CONCEPT QUIZ State True or False. 1. A linear programming problem is infeasible if there are any slack variables in the solution. 2. A solution becomes degenerate whenever there is a tie in the row to be replaced. 3. An optimal solution to a maximisation problem is arrived at when all values in the net evaluation row are positive or zero. 4. An optimal solution to a minimisation problem is arrived at when all values hi the net evaluation row are positive or zero. 5. The number of constraints in the primal is the same as the number of decision variables in the dual. 6. Each constraint, excluding non-negativity constraints, in the mathematical formulation of a linear programme generates one row in the simplex tableau. 7. Every inequality constraint in a linear programme adds exactly one variable to the problem. 8. All the rules and procedures of the simplex method are identical whether solving maximization or a minimization problem. 9. There should be no artificial variables in the final solution. 10. A maximization problem can be easily solved with the Two-phase method, if all constraints are of "less than equal to" type. Tick the correct answer/answers. 1. If the solution contains a variable that has a value of zero the problem is (a) Infeasible. (b) Unbounded. (c) Degenerate. (d) None of the above. 2. Shadow price is: (a) The cost of a unit of a resource. (b) The contribution of one unit of a resource towards the objective function. (c) The amount by which a resource can be increased without changing the solution. (d) The contribution of one unit of a scarce resource towards the objective function. 3. In the case of a maximization problem, the incoming variable has: (a) A value of zero in the net evaluation row. (b) A positive value in the Zj row. (c) The least negative value in the net evaluation row. (d) The highest positive value in the net evaluation row. 4. The outgoing variable in a simplex pivot operation is the variable with: (a) The least replacement ratio. (b) A negative replacement ratio. (c) The maximum positive replacement ratio. (d) The least positive replacement ratio. 5. In a maximization problem, if a constraint is of the 'greater than or equal to' type, the artificial variable is assigned: (a) A very large value. (b) A very small value. (c) A very large positive value, i.e. M. (d) A very large negative value, i.e. -M. 6. Which of the following statements is true about a primal linear programme and its corresponding dual: (a) The optimal value for the primal is greater than that of the dual. (b) The shadow prices of the primal at optimality are the values of the structural variables of the dual at optimality. (c) Each problem does not have a dual. (d) Both the primal and the dual are either maximisation or minimisation problems. 7. Right-hand side ranging is used to determine: (a) The availability of resources. (b) The requirement of resources. (c) The range of the resource availability over which the solution remains the same. (d) The range of the resource availability over which the shadow prices remain constant. Right-hand side ranging can 8. be determined by: (a) Dividing the Cj - Zj row with the Zj row. (b) Dividing the quantity column by the column representing the slack of the variable whose range we are determining. (c) Dividing the quantity column with the column of the incoming variable. (d) None of the above. 9. In the simplex procedure: (a) Each successive tableau presents a solution superior to the one preceding it. The number of tableaus is determined by the number of constraints. (b) The number of tableaus is determined by the number of variables. None of (c) the above. (d) 10. Artificial variables are required: (a) Only in a minimisation problem. (b) Only in a maximisation problem. (c) To convert inequalities of the 'greater than or equal to' type into equalities. (d) To convert inequalities of the 'less than or equal to' type into equalities. QUESTIONS 1. Explain various steps of the simplex method involved in the computation of an optimal solution to a linear programming problem. 2. Define slack and surplus variables in a linear programming 3. problem. Explain the primal dual relationship. 4. Explain the meaning of basic feasible solution and degenerate solution in a linear programming problem. 5. Explain the use of artificial variable in linear programming. 6. What do you understand by shadow prices? What is the managerial implication of shadow prices? 