21 TABLE 6.53 Viewing hours Reduced Cost Table 6 Network A B C D 1–2 p.m. M 0 47 0 2–3 p.m. M 37 0 0 3–4 p.m. 0 28 93 50 4–5 p.m. 78 0 25 0 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 6LQFHWKHQXPEHURIOLQHVFRYHULQJ]HURVLVHTXDOWRWKHDVVLJQPHQWVFDQEHPDGH7KHUHDUHWZRRSWLPDO VROXWLRQVDVJLYHQKHUH D@ $ @ @ ±SP ±SP ±SP ±SP ([DPSOH $FRPSDQ\KDVLQYLWHGELGVRQIRXUFRQVWUXFWLRQMREV7KUHHFRQWUDFWRUVKDYHSODFHGELGV RQWKHVHMREV7KHLUELGV LQODNKVRI5XSHHV DUHJLYHQEHORZDQGµ ¶LQGLFDWHVWKDWWKHSDUWLFXODUFRQWUDFWRU GLGQRWELGIRUWKHJLYHQMRE -RE &RQWUDFWRU $ % & D $VVXPLQJWKDWHDFKFRQWUDFWRUFDQEHDVVLJQHGRQO\RQHMREGHWHUPLQHWKHPLQLPXPFRVWDVVLJQPHQW RIFRQWUDFWRUVWRMREV E &DQWKHVDPHPHWKRG\RXXVHGWRVROYH D EHDSSOLHGLIWKHVWDWHGDVVXPSWLRQLVUHOD[HG" F )RUPXODWHDQGVROYHWKHDVVLJQPHQWSUREOHPZKHQFRQWUDFWRUV%DQG&FDQGRDVPDQ\DVWZRMREV HDFKZKLOHFRQWUDFWRU$FDQRQO\GRRQHMREDVVXPLQJWKDWDFRQWUDFWRUPD\EHDVVLJQHGWZRMREVDV ZHOO D 6LQFHWKLVLVDQXQEDODQFHGSUREOHPZH¿UVWEDODQFHLWE\DGGLQJDGXPP\FRQWUDFWRU)XUWKHU SURKLELWHGDVVLJQPHQWVDUHPDUNHGDV#DVVKRZQLQ7DEOH TABLE 6.54 Total Cost Matrix , - . / # # 5RZUHGXFWLRQVDUHVKRZQLQ5&7DVVKRZQLQ7DEOH&ROXPQUHGXFWLRQVDUHQRWUHTXLUHGVLQFH HYHU\FROXPQKDVDYDOXHRI]HUR ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ TABLE 6.55 Reduced Cost Table 1 , - . / # # 7KHPLQLPXPQXPEHURIOLQHVGUDZQWRFRYHUDOO]HURVLVZKLFKLVVPDOOHUWKDQWKHRUGHURIWKHPDWUL[ 7KXVZHFDQQRWREWDLQWKHRSWLPDOVROXWLRQWRWKHSUREOHPKHUH7RLPSURYHZHVHOHFWWKHVPDOOHVWQRQ QHJDWLYHYDOXHHTXDOWRDQGREWDLQWKHUHYLVHGWDEOHDV5&7VKRZQLQ7DEOH TABLE 6.56 Reduced Cost Table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otal Cost Matrix , - . / 0 # # # # ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ :LWKD]HURLQHYHU\URZWKHURZUHGXFWLRQVDUHQRWUHTXLUHG7KHFROXPQUHGXFWLRQVDUHVKRZQLQ5&7DV VKRZQLQ7DEOH TABLE 6.58 Reduced Cost Table 1 , - . / 0 # # # # 2QO\WZROLQHVDUHFRYHULQJDOO]HURV+HQFHZHLPSURYHWKHVROXWLRQZLWKDVWKHOHDVWXQFRYHUHGYDOXH 7KLVLVVKRZQLQ5&7LQ7DEOH TABLE 6.59 Reduced Cost Table 2 , - . / 0 # # # # 7KHVROXWLRQQHHGVWREHIXUWKHULPSURYHGVLQFHWKHQXPEHURIOLQHVFRYHULQJDOO]HURVLVVWLOOVKRUWRIWKH UHTXLUHGQXPEHURI6HOHFWLQJDQGXVLQJWKHOHDVWXQFRYHUHGYDOXHRIZHUHYLVHWKHWDEOHDVVKRZQLQ 5&7RI7DEOH TABLE 6.60 Reduced Cost Table 3 , - . / 0 0 0 0 0 0 # 0 0 0 # 0 # 0 0 # 0 0 ^ƉĞĐŝĂůůǇ^ƚƌƵĐƚƵƌĞĚ>ŝŶĞĂƌWƌŽŐƌĂŵŵĞƐ//͗ƐƐŝŐŶŵĞŶƚWƌŽďůĞŵഩഩ +HUH¿YHOLQHVDUHVHHQFRYHULQJDOOWKH]HURV+HQFHZHFDQREWDLQWKHRSWLPDOVROXWLRQ,WLVHYLGHQWIURP WKHWDEOHWKDWWKHDVVLJQPHQWVDUH&RQWUDFWRU-RE&RQWUDFWRU-REVDQGDQG&RQWUDFWRU-RE7KLV ZRXOGLQYROYHDWRWDOFRVWRICODNK7KLVVROXWLRQRIFRXUVHLVQRWXQLTXHRSWLPDO 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closer to reality. Formulation and solution of such problems as linear programming problems are based on two simplifying assumptions that there is only a single objective to achieve (such as maximisation of profit) and that the variables involved are continuous so that, for example, solution to a profit maximising problem of a multi-product firm may call for producing in numbers involving fractional values as well, like 18.7, 32.6, so on. It obviously sounds impractical to produce fractional unit, give fractional number of advertisements in a media-mix problem, engage employees in fractional numbers and so forth; and the firm may have multiple objectives to achieve. The present chapter addresses both these issues. To ensure that the solution involves only integer values, we use integer programming, and the other extension of linear programming covered in this chapter is goal programming, which postulates setting up of multiple goals (may be conflicting in nature) instead of a single objective that calls for maximisation or minimisation. The contents of this chapter help a manager to obtain answers to the questions like the following: What is the optimal mix of the products in terms of the exact number of completed units to be produced for each one? What combination of the investment proposals is best to undertake? How to obtain optimal assignment of workers to jobs by using a different approach and how to determine the optimal route for a travelling salesman who wants to cover a number of cities in a tour? How best can the multiple objectives be achieved by assigning penalties for not satisfying each of the objectives and weaving these penalties in the objective function? This chapter requires higher skills in formulating the problem—both integer and goal programming—in the first instance. It will be evident from the variety of IPPs given and discussed. Further, in the context of goal programming, difference between goals and constraints should be understood clearly for the correct formulation of the problems. For solution to both, the integer as well as goal programming problems, while a knowledge of the Simplex method is essential, much more needs to be understood. This is in keeping with the requirements for each of the two types of problems. ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ Learning Objectives After reading this chapter, you should be able to: LO 1 LO 2 LO 3 LO 4 LO 5 LO 6 LO 7 LO 8 /ĚĞŶƟĨǇǀĂƌŝŽƵƐ/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐWƌŽďůĞŵƐ <ŶŽǁŚŽǁƚŽƐŽůǀĞ/WWƵƐŝŶŐĐƵƫŶŐƉůĂŶĞĂůŐŽƌŝƚŚŵĂŶĚďƌĂŶĐŚĂŶĚďŽƵŶĚ ŵĞƚŚŽĚ ǆƉůĂŝŶƐŽůƵƟŽŶƚŽƚƌĂǀĞůůŝŶŐƐĂůĞƐŵĂŶƉƌŽďůĞŵƐ /ůůƵƐƚƌĂƚĞƚŚĞĐŽŶĐĞƉƚŽĨŐŽĂůƉƌŽŐƌĂŵŵŝŶŐǁŝƚŚƐŝŶŐůĞŐŽĂů hŶĚĞƌƐƚĂŶĚŐŽĂůƉƌŽŐƌĂŵŵŝŶŐǁŝƚŚŵƵůƟƉůĞŐŽĂůƐ ^ŽůǀĞŶŽŶͲƉƌĞͲĞŵƉƟǀĞ'WWƐƵƐŝŶŐĞƋƵĂůĂŶĚĚŝīĞƌĞŶƟĂůǁĞŝŐŚƚƐ ^ŽůǀĞƉƌĞͲĞŵƉƟǀĞ'WWƐŐƌĂƉŚŝĐĂůůǇĂŶĚƵƐŝŶŐŵŽĚŝĮĞĚƐŝŵƉůĞǆŵĞƚŚŽĚ ƉƉůǇŵŽĚŝĮĞĚƐŝŵƉůĞǆŵĞƚŚŽĚĨŽƌŐŽĂůƉƌŽŐƌĂŵŵŝŶŐ ,1752'8&7,21 ! " # #$ ,17(*(5352*5$00,1* % %&& LO 1 Identify various % Integer Programming Problems " '()* +), - " . % " / ". 