Chapter 11 Supplement Operational Decision-Making Tools: Transportation and Transshipment Models Operations Management - 6hh Edition Roberta Russell & Bernard W. Taylor, III Copyright 2006 John Wiley & Sons, Inc. Beni Asllani University of Tennessee at Chattanooga Just how do you make decisions? Emotional direction Intuition Analytic thinking Are you an intuit, an analytic, what??? How many of you use models to make decisions?? Copyright 2006 John Wiley & Sons, Inc. Supplement 10-2 Problems Arise whenever there is a perceived difference between what is desired and what is in actuality. Problems serve as motivators for doing something Problems lead to decisions 42 Copyright 2006 John Wiley & Sons, Inc. Supplement 10-4 Model Classification Criteria Purpose Perspective Use the perspective of the targeted decision-maker Degree of Abstraction Content and Form Decision Environment {This is what you should start any modeling facilitation meeting with} Copyright 2006 John Wiley & Sons, Inc. Supplement 10-5 Purpose Planning Forecasting Training Behavioral research Copyright 2006 John Wiley & Sons, Inc. Supplement 10-6 Perspective Descriptive “Telling it like it is” Most simulation models are of this type Prescriptive “Telling it like it should be” Most optimization models are of this type Copyright 2006 John Wiley & Sons, Inc. Supplement 10-7 Degree of Abstraction Isomorphic One-to-one Homomorphic One-to-many Copyright 2006 John Wiley & Sons, Inc. Supplement 10-8 Content and Form verbal descriptions mathematical constructs simulations mental models physical prototypes Copyright 2006 John Wiley & Sons, Inc. Supplement 10-9 Decision Environment Decision Making Under Certainty TOOL: all of mathematical programming Decision Making under Risk and Uncertainty TOOL: Decision analysis--tables, trees, Bayesian revision Decision Making Under Change and Complexity TOOL: Structural models, simulation models Copyright 2006 John Wiley & Sons, Inc. Supplement 10-10 Mathematical Programming Linear programming Integer linear programming some or all of the variables are integer variables Network programming (produces all integer solutions) Nonlinear programming Dynamic programming Goal programming The list goes on and on Geometric Programming Copyright 2006 John Wiley & Sons, Inc. Supplement 10-11 A Model of this class What would we include in it? Copyright 2006 John Wiley & Sons, Inc. Supplement 10-12 Management Science Models A QUANTITATIVE REPRESENTATION OF A PROCESS THAT CONSISTS OF THOSE COMPONENTS THAT ARE SIGNIFICANT FOR THE PURPOSE BEING CONSIDERED Copyright 2006 John Wiley & Sons, Inc. Supplement 10-13 Mathematical programming models covered in Ch 11, Supplement Transportation Model Transshipment Model Not included are: Shortest Route Minimal Spanning Tree Maximal flow Assignment problem many others Copyright 2006 John Wiley & Sons, Inc. Supplement 10-14 Transportation Model A transportation model is formulated for a class of problems with the following characteristics a product is transported from a number of sources to a number of destinations at the minimum possible cost each source is able to supply a fixed number of units of product each destination has a fixed demand for the product Solution (optimization) Algorithms stepping-stone modified distribution Excel’s Solver Copyright 2006 John Wiley & Sons, Inc. Supplement 10-15 Transportation Method: Example Copyright 2006 John Wiley & Sons, Inc. Supplement 10-16 Transportation Method: Example Copyright 2006 John Wiley & Sons, Inc. Supplement 10-17 Problem Formulation Using Excel Total Cost Formula Copyright 2006 John Wiley & Sons, Inc. Supplement 10-18 Using Solver from Tools Menu Copyright 2006 John Wiley & Sons, Inc. Supplement 10-19 Solution Copyright 2006 John Wiley & Sons, Inc. Supplement 10-20 Modified Problem Solution Copyright 2006 John Wiley & Sons, Inc. Supplement 10-21 The Underlying Network Copyright 2006 John Wiley & Sons, Inc. Supplement 10-22 For problems in which there is an underlying network: There are easy (fast) solutions An exception is the traveling salesman problem The solutions are always integer ones {How about solving a 50,000 node problem in less than a minute on a laptop??