7. Health Centre is a clinic specializing in four types of patient care: dermatology cosmetic surgery, obstetrics and neurosurgery. A patient in each of these specialties contributes Rs 4000, 6000, 8000 and 10000, respectively to profits of the clinic. Past data reveals the following time requirements limitations: Hours required per patient Speciality Lab XRay Therapy Surgery Doctonl Dermatology 5 8 10 8 14 j Cosmetics Obstetrics Neurosurgery Hours available per week 5 3 2 200 2 1 4 140 1 0 8 110 4 10 8 12 320 6 16 240 The doctors have access to as many patients as they wish. They have limited their cosmetic and neurosurgery practice to a total of 150 hours a week. Determine the optimal patient mix on a weekly basis. 8. The Annual Handmade Furniture Show and Sale occurs next month and the School for Vocational Training is planning to make furniture for the sale. There are three wood working classes—I year, II year and III year, at the school and they have decided to make three styles of chairs—A, B and C. Each chair must receive work in each class and the time in hours required for each chair in each class is given below: Chair I Year A 2 3 2 B // Year 4 3 1 /// Year 3 2 4 During the next month there will be 120 hours in the I Year class, 160 hours in the II Year class and 100 hours in the III Year class. The teacher of the wood working classes feels that a maximum of 40 chairs can be sold at the show. The teacher has determined that the profit from each type of chair will be: A, Rs 400; B, Rs 350 and C, Rs 300. Determine the optimal mix of chairs using the simplex method. 9. A furniture company can produce 4 types of chairs. Each chair is first made in the carpentry shop and then varnished, waxed and polished in the finishing shop. Man hours required in each shop are: Chair type Shop 7 2 3 4 Carpentry 4 9 7 10 Finishing Contribution 1 120 1 200 3 180 40 400 Total number of man hours available per month in Carpentry and Finishing shops are 6000 and 4000, respectively. Assuming abundant supply of raw materials and unlimited demand for finished products, determine the number of each type of chair to be produced for profit maximisation, using the simplex method. 10. A factory engaged in the manufacture of pistons, rings and valves, for which the profits per unit are Rs 10, 6 and 4 respectively, wants to decide the most profitable mix. It takes 1 hour of preparatory work, 10 hours of machining and 2 hours of packing and allied formalities for a piston. Corresponding requirements for rings and valves are 1, 4 and 2 and 1, 5 and 6 hours, respectively. The total number of hours available for preparatory work, packing and allied formalities is 100, 600 and 300, respectively. Determine the most profitable mix, assuming that whatever is produced can be sold. 11. Three products A, B and C are produced at three machining centres X, Y and Z Each product involves operations at each of the machine centres. The time required for each operation on various products is given in the table below: Machine centres Product X Y Z Profit per unit A B C Available hours 10 2 1 100 7 3 2 77 2 4 1 80 12 3 1 (a) Formulate a linear programming problem on the basis of the above information. (b) Find a suitable product mix so as to maximise profit. (c) Show that the total available hours of X and Y have been fully utilized and there is surplus hours of Z. 12. A company manufactures three models of cars. There is a backlog of orders with the company. Model A requires 60, 100 and 80 worker-days in three production processes. Model B requires 100, 240 and 100 worker-days. Model C requires 200, 360 and 160 worker-days, respectively in the three production processes. The number of workers employed in the three production processes is 15, 30 and 15 respectively, and an average worker is on the job for 200 working days a year. The expected profit for each model is Rs 7500, Rs 15000 and Rs 30000 respectively. With this capacity determine the company's optimal product-mix and total maximum profit. If there is any other optimal product mix, identify that too. 13. A manufacturer has to decide how much finishing to perform on his product prior to sale. He may sell his product as: (a) Raw castings—5 hours for casting, 1 hour for machining and 1 hour for plating. (b) Semi finished—5 hours of casting, 4 hours of machining and 2 hours of plating. (c) Finished—5 hours of casting, 4 hours of machining, and 4 hours of plating. The profit per unit of sale is: Raw casting—Rs 150 per unit; semi finished— Rs 250 per unit and finished—Rs 350 per unit. The manufacturer can sell all the items that he can produce. The weekly production capacity of the casting department is 130 hours. The machining department has a capacity of 88 hours and the plating department has a capacity of 40 hours per week. What should the product mix be to maximise profits? 