0 1 ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ - # 0 234(5 3XUHDQG0L[HG,QWHJHU3URJUDPPLQJ3UREOHPV 0 . 6 7+3(89+ / )(8:+ £;< <(8)+ £=+ (+ ≥3(+ > %&& . 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(J+ #(#+ #(J+#(8+J<#+≥(J+ # & 5 4(J+#(4+J<#+£4(J+ 4(J+#(4+J<#+8#<74(J+/ " % 5" " # Z 5 $ @ $ 4<J: =J: $0 . . dŽŽďƚĂŝŶ'ŽŵŽƌǇ͛ƐĐƵƚ͕ƚŚĞĮƌƐƚƐƚĞƉŝƐƚŽĞǆƉƌĞƐƐĞĂĐŚŽĨƚŚĞĐŽĞĸĐŝĞŶƚƐŽĨƚŚĞĐŽŶƐƚƌĂŝŶƚĂƐƚŚĞƐƵŵŽĨĂŶ ŝŶƚĞŐĞƌĂŶĚĂŶŽŶͲŶĞŐĂƟǀĞĨƌĂĐƟŽŶ͘dŚŝƐŝŶǀŽůǀĞƐĐŚĂŶŐŝŶŐƚŚĞĐŽĞĸĐŝĞŶƚƐŽĨƚŚĞůŝŶĞĂƌĐŽŶƐƚƌĂŝŶƚƐ;ŝŶĐůƵĚŝŶŐ ƚŚĞZ,^ĂƐǁĞůůͿĂƐĨŽůůŽǁƐ͗ Original constraints Gomory’s cut ;ĂͿ/ŶƚĞŐĞƌ;ƉŽƐŝƟǀĞŽƌŶĞŐĂƟǀĞͿ ;ďͿWŽƐŝƟǀĞĨƌĂĐƟŽŶ;&ŽƌĞǆĂŵƉůĞ͕ϮͬϯͿ ;ĐͿEĞŐĂƟǀĞĨƌĂĐƟŽŶ;&ŽƌĞǆĂŵƉůĞ͕оϮͬϯͿ ĞƌŽ hŶĐŚĂŶŐĞĚ;ϮͬϯͿ WŽƐŝƟǀĞĐŽŵƉůĞŵĞŶƚ;ϭͬϯͿ /ŶƚŚĞƐĞĐŽŶĚƐƚĞƉ͕ĐŚĂŶŐĞƚŚĞ͚с͛ƐŝŐŶƚŽ͚ш͛͘ 0 & # :< #( 7 3 #+73# .(J+#(8+J<#+≥(J+# %&& % & %&&! & % ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ %&&% & Z %&&$ x2 3x1 + 4x2 = 18 is cutting plane IPP feasible point 6 3x1 + 3x2 = 15 5 'ŽŵŽƌǇ͛ƐĐƵƚŚĂƐƚǁŝŶƉƌŽƉĞƌƟĞƐ͗;ŝͿŝƚŝƐŶŽƚƐĂƟƐĮĞĚ ďǇ ƚŚĞ ŽƉƟŵĂů ƐŽůƵƟŽŶ ƚŽ ƚŚĞ >W ƌĞůĂǆĂƟŽŶ͕ ĂŶĚ ;ŝŝͿ ǁŚŝůĞŝƚĐƵƚƐƚŚĞŽƉƟŵĂůƐŽůƵƟŽŶƚŽƚŚĞ>WƌĞůĂǆĂƟŽŶ͕ ŝƚ ĚŽĞƐ ŶŽƚ ĐƵƚ ĂŶǇ ĨĞĂƐŝďůĞ ƐŽůƵƟŽŶƐ ƚŽ ƚŚĞ ŝŶƚĞŐĞƌ ƉƌŽŐƌĂŵŵŝŶŐƉƌŽďůĞŵ͘ (J+#(8+J<#+≥(J+ :( 4 Optimal solution of LP relaxation x1 = 3/2, x2 = 7/2 3 3x1 + 4x2 = 18 2 2x1 + 4x2 = 17 1 x1 0 1 2 3 4 5 6 7 8 9 10 % %&& )LJXUH *UDSKLF 5HSUHVHQWDWLRQ RI & ,33³&XWWLQJ3ODQH (7<J++7:J+ # #$ %&&+(8=+8#(7(: <(8<+8#+7()##(7(:4+(4=+#+7()4<(4<+/ (J+#(8+J<#+≥(J+ <(8=+£(9 # # := TABLE 7.4 "" # :< Revised Simplex Tableau 1 ( + #( #+ #< $ + <33 3 ( (J+ 4(J< 3 :J+ 4+(J+ ( +33 ( 3 4(J+ +J< 3 <J+ AJ= #< 3 3 3 4 (J+ 4+J<[ ( 4(J+ <J= +33 <33 3 3 3 <J+ :J+ 3 3 4(J+ 3 3 4)3 4(33J< 3 / D ≠ % . @ # D #< . ! K4 )3∏ 4 (J+ 7 (33L K4(33J<∏4+J<7)3L0 #+ " ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ " " % % ! 1 . 4 +(J+AJ=<J=" " #<##< # :) " 0 # :) D 0 $ 3(8(+8<J=#(83#+4(J+#<7()J= % 3(8(+8<J=#(83#+4#<8(J+#<7<8<J= # <J=#(8(J+#<7<J=8<4+8#< 0 <J=#(8(J+#< ≥<J= 4<J=#(4(J+#< £4 <J= 4 <J=#(4(J+#<8#= 74<J= #=" TABLE 7.5 "" Revised Simplex Tableau 2 ( + # ( #+ # < + <33 3 ( <J= 3 4 (J+ ()J= ( +33 ( 3 4 ( 3 ( ( #+ 3 3 3 <J= ( 4 <J+ <J= +33 <33 3 3 3 / ( ()J= 3 <J= 3 D 3 3 4+) 3 4)3 / # :; ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ TABLE 7.6 "" Revised Simplex Tableau 3 ( + # ( # + # < # = $ + <33 3 ( <J= 3 4 (J+ 3 ()J= ) ( +33 ( 3 4 ( 3 ( 3 ( 4( #+ 3 3 3 <J= ( 4 <J+ 3 <J= ( #= 3 3 3 4 <J= 3 4(J+ ( 4<J= ( ¨ +33 <33 3 3 3 3 / ( ()J= 3 <J= 3 4 <J= 3 3 4 +) 3 4 )3 3 ≠ D $ ' #('4+)∏4 <J=7 (33J< #<'4)3∏4(J+7 (33 0 #( #= " # # :: TABLE 7.7 Revised Simplex Tableau 4: Optimal Solution ( + # ( # + # < # = + <33 3 ( 3 3 4 ( ( < ( +33 ( 3 3 3 )J< 4 =J< + #+ 3 3 3 3 ( 4 ( 4 + 3 #( 3 3 3 ( 3 +J< 4 =J< ( +33 <33 3 3 3 3 / + < ( 3 3 3 D 3 3 3 3 4 (33J< 4(33J< "" 0 %&&% (7++7<7(<33 % % & %&& J$ %&&/(+ %&& # ' # . $ G ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ " 0 "<+(8+£(); <+(8(3+£(); 0 " X-/ X-/ (J+ %UDQFKDQG%RXQG0HWKRG # % % " # "G $ \ G B # " - 34( dŚĞďƌĂŶĐŚĂŶĚďŽƵŶĚŵĞƚŚŽĚŝƐƵƐĞĨƵůůǇĞŵƉůŽǇĞĚŝŶƉƌŽďůĞŵƐǁŚĞƌĞƚŚĞƌĞĂƌĞĂĮŶŝƚĞŶƵŵďĞƌŽĨƐŽůƵƟŽŶƐ͘Ǉ ĂƉƉůǇŝŶŐƐŽŵĞƌƵůĞƐ͕ƚŚĞƐĞƐŽůƵƟŽŶƐĂƌĞĚŝǀŝĚĞĚŝŶƚŽƚǁŽƉĂƌƚƐരͶരŽŶĞƚŚĂƚŵŽƐƚƉƌŽďĂďůǇĐŽŶƚĂŝŶƐƚŚĞŽƉƟŵĂů ƐŽůƵƟŽŶĂŶĚ͕ƚŚĞƌĞĨŽƌĞ͕ƐŚŽƵůĚďĞĞǆĂŵŝŶĞĚĨƵƌƚŚĞƌ͖ĂŶĚƚŚĞƐĞĐŽŶĚƉĂƌƚƚŚĂƚǁŽƵůĚŶŽƚĐŽŶƚĂŝŶƚŚĞŽƉƟŵĂů ƐŽůƵƟŽŶĂŶĚ͕ƚŚƵƐ͕ďĞůĞŌŽƵƚŽĨĨƵƌƚŚĞƌĐŽŶƐŝĚĞƌĂƟŽŶ͘dŚĞƉƌŽĐĞƐƐŝƐĐŽŶƟŶƵĞĚƵŶƟůŽƉƟŵĂůƐŽůƵƟŽŶŝƐŽďƚĂŝŶĞĚ͘ 6ROXWLRQWR$VVLJQPHQW3UREOHPV ! $ 0 - $ C " . <;+9933 ]# $ # ([DPSOH 7KHFRVW LQC WRSHUIRUPGLIIHUHQWMREVE\GLIIHUHQWZRUNHUVLVJLYHQDVIROORZV ! " / 2EWDLQWKHRSWLPDODVVLJQPHQWRIMREVWRZRUNHUVXVLQJEUDQFKDQGERXQGPHWKRG ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ - "> # 6WHS, # =]7+= $ # "# ^ # ." % +38(98 (+8+)7:) ( + < = % . % $ 6WHS,, % " ! ( " # $ ( CA3D $ ($ " . +9(++) / ( $ # # :9 TABLE 7.8 Assignment of Job 1 to Each Worker "" & ' ! ( " (4+4<4=4 A38+98(+8+)7()) (4+4<4=4 :+8(98(+8+)7(+: (4+4<4=4 )<8(98=98+)7(== % (4+4<4=4 +38(98(+8)37(33 % % $ # :+ #:) " % %% (33 " /()) ( " (+: ( " ( " ())\ (== ( " ( # Job 1 to A Job 1 to B 155 Feasible 127 Feasible 75 Job 1 to C Job 1 to D 144 Infeasible 100 Infeasible )LJXUH 7UHH'LDJUDP$VVLJQPHQWRI -REWR(DFK:RUNHU ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 6WHS,,, % %% ( "0 (33 J + $ ( # # :A TABLE 7.9 Assignment of Job 1 to Worker D and Job 2 to Each Worker "" & ' ! ( " (4+4<4=4 +38(98(+8)37(33 % (4+4<4=4 +38+98(+8)37((3 (4+4<4=4 +38A+8(+8)37(:= % # @ (G " +G " <G " =4 " ((3# :<- (33 Job 1 to A Job 1 to B 75 Job 1 to C 155 Feasible 127 Feasible 144 Infeasible Job 2 to A Job 1 to D Infeasible 100 Job 2 to B Job 2 to C 100 Infeasible 110 Feasible 174 Infeasible )LJXUH 7UHH'LDJUDP$VVLJQPHQWRI -REWR'-REWR(DFK:RUNHU # ( " + " (:= ( + ((3 6WHS,9 # ( + (33 $ ! (+ " ' < = < = C +38(989)8:97+3(+38(98(+8 937(<3 # := ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ Job 1 to A Job 1 to B 75 Job 1 to C Job 1 to D 155 Feasible 127 Feasible Job 2 to A 144 Job 2 to B 100 Job 2 to C 100 Job 3 to B Job 4 to C Job 3 to C Job 4 to B 110 Feasible 201 Feasible 130 Feasible 174 Infeasible )LJXUH %UDQFKDQG%RXQG7UHH2SWLPDO$VVLJQPHQW % ((3 " %%%# %%% ((3# ) * " = )3 + +9 < (+ ( +3 # ((3 # $ - 0 6ROXWLRQWR7UDYHOOLQJ6DOHVPDQ3UREOHPV # ' LO 3 Explain solution 0 - to travelling salesman problems % $ . / " ! # @- ! # ([DPSOH 0U,\HULVDVDOHVPDQZLWK'HOLWH0DQXIDFWXULQJ&RPSDQ\+HZDQWVWRYLVLWVL[FLWLHVVD\ DQGVWDUWLQJZLWKFLW\ZKHUHKHLVVWDWLRQHG7KHGLVWDQFHVEHWZHHQYDULRXVFLWLHVDUHJLYHQ LQ7DEOH 0U,\HUZDQWVWRGHYHORSDWRXUWKURXJKWKH¿YHRWKHUFLWLHVDQGUHWXUQWRKLVKRPHFLW\LQVXFKDZD\WKDW KHKDVWRWUDYHOWKHPLQLPXPGLVWDQFH ,WPD\EHQRWHGWKDWWKHPDWUL[RIWKHGLVWDQFHVEHWZHHQGLIIHUHQWFLWLHVLVQRWDV\PPHWULFDORQH)RULQVWDQFH