1 1 Slide 2008 Thomson South-Western. All Rights Reserved 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. Slides by JOHN LOUCKS St. Edwards University INTRODUCTION TO MANAGEMENT SCIENCE, 13e Anderson Sweeney Williams Martin 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. 2 2 Slide 2008 Thomson South-Western. All Rights Reserved 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. Chapter 6, Part A Distribution and Network Models n Transportation Problem n Assignment Problem n Transshipment Problem n Shortest Route Problem n Maximum Flow Problem 3 3 Slide 2008 Thomson South-Western. All Rights Reserved 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. Transportation Problems (p.262) n Origin or Source node n Destination node n arcs n Minimize the total transportation cost meeting the supply availability and demand requirements. n Example (p.261) Production capacity of 3 plants Production capacity of 3 plants Demand forecasts for 4 distribution centers Demand forecasts for 4 distribution centers Network representation (p.262) Network representation (p.262) Transportation costs for each route. Transportation costs for each route. 4 4 Slide 2008 Thomson South-Western. All Rights Reserved 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. Transportation Problem n Linear Programming Formulation Using the notation: Using the notation: x ij = number of units shipped from x ij = number of units shipped from origin i to destination j origin i to destination j c ij = cost per unit of shipping from c ij = cost per unit of shipping from origin i to destination j origin i to destination j s i = supply or capacity in units at origin i s i = supply or capacity in units at origin i d j = demand in units at destination j d j = demand in units at destination j continued 5 5 Slide 2008 Thomson South-Western. All Rights Reserved 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. Transportation Problem n Linear Programming Formulation (p.267) x ij > 0 for all i and j 6 6 Slide 2008 Thomson South-Western. All Rights Reserved 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. Transportation Problems n LP formulation (p.264) n Output (p.264) n What is the solution? n How much is the total transportation cost n Is it dual solution or degenerate solution? Row Slack or Surplus Dual Price Row Slack or Surplus Dual Price Variable Value Reduced Cost X X X X X X X X X X X X X X X X X X X X X X X X 7 7 Slide 2008 Thomson South-Western. All Rights Reserved 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. Transportation Problems n If total supply is greater than the total demand? No change No change n If total supply is less than the total demand? Use dummy origin with 0 cost Use dummy origin with 0 cost example : Lexington demand is 2500 8 8 Slide 2008 Thomson South-Western. All Rights Reserved 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. n LP Formulation Special Cases The objective is maximizing profit or revenue: The objective is maximizing profit or revenue: Minimum shipping guarantee from i to j : (p.266) Minimum shipping guarantee from i to j : (p.266) x ij > L ij x ij > L ij Maximum route capacity from i to j : Maximum route capacity from i to j : x ij < L ij x ij < L ij Unacceptable route: Unacceptable route: Remove the corresponding decision variable. Remove the corresponding decision variable. Transportation Problem Solve as a maximization problem. 9 9 Slide 2008 Thomson South-Western. All Rights Reserved 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. Assignment Problem (p.268) n Assign m workers to m jobs. n Minimize the total assignment cost n Cost info (p.268) n Assignment problem is a special case of transportation problem with all RHS are 1. n Network representation (p.269) 10 Slide 2008 Thomson South-Western. All Rights Reserved 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. n Linear Programming Formulation Using the notation: Using the notation: x ij = 1 if agent i is assigned to task j x ij = 1 if agent i is assigned to task j 0 otherwise 0 otherwise c ij = cost of assigning agent i to task j c ij = cost of assigning agent i to task j Assignment Problem continued 11 Slide 2008 Thomson South-Western. All Rights Reserved 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. n Linear Programming Formulation (continued) Assignment Problem x ij > 0 for all i and j 12 Slide 2008 Thomson South-Western. All Rights Reserved 