14. A trucking company with Rs 4000000 to spend on new equipment is contemplating three types of vehicles. Vehicle A has a 10 ton payload and is expected to average 35 km per hour. It costs Rs 80000. Vehicle B has a 20 ton payload and is expected to average 30 km per hour. It costs Rs 130000. Vehicle C is a modified form of Vehicle B; it carries sleeping quarters for one driver and that reduces its carrying capacity to 18 tons and raises the cost to Rs 150000. Vehicle A requires a crew of one man, and if driven on three shifts per day, could be run for an average of 18 hours per day. Vehicles B and C require a crew of two men each, but whereas B would be driven 18 hours per day with three shifts, C would average 21 hours per day. The company has 150 drivers available each day and would find it difficult to obtain further crews. Maintenance facilities are such that the total number of vehicles must not exceed 30. How many vehicles of each type should be purchased if the company wishes to maximise its capacity in ton-km per day? 15. A firm makes two products X and Y and has total production capacity of 9 tons per day, X and Y requiring the same production capacity. The firm has a permanent contract to supply at least 2 tons of X and at least 3 tons of Y per day to another company. Each ton of X requires 20 machine hours of production time and each ton of Y requires 50 machine hours of production time; the daily maximum possible number of machine hours is 360. All the firms output can be sold, and the profit made is Rs 80 per ton of X and Rs 120 per ton of Y. It is required to determine the production schedule for maximum profit and to calculate this profit. 16. The ABC Candy Company makes three different types of candy bars. The ingredients, in grams, for each candy bar are as follows: Candy bar Per unit Chocolate Nuts Caramel contribution Weekly demand A Rs. 2.00 12 4 15 900 B C Availability Rs. 2.50 Rs. 1.50 6 10 25000 10 2 15000 8 15 30000 Very large Very large What should be the product mix to maximise the profits? 17. The ABC Company sells two types of porch furniture, gliders and chairs. It makes a profit of Rs 100 on each glider and Rs 40 on each chair. Each glider requires 40 square feet of display space and each chair requires 25 square feet of display space. It takes 1% hours to assemble a glider and 2/3 hours to assemble a chair. ABC has 900 square feet space and 30 hours of labour available for assembly. The sales manager wants at least two chairs displayed for every glider displayed. Formulate and solve the LP model. 18. Mr. Jain, the marketing manager of ABC Typewriter Company, is trying to decide how to allocate his salesmen to the company's three primary markets. Market 1 is in an urban area and salesmen can sell on the average 40 typewriters per week. Salesmen in the other two markets can sell on the average, 36 and 25 typewriters per week, respectively. For the coming week three of the salesmen will be on leave, leaving only 12 men for duty. Also because of lack of company cars, maximum of 5 salesmen can be allocated to Market 1. The selling expenses for salesmen per week for salesmen in each area are Rs 800 for Market 1 and Rs 700 and Rs 500 per week for Markets 2 and 3, respectively. The budget for the next week is Rs 7000. The profit margin per typewriter is Rs 150. Determine how many salesmen should be assigned to each area so as to maximise the profits. 19. A certain manufacturer of screw fastenings found that there was a market for packages of mixed screw sizes. His market research data indicated that two mixtures of three screw types (1, 2 and 3) properly priced would be most acceptable to the buyer. The relevant data is: Mixture Specifications Selling price > 50% Type 1 A < 30% Type 2 any quantity of Type 3 B Rs 5 per kg > 35% of Type 1 < 45% of Type 2 any quantity of Type 3 Rs 4 per kg For these screws, plant capacity and manufacturing costs are given below: Screw type Plant capacity (kg /day Manufacturing cost x 100) (Rs/kg) 1 10 4.50 2 3 10 6 3.50 2.70 What production should the manufacturer schedule for greatest profits assuming that he can sell all that he manufactures? 20. A piano manufacturer manufactures three types of pianos. The data below give the production hours per unit in each of the three operations, maximum time available and profit per unit. Operations (hrs) Profit per unit Piano types 1 2 3 (OOs of Rs) A B 2 4 5 2 5 2 30 40 C Max time available 2 600 3 400 10 900 20 How many units of each type of piano should be produced to maximise the total profit? Write the dual and use it to check the optimal solution. 