WKHGLVWDQFHIURPFLW\WRLVQRWJLYHQWREHWKHVDPHDVWKHGLVWDQFHIURPFLW\WR6XFKVLWXDWLRQVPD\ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ GHYHORSEHFDXVHRIVHYHUDOUHDVRQV,QWKHRQHZD\WUDI¿FV\VWHPIRUH[DPSOHWKHGLVWDQFHWREHFRYHUHGWR UHDFKDSRLQWPLJKWEHGLIIHUHQWIURPWKHGLVWDQFHWREHFRYHUHGZKLOHUHWXUQLQJ TABLE 7.10 Inter-city Distances (in km) ( + , - . / 0 1 ( 4 +) (9 <) )3 <A + +( 4 +9 (; <3 (< < ++ +9 4 (= (; +3 = <) (+ (= 4 (+ (+ ) )3 <3 (; (+ 4 9 ; <A () +3 (+ : 4 # $ ! $ \ (+3 " # 6 % " (4+4<4=4)4;4(# +)8+98(=8(+898<A7(+;" D . (+;" # . # (+;" ! ' # #" " 2 5 2 ( ;# % # " # # :((:(+ TABLE 7.11 Reduced Cost Table 1 , - . / 0 1 ( 2 : 3 (: <+ +( + 9 2 () < (: 3 < 9 (= 2 3 + ; = +< 3 + 2 3 3 ) =+ ++ 9 = 2 3 ; <+ 9 (< ) 3 2 ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ TABLE 7.12 Reduced Cost Table 2 City City 1 2 3 4 5 6 1 M 7 0 17 32 21 2 0 M 15 3 17 0 3 0 14 M 0 2 6 4 15 0 2 M 0 0 5 34 22 8 4 M 0 6 24 8 13 5 0 M 0 ' (4<+4(<4==4+)4;;4) / ' (4<4=4+4()4;4) ' ;)" ()" %# :(+ @ .; - (4<+4(<4==4+)4;;4) 93" # # 93" \ % " "(4<4=4+4()4;4) D (+; " 93 " ! 0 / ! " % " (4<4=4+4()4;4) ;) ()" /)4;4) " - G " )4; " ;4) # " # :) Upper Bound = 126 km ble pta e c ac Sub-tours: Un 5– 6 1– 3– 4– 2– 1 (65 km) LB 5– 6– 5 (15 km) = 80 6– 5 Break 5– 6– 5 Un acc ept abl e )LJXUH ,QLWLDO%UDQFKDQG%RXQG7UHH ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 7KHZRUNLQJRIWKHPHWKRGLVOLNHWKLV)LUVWVWDUWZLWKWKHRSWLPDOVROXWLRQREWDLQHGHDUOLHULQVHUW2LQWKH FHOOFRUUHVSRQGLQJWRWKHURXWH±LQWKHWDEOHPDNLQJWKHURXWHLPSRVVLEOH1RZVROYHLWDVDQDVVLJQPHQW SUREOHP,WLVVKRZQLQ7DEOHVDQG TABLE 7.13 Reduced Cost Table 1 , - . / 0 1 2 2 2 2 2 2 2 TABLE 7.14 Reduced Cost Table 2 City City 1 2 3 4 5 6 1 M 7 0 17 32 21 2 0 M 15 3 17 0 3 0 14 M 0 2 6 4 15 0 2 M 0 0 5 30 22 4 0 M M 6 24 8 13 5 0 M $VVLJQPHQWV ±±±±±± 6XEWRXUV ±±DQG±±±± /HQJWK NP NP $VZHREVHUYHWKHRSWLPDOVROXWLRQWRWKLVSUREOHPOHDGVWRWZRVXEWRXUV±±DQG±±±±ZLWKD WRWDOGLVWDQFHYDOXHRINP 1H[WDJDLQVWDUWZLWKWKHRSWLPDOVROXWLRQRIWKHSUREOHPREWDLQHGHDUOLHU 7DEOH DQGSXWDQ2LQWKH FHOOFRUUHVSRQGLQJWRWKHURXWH±7KLVPDNHVWKLVURXWHXQDFFHSWDEOH1RZVROYHWKLVSUREOHPDOVRDVDQ DVVLJQPHQWSUREOHP7KLVLVVKRZQLQ7DEOHV± ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ TABLE 7.15 Reduced Cost Table 1 , - . / 0 1 2 2 2 2 2 2 2 TABLE 7.16 Reduced Cost Table 2 , - . / 0 1 2 2 2 2 2 2 2 TABLE 7.17 Reduced Cost Table 3 City City 1 2 3 4 5 6 1 M 7 0 19 32 23 2 0 M 13 3 15 0 3 0 12 M 0 0 6 4 17 0 2 M 0 0 5 34 20 6 4 M 0 6 19 1 6 0 M M $VVLJQPHQWV 7RXU /HQJWK ±±±±±± ±±±±±± NP ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 7KHRSWLPDOSDWWHUQRIDVVLJQPHQWVLQ7DEOH FOHDUO\LQYROYHVDWRXU±±±±±±ZLWKDWRWDO OHQJWKHTXDOWRNP = 126 km Upper bound Revised upper bound = 87 km Revised lower bound = 84 km &RQVLGHULQJ WKH UHVXOWV RI ERWK WKH EUDQFKHV ZH UHGXFHWKHXSSHUERXQGWR EHFDXVHPRYHPHQW DORQJ WKH EUDQFK PDNLQJ ± DQ XQDFFHSWDEOH URXWHUHVXOWVLQDIHDVLEOHVROXWLRQWRWKHSUREOHP 6LPLODUO\WKHORZHUERXQGZRXOGEHUHYLVHGWR NP:HFDQVKRZWKHPRGL¿HGEUDQFKDQGERXQG WUHHDVLQ)LJXUH le LB = 80 6 5– 84 ab ept Sub-tours: 1–3–1 (40 km) 2– 6–5–4–2 (44 km) acc Un 6–5 U n acc 1RZVLQFHDJDSH[LVWVEHWZHHQWKHORZHUDQGXSSHU ept abl Tour: e ERXQGVZHVKDOODWWHPSWWR¿QGWKHSRVVLELOLW\RI 87 1–3–5–6– 4–2–1 QDUURZLQJLW)RUWKLVDJDLQZHEUHDNWKHVPDOOHVW (87 km) RIWKHVXEWRXUV²LQWKHSUHVHQWFDVHWKHVXEWRXULV ±±7KHUHZRXOGDJDLQEHWZREUDQFKHVRQHWKDW )LJXUH 0RGLILHG%UDQFKDQG%RXQG7UHH PDNHVWKHURXWH±XQDFFHSWDEOHDQGWKHRWKHUWKDW PDNHVWKHURXWH±QRWDFFHSWDEOH)RUWKLVSXUSRVH ZHVWDUWZLWKWKHRSWLPDOVROXWLRQJLYHQLQ7DEOHDQGSXWDQ2LQWKHFHOOLQGLFDWLQJWKHURXWH±7KLV PDNHVWKLVURXWHLPSRVVLEOH7KHSUREOHPLVWKHQVROYHGLQWKHPDQQHURIDQDVVLJQPHQWSUREOHPDVVKRZQ LQ7DEOHV± TABLE 7.18 Reduced Cost Table 1 , - . / 0 1 2 2 2 2 2 2 2 2 TABLE 7.19 Reduced Cost Table 2 , - . / 0 1 2 2 2 2 2 2 2 2 ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ TABLE 7.20 Reduced Cost Table 3 City City 1 2 3 4 5 6 1 M 0 M 10 25 14 2 0 M 13 3 17 0 3 0 14 M 0 2 6 4 15 0 0 M 0 0 5 30 18 2 0 M M 6 24 8 1 5 0 M $VVLJQPHQWV 7RXU /HQJWK ±±±±±± ±±±±±± NP 0DNLQJ WKH URXWH ± XQDFFHSWDEOH UHVXOWV LQ D WRXU ±±±±±± ZLWK D WRWDO GLVWDQFH RI NP 7DEOHV±JLYHWKHVROXWLRQZKHQWKHURXWH±LVWDNHQQRWWREHDQDFFHSWDEOHRQH TABLE 7.21 Reduced Cost Table 1 , - . / 0 1 2 2 2 2 2 2 2 2 TABLE 7.22 Reduced Cost Table 2 , - . / 0 1 2 2 2 2 2 2 2 2 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 7.23 Reduced Cost Table 3 City City 1 2 3 4 5 6 1 M 5 0 21 34 19 2 0 M 17 9 21 0 3 M 8 M 0 0 0 4 15 0 4 M 4 0 5 24 12 0 0 M M 6 20 4 11 7 0 M $VVLJQPHQWV 7RXU /HQJWK ±±±±±± ±±±±±± NP 7KHUHVXOWRIPDNLQJWKHURXWH±XQDFFHSWDEOHLVDOVRDWRXU±±±±±±LQYROYLQJDWRWDOGLVWDQFH RINP &OHDUO\WKHQWKHPRYHPHQWDORQJERWKWKHEUDQFKHVRIWKLVQRGHUHVXOWVLQIHDVLEOHVROXWLRQVEXWHDFKRQH RIWKHPLQYROYHVDSODQWKDWZRXOGPDNHWKHVDOHVPDQWUDYHOORQJHUWKDQKRZPXFKKHZRXOGLIKHDFFHSWV WKHSODQDOUHDG\DWKDQG7RFRQFOXGHWKHUHIRUHWKHXSSHUERXQGFDQQRWEHUHGXFHGDQ\IXUWKHUDQGWKH RSWLPDOVROXWLRQWRWKHSUREOHPLVWKHWRXULQWKHRUGHU±±±±±± 7KH¿QDOEUDQFKDQGERXQGWUHHWRWKLVSUREOHPLVVKRZQLQ)LJXUH Initial UB = 126 km Revised UB = 87 km Initial LB = 80 km Revised LB = 84 km Revised LB le tab Initial 1– 3 84 e bl pta LB = 80 Un ep acc 93 Sub-tours: 1–3–4 (40) 2–6–5–4–2 (44) Break 1–3–1 3–1 Un e acc acc n ept U abl 6 Sub-tours: 5– e 1–3–4–2–1 (65) 5–6–5 (15) Break 5–6–5 6– 5 Un acc ept abl e 87 90 Tour: 1–2–6–5–4–3–1 (93 km) Not to be considered as the current UB is exceeded Tour: 1–3–6–5–4–2–1 (90 km) Revised UB STAGE–I STAGE–II STAGE–III )LJXUH )LQDO%UDQFKDQG%RXQG7UHHRI WKH7UDYHOOLQJ6DOHVPDQ·V3UREOHP ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ START Obtain value of the tour given by visiting cities in numerical order. This is initial upper bound. Formulate an assignment model, taking inter-city distances