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. Assignment Problem n LP formulation (p.270) n Output (p.270) n What is the solution? n What is the total completion time? n Is the solution dual solutions or degenerate? Variable Value Reduced Cost X X X X X X X X X X X X X X X X X X Row Slack or Surplus Dual Price Row Slack or Surplus Dual Price 13 Slide 2008 Thomson South-Western. All Rights Reserved 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. n LP Formulation Special Cases Number of agents exceeds the number of tasks: Number of agents exceeds the number of tasks: Number of tasks exceeds the number of agents: Number of tasks exceeds the number of agents: Add enough dummy agents to equalize the Add enough dummy agents to equalize the number of agents and the number of tasks. number of agents and the number of tasks. The objective function coefficients for these The objective function coefficients for these new variable would be zero. new variable would be zero. Assignment Problem Extra agents simply remain unassigned. 14 Slide 2008 Thomson South-Western. All Rights Reserved 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. Assignment Problem n LP Formulation Special Cases (continued) The assignment alternatives are evaluated in terms of revenue or profit: The assignment alternatives are evaluated in terms of revenue or profit: Solve as a maximization problem. Solve as a maximization problem. An assignment is unacceptable: An assignment is unacceptable: Remove the corresponding decision variable. Remove the corresponding decision variable. An agent is permitted to work t tasks: An agent is permitted to work t tasks: 15 Slide 2008 Thomson South-Western. All Rights Reserved 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. Assignment Problem n If there are two more tasks with completion times (10, 18, 3) and (9, 14, 9) Min=10*x11+15*x12+9*x13+9*x21+18*x22+5*x23+6*x31+14*x32+3*x33 +10*x14+9*x15+18*x24+14*x25+3*x34+9*x35; +10*x14+9*x15+18*x24+14*x25+3*x34+9*x35; x11+x12+x13 +x14 +x15 = 4; ! connect 2-3; x12 +x13 +x23 +x26 +x35 +x45 +x46 +x56 >= 4; ! connect 2-4; x12 +x13 +x23 +x24 +x35 +x45 +x46 +x56 >= 4; ! connect 2-6; x12 +x13 +x23 +x24 +x26 +x45 +x46 +x56 >= 4; ! connect 3-5; x12 +x13 +x23 +x24 +x26 +x35 +x46 +x56 >= 4; ! connect 4-5; x12 +x13 +x23 +x24 +x26 +x35 +x45 +x56 >= 4; ! connect 4-6; x12 +x13 +x23 +x24 +x26 +x35 +x45 +x46 >= 4; ! connect 5-6; x24 +x26 +x35 +x45 +x46 +x56 >= 3; ! connect (123, 4, 5, 6); x24 +x26 +x35 +x45 +x46 +x56 >= 3; ! connect (123, 4, 5, 6); x13 +x23 +x26 +x35 +x45 +x46 +x56 >= 3; ! connect (1-2, 2-4), (124, 3, 5, 6); x13 +x23 +x26 +x35 +x45 +x46 +x56 >= 3; ! connect (1-2, 2-4), (124, 3, 5, 6); x13 +x23 +x24 +x35 +x45 +x46 +x56 >= 3; ! connect (1-2, 2-6), (126, 3, 4, 5); x13 +x23 +x24 +x35 +x45 +x46 +x56 >= 3; ! connect (1-2, 2-6), (126, 3, 4, 5); x12 +x23 +x24 +x26 +x45 +x46 +x56 >= 3; ! connect (1-3, 3-5), (135, 2, 4, 6; x12 +x13 +x26 +x35 +x45 +x46 +x56 >= 3; ! connect (2-3, 2-4), (234, 1, 5, 6); x12 +x13 +x24 +x26 +x45 +x46 +x56 >= 3; ! connect (2-3, 3-5), (235, 1, 4, 6); x12 +x13 +x24 +x35 +x45 +x46 +x56 >= 3; ! connect (2-3, 2-6), (236, 1, 4, 5); x12 +x13 +x23 +x26 +x35 +x46 +x56 >= 3; ! connect (2-4, 4-5), (245, 1, 3, 6); x12 +x13 +x23 +x35 +x45 +x56 >= 3; ! connect (246, 1, 3, 5); x12 +x13 +x23 +x24 +x35 +x45 +x46 >= 3; ! connect (2-6, 5-6), (256, 1, 3, 4); x12 +x13 +x23 +x24 +x26 +x46 +x56 >= 3; ! connect (3-5, 4-5), (345, 1, 2, 6); x12 +x13 +x23 +x24 +x26 +x45 +x46 >= 3; ! connect (3-5, 5-6), (356, 1, 2, 4); x12 +x13 +x23 +x24 +x26 +x35 >= 3; ! connect (456, 1, 2, 3); 29 Slide 2008 Thomson South-Western. All Rights Reserved 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. Minimum spanning tree (p.281) n LP formulation (continued) x26 +x35 +x45 +x46 +x56 >= 2; ! connect (1234, 5, 6); x26 +x35 +x45 +x46 +x56 >= 2; ! connect (1234, 5, 6); x24 +x26 +x45 +x46 +x56 >= 2; ! connect (1235, 4, 6); x24 +x26 +x45 +x46 +x56 >= 2; ! connect (1235, 4, 6); x24 +x35 +x45 +x46 +x56 >= 2; ! connect (1236, 4, 5); x24 +x35 +x45 +x46 +x56 >= 2; ! connect (1236, 4, 5); x13 +x23 +x26 +x35 +x46 +x56 >= 2; ! connect (1245, 3, 6); x13 +x23 +x26 +x35 +x46 +x56 >= 2; ! connect (1245, 3, 6); x13 +x23 +x35 +x45 +x56 >= 2; ! connect (1246, 3, 5); x13 +x23 +x35 +x45 +x56 >= 2; ! connect (1246, 3, 5); x13 +x23 +x24 +x35 +x45 +x46 >= 2; ! connect (1256, 3, 4); x13 +x23 +x24 +x35 +x45 +x46 >= 2; ! connect (1256, 3, 