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Chapter 6: Transportation, Assignment, and Transshipment Problems A network model is one which can be represented by a set of nodes, a set of arcs, and functions (e.g. costs, supplies, demands, etc.) associated with the arcs and/or nodes. Examples include transportation, assignment, transshipment as well as shortest-route, maximal flow problems, minimal spanning tree and PERT/CPM problems. All network problems can be formulated as linear programs. However, there are many computer packages that contain separate computer codes for these problems which take advantage of their network structure. If the right-hand side of the linear programming formulations are all integers, then optimal solution of the decision variables will also be integers. © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 1 Transportation Problem The transportation problem seeks to minimize the total shipping costs of transporting goods from m origins (each with a supply si) to n destinations (each with a demand dj), when the unit shipping cost from an origin, i, to a destination, j, is cij. The network representation for a transportation problem with two sources and three destinations is given on the next slide. © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 2 Transportation Problem Network Representation s1 s2 1 c11 c23 m Sources d1 2 d2 3 d3 c12 c13 c21 2 1 c22 n Destinations © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 3 Transportation Problem Linear Programming Formulation Using the notation: xij = number of units shipped from origin i to destination j cij = cost per unit of shipping from origin i to destination j si = supply or capacity in units at origin i dj = demand in units at destination j continued © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 4 Transportation Problem Linear Programming Formulation (continued) m Min n c ij xij i 1 j 1 n x ij si i 1, 2, ,m Supply ij dj j 1, 2, ,n Demand j 1 m x i 1 xij > 0 for all i and j © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 5 Example: Transportation Problem The Navy has depots in Albany, BenSalem, and Winchester. Each of these three depots has 3,000 pounds of materials which the Navy wishes to ship to three installations, namely, San Diego, Norfolk, and Pensacola. These installations require 4,000, 2,500, and 2,500 pounds, respectively. The shipping costs per pound for are shown on the next slide. Formulate and solve a linear program to determine the shipping arrangements that will minimize the total shipping cost. © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 6 Example: Transportation Problem (Continued) Source Destination San Diego Norfolk Pensacola Albany BenSalem Winchester $12 20 30 $6 11 26 $5 9 28 © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 7 Transportation Problem: Network Representation Source Albany 3000 BenSalem 3000 1 Destination c11 c13 c21 2 c22 1 San Diego 4000 2 Norfolk 2500 3 Pensacola 2500 c23 c31 Winchester 3000 c12 3 c33 c32 © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 8 Example: Transportation Problem (Continued) Define the Decision Variables We want to determine the pounds of material, xij , to be shipped by mode i to destination j. The following table summarizes the decision variables: San Diego Norfolk Pensacola Albany x11 x12 x13 BenSalem x21 x22 x23 Winchester x31 x32 x33 © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 9 Example: Transportation Problem (Continued) Define the Objective Function Minimize the total shipping cost. Min: (shipping cost per pound for each mode per destination pairing) x (number of pounds shipped by mode per destination pairing). Min: 12x11 + 6x12 + 5x13 + 20x21 + 11x22 + 9x23 + 30x31 + 26x32 + 28x33 © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 10 Transportation Problem: Example #2 Define the Constraints Source availability: (1) x11 + x12 + x13 = 3000 (2) x21 + x22 + x23 = 3000 (3) x31 + x32 + x33 = 3000 Destination material requirements: (4) x11 + x21 + x31 = 4000 (5) x12 + x22 + x32 = 2500 (6) x13 + x23 + x33 = 2500 Non-negativity of variables: xij > 0, i = 1, 2, 3 and j = 1, 2, 3 © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 11 Example: Transportation Problem (Continued) Computer Output OBJECTIVE FUNCTION VALUE = 142000.000 Variable Value Reduced Cost x11 1000.000 0.000 x12 2000.000 0.000 x13 0.000 1.000 x21 0.000 3.000 x22 500.000 0.000 x23 2500.000 0.000 x31 3000.000 0.000 x32 0.000 2.000 x33 0.000 6.000 © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 12 Transportation Problem: Example #2 Solution Summary • San Diego will receive 1000 lbs. from Albany and 3000 lbs. from Winchester. • Norfolk will receive 2000 lbs. from Albany and 500 lbs. from BenSalem. • Pensacola will receive 2500 lbs. from BenSalem. • The total shipping cost will be $142,000. © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 13 Transportation Problem LP Formulation Special Cases • Total supply exceeds total demand: No modification of LP formulation is necessary. • Total demand exceeds total supply: Add a dummy origin with supply equal to the shortage amount. Assign a zero shipping cost per unit. The amount “shipped” from the dummy origin (in the solution) will not actually be shipped. Assign a zero shipping cost per unit © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. • Maximum route capacity from i to j: Slide 14 Transportation Problem LP Formulation Special Cases (continued) • The objective is maximizing profit or revenue: Solve as a maximization problem. • Minimum shipping guarantee from i to j: xij > Lij • Maximum route capacity from i to j: xij < Lij • Unacceptable route: Remove the corresponding decision variable. © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 15 Assignment Problem An assignment problem seeks to minimize the total cost assignment of m workers to m jobs, given that the cost of worker i performing job j is cij. It assumes all workers are assigned and each job is performed. An assignment problem is a special case of a transportation problem in which all supplies and all demands are equal to 1; hence assignment problems may be solved as linear programs. The network representation of an assignment problem with three workers and three jobs is shown on the next slide. © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 16 Assignment Problem Network Representation 1 Agents c11 c13 c21 2 1 c12 Tasks c22 2 c23 c31 3 c33 c32 3 © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 17 Assignment Problem Linear Programming Formulation Using the notation: xij = 1 if agent i is assigned to task j 0 otherwise cij = cost of assigning agent i to task j continued © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 18 Assignment Problem Linear Programming Formulation (continued) m Min n c ij xij i 1 j 1 n x ij 1 i 1, 2, ,m Agents ij 1 j 1, 2, ,n Tasks j 1 m x i 1 xij > 0 for all i and j © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 19 Example: Assignment Problem An electrical contractor pays his subcontractors a fixed fee plus mileage for work performed. On a given day the contractor is faced with three electrical jobs associated with various projects. Given below are the distances between the subcontractors and the projects. Subcontractor Westside Federated Goliath Universal Projects A B C 50 36 16 28 30 18 35 32 20 25 25 14 How should the contractors be assigned so that total mileage is minimized? © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 20 Example: Assignment Problem Network Representation West. Subcontractors 50 36 16 28 Fed. 18 35 Gol. Univ. 20 25 A Projects 30 B 32 C 25 14 © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 21 Assignment Problem: Example Linear Programming Formulation Min s.t. 50x11+36x12+16x13+28x21+30x22+18x23 +35x31+32x32+20x33+25x41+25x42+14x43 x11+x12+x13 < 1 x21+x22+x23 < 1 Agents x31+x32+x33 < 1 x41+x42+x43 < 1 x11+x21+x31+x41 = 1 x12+x22+x32+x42 = 1 Tasks x13+x23+x33+x43 = 1 xij = 0 or 1 for all i and j © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 22 Assignment Problem: Example The optimal assignment is: Subcontractor Project Distance Westside C 16 Federated A 28 Goliath (unassigned) Universal B 25 Total Distance = 69 miles © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 23 Assignment Problem LP Formulation Special Cases •Number of agents exceeds the number of tasks: Extra agents simply remain unassigned. •Number of tasks exceeds the number of agents: Add enough dummy agents to equalize the number of agents and the number of tasks. The objective function coefficients for these new variable would be zero. © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 24 Assignment Problem LP Formulation Special Cases (continued) •The assignment alternatives are evaluated in terms of revenue or profit: Solve as a maximization problem. •An assignment is unacceptable: Remove the corresponding decision variable. •An agent is permitted to work t tasks: n x ij t i 1, 2, ,m Agents j 1 © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 25 Transshipment Problem Transshipment problems are transportation problems in which a shipment may move through intermediate nodes (transshipment nodes)before reaching a particular destination node. Transshipment problems can be converted to larger transportation problems and solved by a special transportation program. Transshipment problems can also be solved by general purpose linear programming codes. The network representation for a transshipment problem with two sources, three intermediate nodes, and two destinations is shown on the next slide. © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 26 Transshipment Problem