as costs and placing M in all diagonal positions making assignment of a city to itself impossible. Solve using HAM and write value of solution in node. It represents the lower bound on the value of any solution obtainable by branching from this node. [LB for the whole problem equals the least of all the nodes for which branching is not completed.] Set new UB equal to the value of the Yes solution corresponding to new tour. Does the optimal solution yield a tour with a value less than the current UB? No Is the lower bound for the whole problem equal to UB? No Yes Solution corresponding to UB is optimal STOP Branch from the node with the smallest value by breaking up the shortest sub-tour at that node. Take a branch for each city-to-city trip in this sub-tour. Formulate a new assignment problem with one of the routes unacceptable. 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" 4 ! ! " % " # $ % 0 $ " " # $ ! ! ! %! ! 5 6 ! $ $ ! 7 8 % ! $ ! % '7'7 '''7 2 % $ % " % 1443 9 % " % ! 0 '7'7 ¥ , '''7 ¥ C !5''7 8" ! $ # % % $ 0 " % : $ ! %) C ! $ $ 5 6 " % 6 6 ! !" #$ % &'(## #%$#*' &#+ $ ( £,## ($ ' £### '$ ( £## ) ) - . 2 . 2 / ! 0 1 /!!0 - 2 & 0 C'(## $ & 0 C'(## 2 & " #+ + & " #+ 3 ! ! !" 4 ! ! . & 2$ 2 5 6 # $ % $ 22 $ &'(## #% $#*' $ 22 $ &#+ $ ( $ # &,## ( $ ' $ # &### #% &## ' $ ( $ ###% 2 $ 2 $ ≥# ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ ! 1 ! 0 ! 7 8 922 $ : 0 ; ! / ! 2 2 " ## #% / /**2*% / /*% &'#&*' 2&*<(&,*'# &' #%&'# 6 7 ,*'1 " #' ! ,*' '" 2 '#" 2 / 0 ! TABLE 7.27 Simplex Tableau 1 "" 5 $ 5 4 # # #% $ 5 # % 2 # # # # # '(## ,+*' 2 %<# %<(= # # 2 # # # #+ (( ¨ # # ( # # # # # # ,## '# # # ( ' # # # # # # ### ## #% # ' ( # # # # # # ## %## # # # # # # # 5 # # '(## # #+ # ,## ### ## D 2#%<# # # # # TABLE 7.28 "" 2%<( # ≠ Simplex Tableau 2 5 $ 5 4 # # #% $ 2+<% +<% # # # *> >><, 5 %,<' # 2 # <' # # (<% 2(<% # # # (( 2 # # <' # # # 2,<% ,<%= # # ( >< # # # # # 2#<% #<% # # +# ( #% # *<' # # # 2,<% ,<% # # ,( * # # # # # # # 5 # (( *> # # # ( +# ,( D 2%,<' # # %<% 2+<% # # # ≠ ¨ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 7.29 Simplex Tableau 3 "" 5 $ 5 4 # # #% $ 2 ( # 2 # # 2+ # # ,## '# # < # # # # <( # # '# %## $ # %<(#= # # # 2 %<, # # >< # # %< # # # # # 2'<( # '# '##<% #% # % # # # # # 2 # ,## ## # # # # # # # 5 # '# ,## # # >< # '# ,## D 2 ( # # # # + # # ≠ TABLE 7.30 ,# ¨ Simplex Tableau 4 "" 5 $ 5 4 # # #% $ 2 # # 2 ,#<%= 2,#<% 2+ # # %,# *<( ¨ # # # # #<% 2#<% 2 # # # + # # # # 2(#<% (#<% '< # # ,# 2 # # # # # # # 2# 2' # ,# + #% # # # # # (# 2(# 2*< # (# < # # # # # # # 5 ,# # %,# # # # # ,# (# D # # # 2'*<% ,#<% + # # ≠ TABLE 7.31 Simplex Tableau 5: Optimal Solution "" 5 4 5 4 # # # % 5 # # %<,# 2%<,# 2 2 *<+# # # *<( # # 2 <+ <+ # # '<( # # *' # # <( 2<( # # 2 # # '# # # # # 2%<+ %<+ # # *<( # ' #% # # # 2 %<( %<( # # ' # '# # # # # # # # 5 '# *' # # *<( # # ' '# D # # '*<,# %<,# # *<+# # # ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ *UDSKLF3UHVHQWDWLRQ 320 280 x2 1 5x + 240 2 4x = 4x 1 00 200 12 / ! 1 *> / !7689# ! " ! 7 ? + 5 x2 / ! 160 P = 10 K 00 ! Goal Programming Solution 120 ! 9 80 Q A 0.3 20x1 + 32x2 = 5400 2x x1 + 0 2 $ 2 $ :3 . 1 + 40 4x 75x2 = R 2 = 108 ! 6 S 00 x1 0 : ! 80 120 160 200 240 280 320 360 40 / 8 )LJXUH *UDSKLF3UHVHQWDWLRQRI *RDO 0 3URJUDPPLQJ3UREOHP @ ! 5 &#&## 7 ¥#$(¥##&,# ! ,## # 2 2 0 'LIIHUHQWLDO:HLJKWLQJ ! 6 9 !: ! 7 ! A ! 4 C' 0 !'<'(## ##> 0 " !'<#+ (,% ! / ! " 6 7 6 2<'(## 2<#+ 1 " 0 ! 2¥9<'(##:&<*## 2<#+ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 8 ! !/ ! 7 C## 0 / ! C## " 9 ! ! C## 0 ": ! ! 6 . & 2$## 2 6 ! ! /*%2*%' /*%'&,# &# 0 #¥,#$%¥#&C'#(# 2&%,# ! 0 C'(##!C%,#8 2 7 ? ! #+ ! ,#" 2 (#" 2 TABLE 7.32 Simplex Tableau 1 "" 5 4 5 4 # # # % $ 5 # % 2 # # # # # '(## ,+*' 5 ## %<# %<(= # # 2 # # # #+ (( ¨ # # ( # # # # # # ,## '# # # ( ' # # # # # # ### ## #% # ' ( # # # # # # ## %## # # # ## # # # # 5 # # '(## # #+ # D 2'# # ## # # # # ≠ TABLE 7.33 "" 2#* ,## ### ## Simplex Tableau 2 5 4 5 4 # # # % $ 5 %,<' # 2 2+<% +<% # # # *> >><, # <' # # (<% 2(<% # # # (( 2 # # <' # # # 2,<% ,<%= # # ( >< ¨ # # # # # 2#<% #<% # # +# ( #% # *<' # # # 2,<% ,<% # # ,( * # # # ## # # # # 5 # (( *> # # # ( +# ,( D 2%,<' # # (+<% 2+<% # # # ≠ ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ TABLE 7.34 Simplex Tableau 3 "" 5 $ 5 4 # # #% $ 5 ( # 2 # # 2+ # # ,## '# # # # # <( # # '## '## # < 4 # %<(#= # # # 2 %<, # # >< # # %< # # # # # 2'<( # '# '##<% #% # % # # # # # 2 # ,## ## # # # ## # # # # 5 # '# ,## # # >< # '# ,## D 2( # # ## # + # # ≠ ,# ¨ TABLE 7.35 Simplex Tableau 4: Optimal Solution "" 5 $ 5 4 # # #% 5 # # 2 ,#<% 2,#<% 2+ # # %,# # # # # #<% 2#<% 2 # # # # # # # 2 (#<% (#<% '< # # ,# # # # # # # # 2# 2' # ,# #% # # # # # (# 2(# 2*< # (# # # # ## # # # # 5 ,# # %,# # # # # ,# (# D # # # (#<% ,#<% + # # 3UHHPSWLYH*RDO3URJUDPPLQJ 4 ! ! 5 LO 7 Solve pre-emptive 0 0 GPPs graphically and using modified simplex method ! ! ! ! !/ 0 ! " ! ! !5 ! ! " ! ! /! 0 " 9: 9: 6 0 9 : 6 6BBB6BBB6%BBBBBB6 6C ! D ! ! / 0 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ % / ! ! / 0 6 6 6 " 0 ! 4 ! ! / " ! ! ! !/ " !! ! ! /ŶĐĂƐĞŽĨƉƌĞͲĞŵƉƟǀĞŐŽĂůƉƌŽŐƌĂŵŵŝŶŐ͕ŐŽĂůƐĂƌĞĂƐƐŝŐŶĞĚƉƌŝŽƌŝƚǇůĞǀĞůƐ͘dŽďĞŐŝŶǁŝƚŚ͕ƚŚĞĮƌƐƚͲƉƌŝŽƌŝƚǇŐŽĂů ŝƐĐŽŶƐŝĚĞƌĞĚĂŶĚĂŶĂƩĞŵƉƚŝƐŵĂĚĞƚŽĂĐŚŝĞǀĞŝƚĂƐĐůŽƐĞůǇĂƐƉŽƐƐŝďůĞ͘KŶůǇǁŚĞŶĂŶŽƉƟŵĂůƐŽůƵƟŽŶǁŝƚŚ ƌĞƐƉĞĐƚƚŽƚŚĞĮƌƐƚͲƉƌŝŽƌŝƚǇŐŽĂůŝƐĂĐŚŝĞǀĞĚƚŚĂƚƚŚĞƐĞĐŽŶĚͲƉƌŝŽƌŝƚǇŐŽĂůŝƐĐŽŶƐŝĚĞƌĞĚ͘^ŝŵŝůĂƌůǇ͕ƚŚĞŶĞǆƚͲŝŶͲ ůŝŶĞŐŽĂůƐĂƌĞĐŽŶƐŝĚĞƌĞĚŽŶůǇǁŚĞŶƚŚĞŽƉƟŵĂůŝƚǇǁŝƚŚƌĞƐƉĞĐƚƚŽƚŚĞŚŝŐŚĞƌƉƌŝŽƌŝƚǇŐŽĂůƐŝƐŽďƚĂŝŶĞĚ͘dŚĞ ƉƌŽĐĞƐƐŝƐĐŽŶƟŶƵĞĚƵŶƟůƚŚĞŽŶůǇǁĂǇƚŽĐŽŵĞĐůŽƐĞƌƚŽƐĂƟƐĨǇŝŶŐĂŐŽĂůŝƐƚŽŝŶĐƌĞĂƐĞƚŚĞĚĞǀŝĂƟŽŶĨƌŽŵĂ ŚŝŐŚĞƌƉƌŝŽƌŝƚǇŐŽĂů͘ *UDSKLF6ROXWLRQWR*33 4 ! !1 7 ? 