4); x12 +x23 +x24 +x26 +x46 +x56 >= 2; ! connect (1345, 2, 6); x12 +x23 +x24 +x26 +x45 +x46 >= 2; ! connect (1356, 2, 4); x12 +x13 +x26 +x46 +x56 >= 2; ! connect (2345, 1, 6); x12 +x13 +x35 +x45 +x56 >= 2; ! connect (2346, 1, 5); x12 +x13 +x24 +x45 +x46 >= 2; ! connect (2356, 1, 4); x12 +x13 +x23 +x35 >= 2; ! connect (2456, 1, 3); x12 +x13 +x23 +x24 +x26 >= 2; ! connect (3456, 1, 2); x26 +x46 +x56 >= 1; ! connect (12345, 6); x26 +x46 +x56 >= 1; ! connect (12345, 6); x35 +x45 +x56 >= 1; ! connect (12346, 5); x35 +x45 +x56 >= 1; ! connect (12346, 5); x24 +x45 +x46 >= 1; ! connect (12356, 4); x24 +x45 +x46 >= 1; ! connect (12356, 4); x13 +x23 +x35 >= 1; ! connect (12456, 3); x13 +x23 +x35 >= 1; ! connect (12456, 3); x12 +x23 +x24 +x26 >= 1; ! connect (13456, 2); x12 +x13 >= 1; ! connect (23456, 1); ! choose arcs from unselected nodes to any of selected nodes ! for nodes that can not be connected to any of selected nodes, choose all the outgoing arcs. 30 Slide 2008 Thomson South-Western. All Rights Reserved 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. Minimum spanning tree (p.281) n LP output Objective value: Objective value: Variable Value Reduced Cost Variable Value Reduced Cost X X X X X X X X X X X X X X X X X X Row Slack or Surplus Dual Price Row Slack or Surplus Dual Price 31 Slide 2008 Thomson South-Western. All Rights Reserved 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. Maximal Flow Problem n Maximal Flow Problem (p.284) n The maximal flow problem is concerned with determining the maximal volume of flow from one node (called the source) to another node (called the sink). n In the maximal flow problem, each arc has a maximum arc flow capacity which limits the flow through the arc. 32 Slide 2008 Thomson South-Western. All Rights Reserved 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. Example: Maximal Flow n A capacitated transshipment model can be developed for the maximal flow problem. n We will add an arc from the sink node back to the source node to represent the total flow through the network. n There is no capacity on the newly added sink-to- source arc. n We want to maximize the flow over the sink-to-source arc. 33 Slide 2008 Thomson South-Western. All Rights Reserved 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. Maximal Flow Problem n LP Formulation (as Capacitated Transshipment Problem) (as Capacitated Transshipment Problem) There is a variable for every arc. There is a variable for every arc. There is a constraint for every node; the flow out must equal the flow in. There is a constraint for every node; the flow out must equal the flow in. There is a constraint for every arc (except the added sink-to-source arc); arc capacity cannot be exceeded. There is a constraint for every arc (except the added sink-to-source arc); arc capacity cannot be exceeded. The objective is to maximize the flow over the added, sink-to-source arc. The objective is to maximize the flow over the added, sink-to-source arc. 34 Slide 2008 Thomson South-Western. All Rights Reserved 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. Maximal Flow Problem n LP Formulation (as Capacitated Transshipment Problem) (as Capacitated Transshipment Problem) Max x k 1 ( k is sink node, 1 is source node) Max x k 1 ( k is sink node, 1 is source node) s.t. x ij - x ji = 0 (conservation of flow) i j s.t. x ij - x ji = 0 (conservation of flow) i j x ij < c ij ( c ij is capacity of ij arc) x ij > 0, for all i and j (non-negativity) x ij > 0, for all i and j (non-negativity) (x ij represents the flow from node i to node j) (x ij represents the flow from node i to node j) 35 Slide 2008 Thomson South-Western. All Rights Reserved 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. Maximal Flow Problem n LP Formulation (pp.284285) n Computer output (p.286) n What is the solution? n What is the maximum flow? 36 Slide 2008 Thomson South-Western. All Rights Reserved 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. A production and Inventory application 37 Slide 2008 Thomson South-Western. All Rights Reserved 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. A production and Inventory application n LP formulation (p.289) n Computer output (p.290) n What is the solution? n What is the total production and inventory cost? 38 Slide 2008 Thomson South-Western. All Rights Reserved 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. End of Chapter 6, Part A