} Copyright 2006 John Wiley & Sons, Inc. Supplement 10-23 CARLTON PHARMACEUTICALS Carlton Pharmaceuticals supplies drugs and other medical supplies. It has three plants in: Cleveland, Detroit, Greensboro. It has four distribution centers in: Boston, Richmond, Atlanta, St. Louis. Management at Carlton would like to ship cases of a certain vaccine as economically as possible. Copyright 2006 John Wiley & Sons, Inc. Supplement 10-24 Data Unit shipping cost, supply, and demand From From Cleveland Cleveland Detroit Detroit Greensboro Greensboro Demand Demand Boston Boston $35 $35 37 37 40 40 1100 1100 Richmond Richmond 30 30 40 40 15 15 400 400 To To Atlanta Atlanta 40 40 42 42 20 20 750 750 St. St.Louis Louis 32 32 25 25 28 28 750 750 Supply Supply 1200 1200 1000 1000 800 800 Assumptions Unit shipping cost is constant. All the shipping occurs simultaneously. The only transportation considered is between sources and destinations. Total supply equals total demand. Copyright 2006 John Wiley & Sons, Inc. Supplement 10-25 Sources NETWORK REPRESENTATI ON Destinations D1=1100 Boston Cleveland S1=1200 Richmond D2=400 Detroit S2=1000 Atlanta D3=750 Greensboro S3= 800 Copyright 2006 John Wiley & Sons, Inc. St.Louis D4=750 Supplement 10-26 • The Associated Linear Programming Model The structure of the model is: Minimize <Total Shipping Cost> ST [Amount shipped from a source] = [Supply at that source] [Amount received at a destination] = [Demand at that destination] Decision variables Xij = amount shipped from source i to destination j. where: i=1 (Cleveland), 2 (Detroit), 3 (Greensboro) Copyright 2006 John Wiley & Sons, Inc. j=1 (Boston), 2 (Richmond), 3 (Atlanta),Supplement 10-27 Supply from Cleveland X11+X12+X13+X14 = 1200 Supply from Detroit X21+X22+X23+X24 = 1000 Supply from Greensboro X31+X32+X33+X34 = 800 The supply constraints Boston D1=1100 X11 Cleveland S1=1200 X12 X13 X21 X31 Richmond X14 X22 Detroit S2=1000 D2=400 X32 X23 X24 Atlanta X33 St.Louis Greensboro S3= 800 Copyright 2006 John Wiley & Sons, Inc. D3=750 X34 D4=750 Supplement 10-28 • The complete mathematical programming model Minimize 35X11+30X12+40X13+ 32X14 +37X21+40X22+42X23+25X24+ 40X31+15X32+20X33+38X34 ST Supply constrraints: X11+ X12+ X13+ X14 X21+ X22+ X23+ X24 X31+ X32+ X33+ X34 Demand constraints: X11+ X12+ X13+ X21+ X31 X22+ X32 X23+ X14+ X33 X24+ X34 = 1200 = 1000 = 800 = 1000 = 400 = 750 = 750 All Xij are nonnegative Copyright 2006 John Wiley & Sons, Inc. Supplement 10-29 Excel Optimal Solution CARLTON PHARMACEUTICALS UNIT COSTS BOSTON RICHMOND ATLANTA ST.LOUIS CLEVELAND $ 35.00 $ 30.00 $ 40.00 $ 32.00 DETROIT $ 37.00 $ 40.00 $ 42.00 $ 25.00 GREENSBORO $ 40.00 $ 15.00 $ 20.00 $ 28.00 DEMANDS 1100 400 750 750 SHIPMENTS (CASES) BOSTON RICHMOND ATLANTA ST.LOUIS CLEVELAND 850 350 0 0 DETROIT 250 0 0 750 GREENSBORO 0 50 750 0 TOTAL 1100 400 SUPPLIES 1200 1000 800 750 TOTAL 1200 1000 800 750 TOTAL COST = Copyright 2006 John Wiley & Sons, Inc. 84000 Supplement 10-30 WINQSB Sensitivity Analysis If this path is used, the total cost will increase by $5 per unit shipped along it Copyright 2006 John Wiley & Sons, Inc. Supplement 10-31 Shadow prices for warehouses - the cost resulting from 1 extra case of vaccine demanded at the warehouse Shadow prices for plants - the savings incurred for each extra case of vaccine available at the plant Copyright 2006 John Wiley & Sons, Inc. Supplement 10-32 Transshipment Model Copyright 2006 John Wiley & Sons, Inc. Supplement 10-33 Transshipment Model: Solution Copyright 2006 John Wiley & Sons, Inc. Supplement 10-34 Copyright 2006 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in section 117 of the 1976 United States Copyright Act without express permission of the copyright owner is unlawful. Request for further information