21. Four products have to be processed through the plant. There are three production lines on which the products could be processed. The rates of production in units per day, the total available capacity and the demand are given in the following table: Product IVj Production line 1 2 3 A 150 100 500 4 400 B 200 100 760 400 20 C 160 80 890 600 18 Demand 2000 3000 3000 6000 tax. line capacity (days) 20 Formulate the above as a linear programming problem to minimise the cost of operations, if Line A requires Rs 3000 per day, Line B Rs 5000 per day and Line C Rs 4000 per day. Solve using suitable software package. 22. Formulate and solve the dual of Question 21. 23. Product A offers a profit of Rs 25 per unit and product B yields a profit of Rs 40 per unit. To manufacture the products, leather, wood and glue are required in the amount as shown below: Resources required for one unit Product Leather (kg) Wood (sq m) Glue (litres) A B Availability 0.50 0.25 2200 4 7 28000 0.2 0.2 1400 (a) Formulate the LP model and find the optimal solution. (b) Which resources are fully consumed? How much of each resource remains unused? (c) What are the shadow prices of the resources? 24. A company manufactures two different kinds of machines, each requiring a different manufacturing technique. The Deluxe machine requires 18 hours of labour, 9 hours of testing, and yields a profit of Rs. 400. The Standard machine requires 3 hours of labour, 4 hours of testing, and yields a profit of Rs. 200. There are 800 hours of labour and 600 hours of testing available each month. A marketing forecast has shown that the monthly demand for the Standard machine to be no more than 250. Management wants to know the number of each model to produce monthly that will maximise total profits. Formulate and solve this as a linear programming problem. 25. A metal product company produces waste cans, filing cabinets, file boxes for correspondence and lunch boxes. Its inputs are sheet metals of two different thicknesses, called A and B and manual labour. Input output relationships for the company are shown in the table. Sheet metal A Sheet metal B Labour Waste cans Filing cabinets Correspondence boxes Lunch boxes 6 0 4 0 10 8 2 0 2 3 0 3 The sales revenue per unit of waste cans, filing cabinets, correspondence boxes and lunch boxes are Rs 20, Rs 400, Rs 90 and Rs 20, respectively. There are 225 units of Sheet metal A available in the company's inventory, 300 of Sheet metal B, and 190 units of labour. What is the company's optimal sales revenue? 26. A knitting machine can produce 1000 trousers or 3000 shirts (or a combination of the two) each day. The finishing department can handle either 1500 trousers or 2000 shirts (or a combination of the two) each day. The marketing department requires that at least 400 trousers be produced each day. The company's objective is profit maximisation. If the profit from the trousers is Rs 40 and that from a shirt is Rs 15, how many of each type should be produced? 118 Quantitative Techniques for Decision Making 27. A manufacturing company makes three products, each of which requires three operations as part of the manufacturing process. The company can sell all of the products it can manufacture but its production capacity is limited by the capacity of its operations centres. Additional data concerning the company is as follows: Manufacturing requirements hour/unit Cost Selling price (Rs) (Rs) Product Centre 1 Centre 2 Centre 3 A 1 3 2 11 15 B 3 4 1 12 20 C 2 2 2 10 16 Hours available 160 120 80 - What should the product mix be? Write the dual of the given problem. 28. Three products A, B and C are to be manufactured. Product A requires 2.4 minutes of punch press time and 5.0 minutes of assembly time. Product B requires 3.0 minutes of punch press time and 2.5 minutes of welding time. Product C requires 2.0 minutes of punch press time and 1.5 minutes of welding time and 2.5 minutes of assembly time. Profit per unit of A, B and C is Re 0.60, Re 0.70 and Re 0.50, respectively. The capacity of the punch press is 20 hours per week, of welding 10 hours per week and of assembly section 25 hours per week. Find the optimal feasible solution and the maximum profit. Also determine the shadow prices of the three resources. 29. A chemical manufacturing company produces two products A and B. Each product passes through three processes. The processing time in hours for each of the two products in each process is given below: Process Product A Product B 1 2 5 2 7 2 3 4 3 The total hours available for each process in a week are 30, 24 and 20, respectively. Product A gives a profit of Rs 18 per unit and one unit of Product B gives Rs 15. Find the quantities of A and B to be produced in the next week so as to maximise profits. A timber merchant manufactures three types of plywood. The data given in the following table shows the production hours per unit in each of three production operations, maximum time available and profits per unit. Operations (hrs) Plywood / II III Profit per unit (Rs) Grade A 2 2 4 40 Grade B Grade C Max time available 5 10 900 5 3 400 2 2 600 30 20 How many units of each grade of plywood should be produced to maximise the total profit? 