Network Representation s1 c15 Supply s2 3 c13 1 c37 c14 Sources c25 6 c46 c47 4 c23 2 c36 c56 c24 5 Demand 7 c57 d1 d2 Destinations Intermediate Nodes © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 27 Transshipment Problem Linear Programming Formulation Using the notation: xij = number of units shipped from node i to node j cij = cost per unit of shipping from node i to node j si = supply at origin node i dj = demand at destination node j continued © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 28 Transshipment Problem Linear Programming Formulation (continued) Min cij xij all arcs s.t. xij arcs out Origin nodes i xij 0 Transhipment nodes xij d j Destination nodes j arcs in xij arcs out arcs in arcs in xij si xij arcs out xij > 0 for all i and j continued © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 29 Transshipment Problem LP Formulation Special Cases • Total supply not equal to total demand • Maximization objective function • Route capacities or route minimums • Unacceptable routes The LP model modifications required here are identical to those required for the special cases in the transportation problem. © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 30 Transshipment Problem Example The Northside and Southside facilities of Zeron Industries supply three firms (Zrox, Hewes, Rockrite) with customized shelving for its offices. They both order shelving from the same two manufacturers, Arnold Manufacturers and Supershelf, Inc. Currently weekly demands by the users are 50 for Zrox, 60 for Hewes, and 40 for Rockrite. Both Arnold and Supershelf can supply at most 75 units to its customers. Additional data is shown on the next slide. © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 31 Transshipment Problem Example Because of long standing contracts based on past orders, unit costs from the manufacturers to the suppliers are: Zeron N Arnold 5 Supershelf 7 Zeron S 8 4 The costs to install the shelving at the various locations are: Zrox Thomas 1 Washburn 3 Hewes Rockrite 5 8 4 4 © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 32 Transshipment Problem Example Network Representation ZROX 75 ARNOLD Arnold 5 Zeron N 8 75 4 50 Hewes HEWES 60 RockRite 40 5 8 3 7 Super Shelf 1 Zrox Zeron WASH BURN S 4 4 © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 33 Transshipment Problem: Example Linear Programming Formulation • Decision Variables Defined xij = amount shipped from manufacturer i to supplier j xjk = amount shipped from supplier j to customer k where i = 1 (Arnold), 2 (Supershelf) j = 3 (Zeron N), 4 (Zeron S) k = 5 (Zrox), 6 (Hewes), 7 (Rockrite) • Objective Function Defined Minimize Overall Shipping Costs: Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 8x37 + 3x45 + 4x46 + 4x47 © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 34 Transshipment Problem: Example Constraints Defined Amount Out of Arnold: Amount Out of Supershelf: Amount Through Zeron N: Amount Through Zeron S: Amount Into Zrox: Amount Into Hewes: Amount Into Rockrite: x13 + x14 < 75 x23 + x24 < 75 x13 + x23 - x35 - x36 - x37 = 0 x14 + x24 - x45 - x46 - x47 = 0 x35 + x45 = 50 x36 + x46 = 60 x37 + x47 = 40 Non-negativity of Variables: xij > 0, for all i and j. © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 35 Transshipment Problem: Example Computer Output Objective Function Value = 1150.000 Variable Value Reduced Costs X13 X14 X23 X24 X35 X36 X37 X45 X46 X47 75.000 0.000 0.000 75.000 50.000 25.000 0.000 0.000 35.000 40.000 0.000 2.000 4.000 0.000 0.000 0.000 3.000 3.000 0.000 0.000 © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 36 Transshipment Problem: Example Solution ZROX 75 ARNOLD Arnold 5 75 Zeron N 8 75 4 50 Hewes HEWES 60 RockRite 40 5 8 3 4 7 Super Shelf 1 Zrox Zeron WASH BURN S 4 © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 37 Transshipment Problem: Example Computer Output (continued) Constraint 1 2 3 4 5 6 7 Slack/Surplus 0.000 0.000 0.000 0.000 0.000 0.000 0.000 Dual Values 0.000 2.000 -5.000 -6.000 -6.000 -10.000 -10.000 © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 38 Transshipment Problem: Example Computer Output (continued) OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit X13 X14 X23 X24 X35 X36 X37 X45 X46 X47 3.000 6.000 3.000 No Limit No Limit 3.000 5.000 0.000 2.000 No Limit 5.000 8.000 7.000 4.000 1.000 5.000 8.000 3.000 4.000 4.000 7.000 No Limit No Limit 6.000 4.000 7.000 No Limit No Limit 6.000 7.000 © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 39 Transshipment Problem: Example Computer Output (continued) RIGHT HAND SIDE RANGES Constraint 1 2 3 4 5 6 7 Lower Limit 75.000 75.000 -75.000 -25.000 0.000 35.000 15.000 Current Value Upper Limit 75.000 No Limit 75.000 100.000 0.000 0.000 0.000 0.000 50.000 50.000 60.000 60.000 40.000 40.000 © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 40