should be addressed to the Permission Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages caused by the use of these programs or from the use of the information herein. Copyright 2006 John Wiley & Sons, Inc. Supplement 10-35 DEPOT MAX A General Network Problem Depot Max has six stores. Stores 5 and 6 are running low on the model 65A Arcadia workstation, and need a total of 25 additional units. Stores 1 and 2 are ordered to ship a total of 25 units to stores 5 and 6. Stores 3 and 4 are transshipment nodes with no demand or supply of their own. Copyright 2006 John Wiley & Sons, Inc. Supplement 10-36 Other restrictions There is a maximum limit for quantities shipped on various routes. There are different unit transportation costs for different routes. Depot Max wishes to transport the available workstations at minimum total cost. Copyright 2006 John Wiley & Sons, Inc. Supplement 10-37 • DATA: 20 10 1 7 3 5 Arcs: Upper bound and lower bound constraints: 6 5 12 0 X ij U ij 2 15 4 11 7 15 –Supply nodes: Network presentation 6 Net flow out of the node] nodes: = [Supply at the node] –Intermediate transshipment Transportation X12 + X13 X15node] - X21= =[Total 10 flow into the (Node 1) [Total flow out of+ the node] –Demand nodes: unit cost X21 - X12 = 15 (Node [Net flow into +the node] = [Demand for the node] X34+X35 =X24 X13 (Node 3) 2) X15 +X46 X35= +X65 X56 = 12 (Node 5) X24 +- X34 (Node 4) X46 +X56 - X65 = 13 (Node 6) Supplement 10-38 Copyright 2006 John Wiley & Sons, Inc. The Complete mathematical model Minimize 5X12 10X13 20X15 6X21 15X24 12X34 7X35 15X46 11X56 7X65 ST X12 + X13 + X15 - X21 - X12 = 10 + X21 + X24 - X13 = 15 + X34 + X35 - X24 - X15 - X34 = 0 + X46 - X35 = 0 + X56 - X65 = -12 - X46 - X56 + X65 = -13 0 X12 3; 0 X13 12; 0 X15 6; 0 X21 7; 0 X24 10; 0 X34 8; 0 X35 8; 0 X46 17; 0 X56 7; 0 X65 5 Copyright 2006 John Wiley & Sons, Inc. Supplement 10-39 WINQSB Input Data Copyright 2006 John Wiley & Sons, Inc. Supplement 10-40 WINQSB Optimal Solution Copyright 2006 John Wiley & Sons, Inc. Supplement 10-41 MONTPELIER SKI COMPANY Using a Transportation model for production scheduling Montpelier is planning its production of skis for the months of July, August, and September. Production capacity and unit production cost will change from month to month. The company can use both regular time and overtime to produce skis. Production levels should meet both demand forecasts and end-of-quarter inventory Supplement 10-42 Copyright 2006 John Wiley & Sons, Inc. Data: Initial inventory = 200 pairs Ending inventory required =1200 pairs Production capacity for the next quarter = 400 pairs in regular time. = 200 pairs in overtime. Holding cost rate is 3% per month per ski. Production Production Costs ProductionForecasted capacity, and forecasted demand Forecasted Production Production Costs for this Month Demand Capacity Month Demand Capacity Regular RegularTime Time Overtime Overtime quarter July 400 1000 25 30 Julypairs of skis), 400 and production 1000 25 per unit 30 (by (in cost August 600 800 26 32 August 600 800 26 32 months) September 1000 400 29 37 September 1000 400 29 37 Copyright 2006 John Wiley & Sons, Inc. Supplement 10-43 Analysis of demand: Net demand to satisfy in July = 400 - 200 = 200 pairs Initial inventory • Analysis of Unit costs Unit cost = [Unit production cost] + Net demand in August = 600 [Unit holding cost per month][the number of months stays in Net demand in September = 1000In+house 1200 = 2200 Forecasted demand inventory inventory] pairs Example: A unit produced in July in Regular time and sold in September costs 25+ (3%)(25)(2 months) = $26.50 Analysis of Supplies: Production capacities are thought of as supplies. There are two sets of “supplies”: Set 1- Regular time supply (production capacity) Supplement 10-44 Copyright 2006 John Wiley & Sons, Inc. Production Month/period 1000 800 July O/T Aug. R/T 25 25.75 26.50 0 30 30.90 31.80 +M 0 26 26.78 400 Aug. O/T Month sold July +M +M 32.96 200 Sept. R/T 0 