7 / 9 : 0 ! @ @ 9: ! !3 ? ! !/ ! 3 9 : ! ! 0/ )EE1 ! 7 ([DPSOH ; ! C C ! " < 0 $ # 0 $ 6 ! $ 0 # # $ " ! $ % ! % % ! " # = % >" ? ) ;" " ? ) - % 8" ! )! " ? ) * " @ 0 % " ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ - &RQVWUDLQWVDQG*RDOV 5DZ0DWHULDO " 7 $£%## ! ! !%##" 3URFHVVLQJ&DSDFLW\5 ! $$22 $&(## 2 ! $ ! 0DUNHW5 ! $ 2&'# $ %2&%'# 3 ! 66 6% ! ! . &6 2$6 9( 2$% %2 :$6% $ 5 6 $£ %## $$22 $& (## $ 2& '# $ %2& %'# 2 $ 2 %2≥ # / ! 0 1 *# ! + / 9! 7 ? : 5 ! ! ! 2 $ / /0 2 &#/ @ FA/ @ 9 & %'#: ! 4 ! * 9 * , * + 2? 3 * + 2? 3 , * + 2? 3 ? - , # * , * + - , ,, ( 7 * )LJXUH *UDSKLF6ROXWLRQWR*333UHHPSWLYH ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ ! 5 ! 0 C(## ! C%## ! G ! ! 2 %2 6 5 %5 ;( 54% %5 # %## '# '# *'# ## ## '# '# ,'# '# '# # ## ,## ¨. / &'# &'#5 7 ! ¥ '# $ '# 2 $ & (## $ & '# ! '# ! " A ! ! ! ! / " % 2 7 ? $ 7 ? $ 6 . &6 2$6 $$6%9( 2$% %2:H 2 $ 7 ? 9 : ! ! ! # " : 7 ! 4 ! &## &## 2&'# %2&'# &,'# @ &'# &## 2&# %2&'# &*'#/ ## ## 2&'# %2 &'# 02',),('6,03/(;0(7+2')25*2$/352*5$00,1* E 0 G LO 8 Apply modified simplex method for goal 6 programming ! ! 666 9 7 !: / 8 0 ! ! 9 ! ! " :/! /D 9 " : 7 ! 7 / 0! "! ! 5 6BBB6$ 6 ! - ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ ([DPSOH -" % " 7! %) 8 + , A , 2, A , , A 3 , , , * -%. * , * £ * , * , , A A , , + * , , A + * , , A + A , A *! *! , ! , ! , , A ≥ / 0 /*%,@ 5 / 9/*%,: #2 ± %2 7 %##(## '# %'# !A ! 7 ? ? 2 ± %2 / ! !6 "6 6% / D - D 9& 2: 1 9 0 : 666% 1 " 5 69 $: &¥& 6 &# D &#2&2 "63 6 # ( % &¥($#¥%&( &# D&#2(&2( "6/ 6% D &# "6% 4 $5 2 2 &¥92:&2 64 !D "6 #292:&5 $ ? &(¥#$%¥#&# D &#2#&# "61 ! !6%&# D &2#& TABLE 7.36 Simplex Tableau 1: Non-optimal Solution "" # 5 4 5 %5 $ # # # # # # %## %## 6 # 2 # # (## ## (6 = # # # # # '# '# 5 5 %5 %6 # # # # # %'# 2 # # # 6 6% (6 %6 # # # # # # # 2( 2% # # # # # ,'# 2 2 # # # # (## ≠ È 63 Í Í 62 ÍÎ 61 D ¨ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ / "66 6% ! / 29 !: (## 6 (##¥&(##5 ! 2&'# 2%&%'# ( % ! ! " '#¥($%'#¥%&,'#1 ! !6% $&# !7 ¥#&# 7HVWRI2SWLPDOLW\DQG'HULYDWLRQRI5HYLVHG7DEOHDX / !0 " D 6/6 ! D C 7 ? / D 6 5 / 7 2 L < '# 2/ 2 " ! D ! D 9 : ! 9 :D @ 9"! "! : ! / /*%* &'#M #&'#"M %2&%'# ! ! 1 D ! /*%,4 D 9&2: " "! 2 / /*%+ TABLE 7.37 "" Simplex Tableau 2: Non-optimal Solution # 5 4 5 %5 $ # # 2 # '# '# # # # 5 6 # = # 2 2 # ## ## # # # # # # '# 2 %5 %6 # # # # # %'# %'# # # # 6 6% (6 %6 # # # # # # # D # 2% # # # ( # #'# È 63 Í Í 62 ÍÎ 61 # 2 # # # ## ≠ ¨ ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ TABLE 7.38 Simplex Tableau 3: Non-optimal Solution "" # 5 4 5 %5 $ # # # # 2 = # '# '# ¨ # # # 2 2 # ## 2 # # # # # # '# 2 %5 %6 # # # 2 '# '# # # # 6 6% (6 %6 # # # # # # D # # # % 2% 2 # *'# È 63 Í Í 62 ÍÎ 61 # # # # # # # # ≠ TABLE 7.39 Simplex Tableau 4: Optimal Solution "" # 5 4 5 %5 4 6% # # 2 # '# # # # # 2 # '# # # # # # # '# %5 %6 # # 2 # # ## # # # 6 6% (6 %6 # # 2 # # # # '# È 63 Í D Í 62 ÍÎ 61 # # % # # # ,## # # # # # # # 1 /*%+ D "6 ! ! !8 D "68 2% 2 $ 2 ! 2%/ $ # 1 ! /*%> ! ! ! D C 7 ? "6 68 D9&2: # 6% 4 ! ! D ! # G ! ! ! D 7 % 6/ D ! D ! 1 5 / ( ! &'#&'# $&'# %2&## '# ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ %'# !## "'# 8VLQJ6ROYHUIRU,QWHJHU3URJUDPPLQJ0RGHOV / / 7 / ([DPSOH ; ! # 0 " % % " ,& : & ;0 B 2$3 ' 6 % 7 6 4 2C3 5 9% %" %! % 2 3 ; % 2%3 ; % D 6ROXWLRQDVD/LQHDU3URJUDPPLQJ3UREOHP / 0 " 1 */ G%G( G' 7 G,F+/ ),)* )+E 8 N5L.ED@FL/9G%F%G#F#:C ! A,A* A+ !/ A, N5L.ED@FL/9G,F,G#F#:C )LJXUH 3URGXFWLRQ3ODQQLQJ3UREOHP ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ 3 5 " 7 6HW7DUJHW&HOO1 " #+ " 8 4 O8O (TXDO7R5 . H 2 . %\&KDQJLQJ&HOOV5 7 7 ! @ / " ! " G#F# OGO#OFO# @ # ! " " 1 6XEMHFWWRWKH&RQVWUDLQWV1 " " 5 " / A,A* A+ 7 ),)* )+ !/ " 9 A,A+ " " ),)+ 1 P&9 7 :/ "OAO,OAO+P&O)O,O)O+ 1 *4 7: " # 36 " /5 1 * )LJXUH 6ROYHU3DUDPHWHUV'LDORJ%R[ 2SWLRQV/ 7 " # 37 " " "" & 2 "" < < 3/ 2 !4 7: # 36 " 6ROYH1 ! " # 3 5 D ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ " : # 3# 1 * 1 *%/ / '*' +*' ! C+%*' )LJXUH 2SWLPDO6ROXWLRQWR/33 E 6ROXWLRQDVDQ,QWHJHU3URJUDPPLQJ3UREOHP 1 ! 7 G#F# 1 " # ! " " # 36 " / " "N 9 C G#F#4 " NC NOGO#OF#& C 1 *( )LJXUH 5HYLVHG6ROYHU3DUDPHWHUV 4 "" & 2 "" < < 3 7 " " # 3 1 *' ! 7 ,*, +' ! '*' +*' ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ ! -E / 0 C+%% C+%*'4 -E E )LJXUH 2SWLPDO6ROXWLRQWR,33 6ROXWLRQWR²,QWHJHU3URJUDPPLQJ3UREOHPV / #2 D A*(/ A 1 *, 5 6 3EH ( N5L.ED@FL/9G%)%G()(:C/D85 ' ! 8'81 8' N5L.ED@FL/9G')'OGO(O)O(:C )LJXUH ,QYHVWPHQW3UREOHP'DWD 3 " / 6HW7DUJHW&HOO1 " #+ " OO( ( 4 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ (TXDO7R5 3EH 2 %\&KDQJLQJ&HOOV " ! " G()( OGO(O)O( 6XEMHFWWRWKH&RQVWUDLQWV1 " " 5 / 1 ** G()( NC NOGO(O)O( & !C )LJXUH 6ROYHU3DUDPHWHUV'LDORJ%R[ 2SWLRQV/ 7 " " "" & 2 "" < < 3 5 E 6ROYH1 ! " # 3 / 1 *+ )LJXUH 6ROXWLRQWR,QYHVWPHQW3UREOHP A ! ( 3EHC''### ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ 5(9,(:,//8675$7,216 ([DPSOH & # . 23 . % ! ! ! ! ! '! 7! 5! ( . 2! ! 3 % 23 4 . % % 8" ! % % 23 C . 2! ! ! 7! (3 % 2"3 : % ! % 5 % 2"3 4 . % % % % . - 9&Q#: 6 / %- 6 A " # ! 9&Q#:& 6 &# 3 9 : 5 '-; -! ! 6 $$%& 9 : / 7 #2≥# 6 # 6 M 6 6 # 9 #2&2: 9 : / 6 9%'*>: $%$'$*$>£ 9 : ($+£ ! 6 ( + 9: 6 6 7 ? / (2#&# ([DPSOH & ? / ) 2 3 2 x1 + 2%3 23 16 2 27 x2 - S1 + 0S2 = 5 5 4 13 3 1 2 9 x1 + x2 + S1 - S2 = 4 5 4 3 5 7 2 3 1 7 x1 - x2 - S1 S2 = 16 3 8 32 3 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ ) !C 9: 0 x1 + 3 1 3 x2 + S1 + 0 S2 ≥ 5 5 4 9: 3 3 1 1 4 x1 + x2 + S1 + S2 ≥ 4 5 4 3 5 9 : 7 1 5 31 1 x1 + x2 + S1 + S2 ≥ 16 3 8 32 3 ([DPSOH % % " ) * < 8 + * , 0 -%. * , 0 £ 7N * £ N *! 0 ≥ &; 0 £ 7N ! * 0 ( ( ( * ± ( ± 0 < A 8< ± ± 2 3 O %." 8 + * , 0 5 2%3 6 " " % * % - 9: 7 '-! 