31. A local travel agent is planning a charter trip to a major sea resort. The eight day seven night package includes the fare for the round trip travel, surface transportation, board and lodging and selected tour options. The charter trip is restricted to 200 persons and past experience indicates that there will be no problem in getting 200 persons. The problem for the travel agent is to determine the number of Deluxe, Standard and Economy tour packages to offer for this charter. These three plans each differ according to seating and service on the flight, quality of accommodation, meal plans and tour options. The following table summarises the estimated price for the three packages and the corresponding expenses for the travel agent per person. The travel agent has hired an aircraft for a flat fee of Rs 200000 for the entire trip. Tour plan Price (Rs) Hotel costs (Rs) Meals and other expenses (Rs) Deluxe 10000 3000 4750 Standard Economy 7000 6500 2200 1900 2500 2200 In planning the trip the following considerations must be taken into account: (a) At least 10 per cent of the packages must be of the deluxe type. (b) At least 35 per cent but not more than 70 per cent must be of the Standard type. (c) At least 30 per cent must be of the Economy type. (d) The maximum number of Deluxe packages available in any aircraft is restricted to 60. (e) The hotel desires that at least 120 tourists should be on the Deluxe and Standard packages together. Determine the number of packages to offer in each type so as to maximise profits. 32. XYZ Company Y\as three departments—assembly, painting and packing, and Can make three types of almirahs. An almirah of Type 1 requires one hour assembly, 40 minutes of painting and 20 minutes of packing time, respectively Similarly almirah of Type II 80 minutes, 20 minutes and 1 hour, respectively. The Type III requires 40 minutes each of assembly, painting and packing time .The total time available at assembly, painting and packing departments is 600 hours, 400 hours and 800 hours, respectively. Type I, Type II and Type III almirahs yield Rs 1000, 1500 and 2000 as profit respectively. (a) Determine the number of almirahs of each type that should be produced in order to maximise the profits. (b) If XYZ Company decides to rent out its capacity to another almirah manufacturer—ABC Company, what should it charge as rental rates? 34. A company makes three products X, Y and Z, which flow through three departments: Drill, Lathe and Assembly. The hours required by each product in each department, the profit contribution of the products and the total time available in each department are shown in the table below: Time required per unit Product Drill Lathe Assembly Profit per unit X 3 3 8 9 Y 6 5 10 15 Z 7 4 12 20 Hours available 210 240 260 The marketing department indicates that the sales potential for Products X and Y is unlimited but for the Product Z it is only 30 units. Determine the optimal j product mix. 35. A manufacturer produces four products, A, B, C and D, each of which is processed on three machines, X, Y and Z. The time required for manufacturing one unit of each of the four products and capacity of each machine is indicated as follows: Product A B C D Capacity Processing time in hrs Profit per unit Machine X Machine Y Machine Z 1.5 2 4 3 550 4 1 2 1 700 2 3 1 2 200 4 6 3 1 (a) What is the optimal product mix? What is the maximum profit? (b) Which machine (s) has(have) excess capacity? How much? (c) If the profit contribution from Product B increases by Rs 2 per unit, will the optimal product mix change? (d) If machine Y is to be shut down for 50 hours due to repairs, will the product mix change? (e) What are the shadow prices of the machine hours on the three machines? 36. ABC Foods Company is developing a low calorie high protein diet supplement called Hi-Pro. The specifications for Hi-Pro have been established by a panel of medical experts. These specifications along with the calorie, protein and vitamin content of the three basic foods are given in the following table: Nutritional elements Units of nutritional elements (per 100 gm basic food) Food 1 Food 3 Hi-Pro Calories 350 250 Food 2 200 300 Proteins Vitamin A Vitamin B 250 100 75 300 150 125 150 75 150 200 100 100 Cost per serving (Rs) 1.50 2.00 1.20 What quantities of Food 1, Food 2 and Food 3 should be used?