0 Aug. 600 Sept. 2200 Dummy 300 +M 0 29 400 +M +M 32 200 Demand Production Capacity 500 July July R/T R/T Network representation 37 0 Sept. O/T Copyright 2006 John Wiley & Sons, Inc. Supplement 10-45 Source: July production in R/T Destination: July‘s demand. Unit cost= $25 (production) Source: Aug. production in O/T Destination: Sept.’s demand 32+(.03)(32)=$32.96 Unit cost =Production+one month holding cost Copyright 2006 John Wiley & Sons, Inc. Supplement 10-46 Copyright 2006 John Wiley & Sons, Inc. Supplement 10-47 Summary of the optimal solution In July produce at capacity (1000 pairs in R/T, and 500 pairs in O/T). Store 1500-200 = 1300 at the end of July. In August, produce 800 pairs in R/T, and 300 in O/T. Store additional 800 + 300 - 600 = 500 pairs. In September, produce 400 pairs (clearly in R/T). With 1000 pairs retail demand, there will be (1300 + 500) + 400 - 1000 = 1200 pairs available for shipment to Ski Chalet Inventory + .Production Copyright 2006 John Wiley & Sons, Inc. Demand Supplement 10-48 Problem 4-25 Copyright 2006 John Wiley & Sons, Inc. Supplement 10-49 Copyright 2006 John Wiley & Sons, Inc. Supplement 10-50 Copyright 2006 John Wiley & Sons, Inc. Supplement 10-51 Copyright 2006 John Wiley & Sons, Inc. Supplement 10-52 Copyright 2006 John Wiley & Sons, Inc. Supplement 10-53 6.3 The Assignment Problem Problem definition m workers are to be assigned to m jobs A unit cost (or profit) Cij is associated with worker i performing job j. Minimize the total cost (or maximize the total profit) of assigning workers to job so that each worker is assigned a job, and each job is performed. Copyright 2006 John Wiley & Sons, Inc. Supplement 10-54 BALLSTON ELECTRONICS Five different electrical devices produced on five production lines, are needed to be inspected. The travel time of finished goods to inspection areas depends on both the production line and the inspection area. Management wishes to designate a separate inspection area to inspect the products such that the total travel time is minimized. Copyright 2006 John Wiley & Sons, Inc. Supplement 10-55 Data: Travel time in minutes from assembly lines to inspection areas. Assembly Assembly Lines Lines 11 22 33 44 55 AA 10 10 11 11 13 13 14 14 19 19 Copyright 2006 John Wiley & Sons, Inc. BB 44 77 88 16 16 17 17 Inspection Inspection Area Area CC 66 77 12 12 13 13 11 11 DD 10 10 99 14 14 17 17 20 20 EE 12 12 14 14 15 15 17 17 19 19 Supplement 10-56 NETWORK REPRESENTATION Assembly Line S1=1 1 Inspection Areas A D1=1 S2=1 2 B S3=1 3 C D3=1 S4=1 4 D D4=1 S5=1 5 E D5=1 Copyright 2006 John Wiley & Sons, Inc. D2=1 Supplement 10-57 Assumptions and restrictions The number of workers equals the number of jobs. Given a balanced problem, each worker is assigned exactly once, and each job is performed by exactly one worker. For an unbalanced problem “dummy” workers (in case there are more jobs than workers), or “dummy” jobs (in case there are more workers Copyright 2006 John Wiley & Sons, Inc. Supplement 10-58 Computer solutions A complete enumeration is not efficient even for moderately large problems (with m=8, m! > 40,000 is the number of assignments to enumerate). The Hungarian method provides an efficient solution procedure. Special cases A worker is unable to perform a particular job. A worker can be assigned to more than one job. Copyright John Wiley & Sons, Inc. assignment problem. Supplement 10-59 A2006 maximization 6.5 The Shortest Path Problem For a given network find the path of minimum distance, time, or cost from a starting point, the start node, to a destination, the terminal node. Problem definition There are n nodes, beginning with start node 1 and ending with terminal node n. Bi-directional arcs connect connectedSupplement nodes Copyright 2006 John Wiley & Sons, Inc. 10-60i Fairway Van Lines Determine the shortest route from Seattle to El Paso over the following network highways. Copyright 2006 John Wiley & Sons, Inc. Supplement 10-61 1 180 Seattle 497 3 432 