5 . &$ 5 6 $£*M £+M £* ≥# ! " 5 / /*(# ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ TABLE 7.40 Simplex Tableau 1: Non-optimal Solution "" # # #% $ # # = # # * * # # # # # + > #% # # # # * 2 # # # 5 D # # * + * # # # 4 /*( 4 ! 6 &%' &%' &( &#5 ! 6 #' ( TABLE 7.41 Simplex Tableau 2: Optimal Solution "" # # #% # # * # # # 2 2 # ( #% # # # # * # # # * # # ( * # 2 2 # # 5 D &( 9: 3 " 0 " ) ! 0 x + 0 y + 0 S1 + 0 S2 + 1 1 S3 ≥ 2 2 ! / /*(/ #% #( ! 0 x + 0 y + 0 S1 + 0 S2 - 1 1 S3 + S 4 = 2 2 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 7.42 "" Simplex Tableau 1: Non-optimal Solution # # #% #( $ 2< # *< 2 # # # # # # 2 # ( ( # # # < # *< * #( # # # # # 2<= 2< 5 D # # # # *< *< # ( # 2< # # 2 # 2< # / /*(% 0 9 7 : / &(&% &# TABLE 7.43 "" Simplex Tableau 2: Optimal Solution # # #% #( # # # 2 ( # # # # 2 # % # # # # % #% # # # # # 2< # # # # ( % # % # # # 2 # 2< # 5 D &# ([DPSOH ; ! 0 ! % # " % " %) ? ,&; &&0=> ,& ,& - 7 ;% 5 4 ( 4 " % % O C C ! " O > 2C> % % 3 %- '-; !/ ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ - . ! !"/EE &'#$# / 0 1 2 + £ 50 75 5 1 2 + £ 40 80 1 2 + £ 90 45 ≥# 4! 5 6 / E 0 . &'#$# 5 6 %$ £'# $ £+# $ £># ≥# -E 4 ! ## #% / . &'#$#$##$##$##% 5 6 %$$# &'# $$#&+# $$#% &># ###% ≥# " /*((2*(, TABLE 7.44 "" Simplex Tableau 1: Non-optimal Solution # # #% $ # # % # # '# '# # # = # # +# (# ># ># #% # # # '# # # # # 5 # # '# +# ># D '# # # # # ≠ ¨ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 7.45 Simplex Tableau 2: Non-optimal Solution "" # # #% $ # # # < 2%< # %# ,# '# < # < # (# +# #% # # %<= # 2< '# ##<% '# # # # # 5 (# # %# # '# D # (' # 2*' # ¨ &,### ≠ TABLE 7.46 Simplex Tableau 3: Optimal Solution "" # # #% # # # # 2(<% 2<% (#<% '# # # <% 2<% *#<% # # # 2<% <% ##<% '# # # # # 5 *#<% ##<% (#<% # # D # # # 2,# 2%# &*'## 1 /*(, -E &*#<% &##<% EE /! 0 /*(,/ / ! 4 1 40 #$#$#2 #2 - #3& 3 3 3 2ˆ 2ˆ 1 Ê Ê #$#$9$##:$ Á - 2 + ˜ #2 + Á - 1 + ˜ #3&13 Ë Ë 3¯ 3¯ 3 D ! 4 ! -85 D85 2 2 1 #2 + #3 & + (13 - 1#1 + 2 #2 + #3 ) 3 3 3 2 2 1 # 2 + #3 ≥ 3 3 3 1 2 2 - #2 - #3 £2 3 3 3 ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ 4 2 2 1 2 #2 - #3 + #4 &3 3 3 TABLE 7.47 5 / %/*(, /*(* Revised Simplex Tableau 1 "" # # #% #( $ # # # # 2(<% 2<% # (#<% 2 '# # # <% 2<% # *#<% 2 # # # 2<% <% # ##<% '# #( # # # # 2<% 2<%= 2<% < ¨ '# # # # # # 5 *#<% ##<% (#<% # # 2<% D # # # 2,# 2%# # ≠ #( TABLE 7.48 "" Revised Simplex Tableau 2 # # #% # # # # 2 # 2< *< '# # # # 2< (*< # # # 2 # %% #% # # # # 2%< < '# # # # # # 5 (*< %% *< # < # D # # # 2%# # 2(' &*(+' / /*(+ D 1 0 /*(+ 1 27 #$#$#2#$##%2 #(& 2 2 0 1 1 #(& $[%2#$#$#(\ 2 2 1 1 #(£2 2 2 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 1 1 2 #($#'&2 2 2 5 / /*(> TABLE 7.49 Revised Simplex Tableau 3 "" # # #% #( #' $ # # # # 2 # 2< # *< 2 '# # # # 2< # (*< 2 # # # 2 # # %% %% #% # # # # 2%< # < 2 #' # # # # # # 2<= 2< ¨ '# # # # # # # 5 (*< %% *< # < # 2< D # # # 2%# # 2(' # #' ≠ / /*'# TABLE 7.50 Revised Simplex Tableau 4 "" # # #% #( # # # # 2 # # 2 ( '# # # # # 2 ( # # # 2 # # % #% # # # # # 2% #( # # # # # # 2 '# # # # # # # ( % ( # # # # # 2%# # # 2># 5 D &*((# / EE 4 !&(&% *((# ([DPSOH ; " # $ " % " # % 2 $3 % " # P " ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ $ &0 # A 7 ' ( 7 A 7 ' A ' A ( 7 A # 7&0 2%- '-!/3 / /*' 1 !2 ! / /*' *'% TABLE 7.51 Distance Profile 2 %*' %*' 2 ,## %## '# %'# ># *' ,## %## 2 %'# '## ,# %'# %'# 2 %## ># *' '## %## 2 TABLE 7.52 Reduced Cost Table 1 2 ## ' 2 ('# ' # *' (# # %## # 2 '# ## # ># ># 2 (# ' # %' ' 2 TABLE 7.53 Reduced Cost Table 2 2 ## ' 2 %' # # *' (# # %## # 2 '# ## # ># ,' 2 (# ' # ## ' 2 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ D/ ? / /*'(5 7 1 ! /*'* TABLE 7.54 Reduced Cost Table 3 2 (# %' # (# ## 2 # *' # +' # 2 %' +' # #' ,' 2 (# # # +' # 2 Table 7.55 Reduced Cost Table 4 2 *' %' # (# %' 2 # *' # +' # 2 # '# # #' %# 2 #' # # '# *' 2 TABLE 7.56 Reduced Cost Table 5 2 *' >' # # ,' 2 # #' # +' # 2 # # # #' # 2 *' # # # *' 2 TABLE 7.57 Reduced Cost Table 6 2 *' >' 0 # ,' 2 0 #' 0 0 2 0 # +' 0 #' 0 2 *' 0 0 # *' 2 ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ / / *'* 2 2 2 2 2/ 22222 / 4 ! '#$%'#$%##$*'$>#&,'" ([DPSOH ; ! ! % 0 #! # > 57 ' ' 7' 5 ' 5 7 ' # # 55 ' 5 ' 7' 5 7 8 23 " 23 " Q 40 2 ! / /*'+2*, TABLE 7.58 Distance Profile 2 #% ++ %, %+ #% 2 , *, ' +* , 2 +' *' %, *, +' 2 , %+ ' *' , 2 TABLE 7.59 Reduced Cost Table 1 2 ,' (> >+ # ' 2 # ( # # ** 2 # ># ' > # 2 ** # ( %* ( 2 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 7.60 Reduced Cost Table 2 2 ' (> >+ # ' 2 # ( # # ,% 2 # ># ' ** # 2 ** # # %* ( 2 TABLE 7.61 Reduced Cost Table 3 2 0 (> >+ 0 0 2 # ( 0 ' 2 0 ># 0 , 0 2 ** 0 0 ++ *' 2 4 5 - 22222 22222 >%$*#&%,% A ! 222 227 %,% 3 22222 ,'#/ ! / - L %,% ,'# 3 " 22 ! 9 :1 D/% /*, 2 % 2 ! 2 / /*, % /*,%4 ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ TABLE 7.62 Reduced Cost Table 4: Sub-problem 2 2 # (> >+ # # 2 # ( # ' 2 2 ># # , # 2 ** # # ++ *' 2 4 5 / TABLE 7.63 22222 22222 ' Reduced Cost Table 5: Sub-problem 3 2 # (> >+ # # 2 # ( # ' 2 # ># # , 2 2 ** # # ++ *' 2 4 5 / 22222 22222 '% 5 ! ' " 4 1 *> 0 2 9 (: 2 9 ':/ /*,( *,' ! abl ept LB = 363 DU C- c nac DCU nac e 512 Sub-tours: A-D-C-A, B-E-B Total distance = 512 513 Sub-tours: A-C-D-A, B-E-B Total distance = 513 cep tab le Initial Lower Bound = 363 Revised Lower Bound = 512 Upper Bound = 650 )LJXUH %UDQFKDQG%RXQG7UHH ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 7.64 Reduced Cost Table 6: Sub-problem 4 2 # (> # # # 2 # , 2 # , 2 2 ># # , # 2 ** # # ++ ** 2 4 / / TABLE 7.65 22222 22222 '%+ Reduced Cost Table 7: Sub-problem 5 2 # (> # # # 2 # , # # , 2 2 ># # , # 2 ** # 2 ++ ** 2 4 / / 22222 22222 '*% 3 ! % /*,, , ! 2 5 !/*,* *" 2 / TABLE 7.66 Reduced Cost Table 8: Sub-problem 6 2 # # >+ # # 2 , ( 2 ' 2 # ># # , 2 2 ** # # %> *' 2 ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ 4 / / TABLE 7.67 22222 25222 '*% Reduced Cost Table 9: Sub-problem 7 2 # # >+ # # 2 , ( # ' 2 # ># # , 2 2 ** # 2 %> *' 2 4 / / 22222 22222 '%> ! ! / @ '%+/ '%+/ 1 *# 1 '%+ LB = 363 UB = 650 22222 LB = 512 UB = 650 LB = 538 UB = 538 Tour: A–E–B–D–C–A Sub-problem 4 le Revised E B– ab ept acc Un 538 LB = 512 E–B Un le acc tab Sub-problem ept abl e 2 Initial ep acc n DU C– Tour: A–D–C–B–E–A Sub-problem 5 573 LB = 363 Sub-problem D– CU 1 n acc ept EU B– abl e 513 Tour: A–D–C–B–E–A Sub-problem 6 ble pta ce nac 573 E– BU nac cep Sub-problem tab 3 le Tour: A–C–D–B–E–A Sub-problem 7 539 )LJXUH )LQDO%UDQFKDQG%RXQG7UHH ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ ([DPSOH ; ! ; 4 ; ! 0 ! "! # $ # $ $ ! ! ! " # " % 23 ;" 23 * " " $ 23 ;" " % 2"3 ;" ! 2"3 * " % 6 % - % !L ! *RDO - 2 ! $ 9 : ! $%$%$ 22 $&(# *RDO - 2 ! ! %5 $ 2&* %$ %2&' / *RDO !0 $ (2& $(& 4 ! 3 # # $$ (22 $(&# 4 ! ! ! / $%$%$22 $&(# $%$%$ 229#2 (2$ $(:&(# $%$%$ (52 ($&'# 9 : 7 ! *RDO '2 2, *2 ! $ '2&#M$ ,2&# %$*2& 3 ! . 5 6 &6 2$6 2$6 %2$6% (2$6( '2$6( ,2$6( 2*$6' $ $ 2&* % $ %2&' ! $%$%$ 22 $&(# ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ $%$%$ 2(2 ($&'# $ '2&# $ ,2&# %$ *2& 2 $ 2 2 2 $ % % ( ( '2 ,2 2*≥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x1 + 65 x2 + 70 x3 + 70 x4 + 70 x5 + d1- - d1+ &### 40 x1 + 45 x2 + 40 x3 + 50 x4 + 40 x5 + d 2- - d 2+ &## 20 x1 + 20 x2 + 15 x3 + 25 x4 + 15 x5 + d3- - d3+ &(## 30 x1 + 20 x2 + 30 x3 + 20 x4 + 30 x5 + d 4- - d 4+ &%## 20 x1 + 25 x2 + 25 x3 + 25 x4 + 25 x5 + d5- - d5+ &%## 5 x1 - 5 x2 + 10 x3 + 10 x4 + 15 x5 + d 6- - d 6+ &# 10 x1 + 15 x2 + 20 x3 + 15 x4 + 15 x5 + d 7- - d 7+ &# 2 $≥#&º,' ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ 1RWHV / 9 : 7 6 6,! %## 6, 6 %#! / 7 / 9]:2/ 9],:≥%## / 9],:2/ 9]%#:≥%## 9: 6>,( %# 5 6, %# ! 9 : / 9]>:2/ 9],:≥# / 9](:2/ 9]%#:≥# ([DPSOH $ ¿UP SURGXFHV WZR SURGXFWV DQG B ZKLFK \LHOG D FRQWULEXWLRQ PDUJLQ RI C DQG CSHUXQLWUHVSHFWLYHO\7KH¿UPKDVDOLPLWHGFDSDFLW\LQWKHWZRGHSDUWPHQWVZKHUHWKHVHSURGXFWVQHHG SURFHVVLQJ7KHDYDLODELOLW\DQGUHTXLUHPHQWVDUHJLYHQEHORZ & 3 ' ,& ,&B 0;0 ' , ,, 7KHPDQDJHPHQWRIWKH¿UPKDVVSHFL¿HGWKHIROORZLQJJRDOV C 0 3URGXFHDSURGXFWPL[WRPDNHDGDLO\SUR¿WRIDWOHDVWC $FKLHYHDGDLO\VDOHVRIDWOHDVWXQLWVRISURGXFWB $FKLHYHDGDLO\VDOHVRIDWOHDVWXQLWVRISURGXFW )RUPXODWHDQGVROYHLWDVDJRDOSURJUDPPLQJSUREOHP - ! 0 6 8 !1 2& ! 0 $& ! 0 2& ! 8 $& ! 8 %2& ! 6 $%& ! 6 4 # # !/ ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ . &6 5$6 2$6% %2 5 6 ##$%##$ 22 &(+## $ 22 $&' $ %22 %$&' ($$#&(' ($($#&*# 2 $ 2 $ 2% 4%##≥# / 0 /*,+2** TABLE 7.68 E 0 E 8 E 6 F F "" Simplex Tableau 1: Non-optimal Solution 5 4 5 4 %5 %4 # # $ 2 6 ## %## 2 # # # # # # (+## 2 6 # = # # 2 # # # # ' ' ¨ %2 6% # # # # # 2 # # ' 2 # # ( # # # # # # # (' ('< # # ( ( # # # # # # # *# %'< # # 6 # 6 # 6% # # # È P3 Í D Í P2 ÍÎ P1 2 # # # # # # # # # 2 # # # # # # # 2## 2%## # # # # # # # TABLE 7.69 "" , ≠ Simplex Tableau 2: Non-optimal Solution 5 4 5 4 %5 %4 # # $ 2 6 ## # 2 2%## %##= # # # # %## # # # # 2 # # # # ' 2 %2 6% # # # # # 2 # # ' 2 # # ( # # # 2 # # # ' '< # # ( # # # 2( ( # # # # '< # # 6 # 6 # 6% # # # È P3 Í D Í P2 ÍÎ P1 2 # # # # # # # # # # # # # # # # # 2## # # %## 2%## # # # # ≠ ¨ ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 7.70 "" Simplex Tableau 3: Non-optimal Solution 5 4 5 4 %5 %4 # $ # <%= # <%## 2<%## 2 # # # # <% <%## 2<%## # # # # # 2% 6% # # $ # %< ¨ # , ( # # # # 2 # # ' ' # # +<% # 2<'# <'# # # # # # % %><+ # # (<% # 2<*' <*' # # # # # , >< # # 6 # 6 # 6% # # # È 63 Í Í 62 ÍÎ 61 2 # # # # # # # # D # # # # # # # # # # # # # # # # # # ≠ TABLE 7.71 Simplex Tableau 4: Non-optimal Solution "" 5 4 5 4 %5 %4 # # $ <## 2<## 2%< # # %< # # # # %< 2 # # # # 2 # # # # ' 2 2% 6% # # 2<## <## %< 2%< $ 2 # # *< *## # # # # 2<'# <'# ( 2( # # # > ('# # # # # 2<'# <'#= 2 # # # ( ## ¨ # # 6 # 6 # 6% # # # È 63 Í D Í 62 ÍÎ 61 # # <## 2<## 2%< %< # # # # # # # # # # # # # # # # # # # # # ≠ TABLE 7.72 "" Simplex Tableau 5: Optimal Solution 5 4 5 4 %5 %4 # # # # # # 2 # # # <( '< # # # # 2 # # # # ' %2 6% # # # # 2 2 # 2<( '< # # # # # # 2 # # 2 ' $ # # # 2 ## 2## # # # '# ## # # 6 # 6 # 6% # # # È 63 Í Í 62 ÍÎ 61 # # # # 2 # # <( # # # # # # # # # # # # # # # # # # D ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ 1 5 / ' '< 6 ' 8 !/ 0 0 9 :% 36KP2TCEVKEG 1RVKOCN2TQFWEV/KZ4CLC,K'NGEVTQPKEU .<5D66 A " EA 91.: 9F: ! 955: 7 ! / 7 ! 4 ! 0 + " " + 7 ## #(# #%# #,# (## 5 #,# #,# #,# %,## 4! (# ## ## >,## ### +# '# (## '## ## ## 0 9C: ( %# (# . L + 3 (2 / 8 ! ! 0 ^ 0 ^ 8 # ^F! " 0 # 0 ! ^ % ^ 7(67<28581'(567$1',1* 0DUNWKHIROORZLQJVWDWHPHQWVDV7 7UXH RU) )DOVH 4 4 ! % 4 EE ( 4 EE 7 ? N#2EEC ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ ' ! #2 ! , / ! ! ! < * ! ! ! ! ! + EE " ! 0 > ! ! # D85 ! D85 4 ! ! ! EE G ! % G ! " ( 4 ! ! ! ! 8 4 . ' ! ! ! ! 22%22/ , ) 6 " * / 0 ! ; + / " > A # / ! ! 6 0 / 7 ; G _ _ % @ 7 ? ( @ ! ! ' 3 6 , E ! ! " * / " C ! 6 ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ + E ! > H 7 %# D 7 ? 0 ! (;(5&,6(6 F ) N ! ! C % Â N C -E ( 5 F 5 ' F ! , ) #2 ! * F 8 ^ + N N !C ! 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(## / 0 C#### 55##C'### 55'# C'### 55'# / ! 9 : 4 ! 9 : . _%# 55### 55'# '# 55'# 9 : 4 ! 9 : . 0 1 B2 ! ,FF/C /2 * /5 4 ! 4 5 4 /H 4 0 ) D #" 98.: ) D %#" 9.1: )% D '" 981: / < 3'" B !"C AI2 2I( AI( " " B C 5 ( ' ##### 5 % + ( %##### " ǆƚĞŶƐŝŽŶƐŽĨ>ŝŶĞĂƌWƌŽŐƌĂŵŵŝŶŐ͗/ŶƚĞŐĞƌWƌŽŐƌĂŵŵŝŶŐĂŶĚ'ŽĂůWƌŽŐƌĂŵŵŝŶŐഩഩ 1 ! C" " ! C" 5 ! ! % C C#+" !8 5 4 4 ^1 /2 + 4 6 ! / ! # ! / " !