Portland 5 Sac. 599 691 Boise 420 4 138 Reno 6 345 Bakersfield 114 13 Los Angeles 621 Denver 9 Las Vegas 11 108 155 Barstow 14 452 Kingman 469 15 207 Albuque. Phoenix 386 425 Copyright 2006 John Wiley & Sons, Inc. 12 403 16 17 8 102 432 118 San Diego 440 7 526 280 Cheyenne Salt Lake City 291 10 Butte 2 Tucson 18 314 19 Supplement 10-62 El Paso Solution - a linear programming approach Decision variables 1 if a truck travels on the highway from city i to city j X ij 0 otherwise Objective = Minimize S dijXij Copyright 2006 John Wiley & Sons, Inc. Supplement 10-63 Subject to the following constraints: 1 180 Seattle 497 3 432 Portland Butte 599 2 Boise 4 345 Salt Lake City 7 [The number of highways traveled out of Seattle (the start node)] = 1 X12 + X13 + X14 = 1 In a similar manner: [The number of highways traveled into El Paso (terminal node)] = 1 X12,19 + X16,19 + X18,19 = 1 [The number of highways used to travel into a city] = [The number of highways traveled leaving the city]. For example, in Boise (City 4): X14 + X34 +X74 = X41 + X43 + X47. Supplement 10-64 Nonnegativity constraints Copyright 2006 John Wiley & Sons, Inc. WINQSB Optimal Solution Copyright 2006 John Wiley & Sons, Inc. Supplement 10-65 Solution - a network approach The Dijkstra’s algorithm: Find the shortest distance from the “START” node to every other node in the network, in the order of the closet nodes to the “START”. Once the shortest route to the m closest node is determined, the shortest route to the (m+1) closest node can be easily determined. This algorithm finds the shortest route from the start to all the nodes in the network. Copyright 2006 John Wiley & Sons, Inc. Supplement 10-66 An illustration of the Dijkstra’s algorithm + 420 SLC.= SLC 599 BUT. BUT 691 + CHY. = 345 = + SLC SLC. SLC 497 SEA. BOI BOI BOI. 1 Seattle 497 180 138 POR. POR 180 180 + 432 = BOIBOI 2 Butte 691 Boise 420 3 432 Portland 5 Sac. 599 4 Reno 6 345 Salt Lake City 440 7 526 621 291 10 Bakersfield Denver 9 … and so on until the Kingman Barstow whole network 15 12 14 Albuque. isPheonix covered. 11 Las Vegas 280 108 452 155 469 207 Copyright 2006 John Wiley & Sons, Inc. 8 102 432 114 + 602 = SACSAC. Cheyene 13 Los Angeles 386 San Diego 17 403 16 118 425 Tucson 18 19 314 Supplement 10-67 El Paso 6.6 The Minimal Spanning Tree This problem arises when all the nodes of a given network must be connected to one another, without any loop. The minimal spanning tree approach is appropriate for problems for which redundancy is expensive, or the flow along the arcs is considered Copyright 2006 John Wiley & Sons, Inc. Supplement 10-68 instantaneous. THE METROPOLITAN TRANSIT DISTRICT The City of Vancouver is planning the development of a new light rail transportation system. The system should link 8 residential and commercial centers. The Metropolitan transit district needs to select the set of lines that will connect all the centers at a minimum total cost. Copyright 2006 John Wiley & Sons, Inc. Supplement 10-69 SPANNING TREE NETWORK North Side PRESENTATION 3 34 University 50 5 Business District 39 4 West Side 45 1 8 35 2 City Center 41 7 Copyright 2006 John Wiley & Sons, Inc. 6 Shopping Center East Side South Side Supplement 10-70 Solution - a network approach The algorithm that solves this problem is a very easy (“trivial”) procedure. It belongs to a class of “greedy” algorithms. The algorithm: Start by selecting the arc with the smallest arc length. At each iteration, add the next smallest arc length to the set of arcs already selected (provided no loop is constructed). Finish when all nodes are connected. Copyright 2006 John Wiley & Sons, Inc. Supplement 10-71 WINQSB Optimal Solution Copyright 2006 John Wiley & Sons, Inc. Supplement 10-72 OPTIMAL SOLUTION NETWORK REPRESENTATION 3 North Side 34 West Side University 50 5 Business District 39 4 45 Loop 1 8 35 2 City Center 41 6 Shopping Center East Side Total Cost = $236 million 7 Copyright 2006 John Wiley & Sons, Inc. South Side Supplement 10-73