/ ! 9 : 4 ! ## 9 : 4 ! ! 9 : 4 ! " 9 ! : 9 : . ! 1 B2 ! ,FF/C 8 7KHRU\RI*DPHV &KDSWHU2YHUYLHZ There are several situations in which the outcome of a particular course of action is dependent upon the action taken by another party. Examples include a chess game being played by two players, a war between two countries, competing firms operating in a market, and the management and workers’ union negotiations, and so on. In such cases, the parties involved are the players who have conflicting interests so that a gain to one party is a loss to the other, with each of them having a certain number of strategies to choose from, and the outcome depends upon the particular pair of strategies chosen by the parties. All these ingredients make it a game situation, hence the name Theory of Games. The theory helps to determine the best decision for each of the parties involved. An understanding of the theory helps the manager to answer questions like: What are the consequences of the interplay of each combination of strategies of both the players? Is there one (or more) strategy for each of the players that is clearly the best one to play in the given situation? In the language of theory, does the game have a saddle point so that players play pure strategies? If there is no clear cut strategy for each player, then what combination of strategies should be played by each one? Thus, when no saddle point exists, what is the optimal mix of strategies for the players and what is the expected pay-off resulting from the game? Is a particular strategy or a combination of strategies for a player superior to another strategy, so that the dominated strategy can be eliminated? By this process, if the game reduces to a small size, then how to solve it analytically? If the game size cannot be reduced to a small size, then how to formulate the game as a linear programming problem and solve accordingly? How to use the primal–dual relationship to obtain the answers? For this chapter, you should know the concept and calculation of expected value and an application of simplex method to solve linear programming problems, together with the primal–dual connection. The ideas and concepts to focus on in this chapter include developing a pay-off matrix using the given information, obtaining a saddle point if it exists, the reduction of game size by the principle of dominance and formulation of a game as an LPP (especially when some negative pay-offs are involved) and its solution by simplex algorithm. dŚĞŽƌǇŽĨ'ĂŵĞƐഩഩ Learning Objectives After reading this chapter, you should be able to: LO 1 LO 2 LO 3 LO 4 LO 5 LO 6 LO 7 LO 8 <ŶŽǁƚŚĞĐŽŵŵŽŶůǇŽďƐĞƌǀĞĚŐĂŵĞŵŽĚĞůƐ /ĚĞŶƟĨǇŚŽǁƚŽŽďƚĂŝŶŽƉƟŵĂůƐƚƌĂƚĞŐŝĞƐŽĨƚŚĞƉůĂǇĞƌƐŝŶĂ ƚǁŽͲƉĞƌƐŽŶǌĞƌŽͲƐƵŵŐĂŵĞ hŶĚĞƌƐƚĂŶĚƚŚĞĐŽŶĐĞƉƚŽĨ^ĂĚĚůĞWŽŝŶƚĂŶĚŝƚƐŝŵƉůŝĐĂƟŽŶ ĞƚĞƌŵŝŶĞƐŽůƵƟŽŶƚŽĂƚǁŽͲƉĞƌƐŽŶǌĞƌŽͲƐƵŵŐĂŵĞǁŝƚŚŶŽƐĂĚĚůĞƉŽŝŶƚ /ůůƵƐƚƌĂƚĞƚŚĞZƵůĞŽĨŽŵŝŶĂŶĐĞĂŶĚŝƚƐƵƟůŝƚǇ ŝƐĐƵƐƐƚŚĞƐŽůƵƟŽŶƚŽϮ¥nĂŶĚm¥ϮŐĂŵĞƐƵƐŝŶŐŐƌĂƉŚŝĐĂƉƉƌŽĂĐŚ ǆƉůĂŝŶƚŚĞĨŽƌŵƵůĂƟŽŶĂŶĚƐŽůƵƟŽŶŽĨĂŐĂŵĞĂƐĂŶ>WW ZĞǀŝĞǁƚŚĞůŝŵŝƚĂƟŽŶƐŽĨƚŚĞdŚĞŽƌǇŽĨ'ĂŵĞƐ ,1752'8&7,21 ! " # " ! # $ % % VWUDWHJLHV # ! SD\RIIV & ! 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D0 D6 \7 E0 E: ]0-:-- G-:-- 200 - 400 = 0.4 ( -100 + 200) - (400 + 200) 400 400 300 UPPER ENVELOPE 200 P 300 200 Q a3 100 100 0 0 a2 – 100 – 200 a1 – 100 – 200 + 4. D: D6 \7 E0 E: 6-:-- >:-:-- )LJXUH *UDSKLF6ROXWLRQWRWKH*DPH 200 - ( -200) = 0.8 (300 + 200) - ( -200 + 200) ! $ --0 % \0>\ -G£\£-; !Y7:-- ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ ([DPSOH)RUWKHIROORZLQJSD\RIIPDWUL[GHWHUPLQHWKHYDOXHRIWKHJDPHDQGWKHVWUDWHJLHVRISOD\HUV $DQG%XVLQJ D $OJHEUDLFPHWKRG E /LQHDUSURJUDPPLQJ 3OD\HU% 3OD\HU$ 6WUDWHJ\ í í ' % 1 1 1 1 1 1 ( - 3) + (2) = - < 1 (5) + ( - 4) = < 6 2 2 2 2 2 2 U % :¥: ' % 5 [ 7 D22 - D21 -4 - 5 9 = = (D11 + D22 ) - (D12 + D21 ) ( - 3 - 4) - (2 + 5) 14 \ 7 D22 - D12 -4 - 2 6 3 = = = (D11 + D22 ) - (D12 + D21 ) ( - 3 - 4) - (2 + 5) 14 7 9 7 (D11 ¥ D22 ) - (D12 ¥ D21 ) ( -3)( -4) - (2 ¥ 5) 1 = = (D11 + D22 ) - (D12 + D21 ) ( - 3 - 4) - (2 + 5) 7 # $ 820G,20G % -62@G2@ ]02@ 1 < N , ! ' . ( % < ( $ 0 : 6 0 ? : @ : 8 0- 0 1 [0[: $ 0: \0\:\6 % 0:6 ! = . dŚĞŽƌǇŽĨ'ĂŵĞƐഩഩ + $= . + %= . 0 7;0I;: 8 H H' <" ?;0I8;:≥0 <" 0 7<0I<:I<6 9 ?<0I:<:I@<6£0 :;0I0-;:≥0 8<0I0-<:I<6£0 @;0I;:≥0 <0<:<6≥- ;0;:≥- < \L 9 ;0 [08;: [: 8 1 %= ! 606: . 0 H' 7 <0I<:I<6I-60I-6: 9 <" ?<0I:<:I@<6I6070 8<0I0-<:I<6I6:70 <0<:<6606:≥! !;,>;; TABLE 8.5 %DVLV Simplex Tableau 1: Non-optimal Solution <0 < : < 6 6 0 6: E L EL DLM 60 - ? : @ 0 - 0 02? 6: - 8E 0- 0 - 0 0 028 Ä FM 0 0 0 - - < - - - 0 0 0 0 0 - - DM ≠ TABLE 8.6 %DVLV Simplex Tableau 2: Non-optimal Solution <0 < : < 6 60 6: E L EL DLM 0208 Ä 60 - - >0G26 0826E 0 >:26 026 <0 0 0 0-28 028 - 028 028 FM 0 0 0 - - < 028 - - 026 - - >028 ;28 - >028 DM ≠ 0 ഩഩYƵĂŶƟƚĂƟǀĞdĞĐŚŶŝƋƵĞƐŝŶDĂŶĂŐĞŵĞŶƚ TABLE 8.7 %DVLV Simplex Tableau 3: Non-optimal Solution < 0 < : < 6 6 0 6 : E L EL DLM > <6 0 - >0G208 0 6208 >:208 0208 <0 0 0 ?;2,@E - >02,@ @2,@ :208 FM 0 0 0 - - < :208 - 0208 - - DM - 602,@ - >;2,@ >02,@ ≠ TABLE 8.8 %DVLV 626GÄ Simplex Tableau 4: Optimal Solution < 0 < : < 6 6 0 6: EL <6 0 G:2?; - 0 ,26G >026G :20@ <: 0 ,@2?; 0 - >02?; @2?; 626G FM 0 0 0 - - - 626G :20@ - - >602?; - - >82?; >,2?; < DM ! <0<:<6 -626G:20@ + 0 3 2 7 7-I + = 9 34 17 34 ! 976G2@< 6 ) 9>6 6G2@>67062@+ \L7<L9\07-¥6G2@7-\:7626G 6G2@ 762@ \67:20@ 6G2@ 7G2@ ! ;0;: DM < '!G+ ;0 7 82?; ;: 7 ,2?; + 028 7 82?; I ,2?; 7 @26G ! 8 7 9 7 6G2@ ! [0782?; 6G2@ 7820G[:7,2?; 6G2@ 7,2

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