Transportation, Assignment, and Network Models 9 To accompany Quantitative Analysis for Management, Twelfth Edition, by Render, Stair, Hanna and Hale Power Point slides created by Jeff Heyl Copyright ©2015 Pearson Education, Inc.

After completing this chapter, students will be able to: LEARNING OBJECTIVES Copyright ©2015 Pearson Education, Inc.9 – 2 1.Structure LP problems for the transportation and assignment models. 2.Solve facility location and other application problems with transportation models. 3.Use LP to model shortest-route problems. 4.Solve minimal-spanning tree problems.

9.1Introduction 9.2The Transportation Problem 9.3The Assignment Problem 9.6Shortest-Route Problem 9.7Minimal-Spanning Tree Problem CHAPTER OUTLINE Copyright ©2015 Pearson Education, Inc.9 – 3

Introduction LP problems modeled as networks –Helps visualize and understand problems Transportation problem Assignment problem Shortest-route problem Minimal-spanning tree problem –Specialized algorithms available Copyright ©2015 Pearson Education, Inc.9 – 4

Introduction Common terminology for network models –Points on the network are referred to as nodes Typically circles –Lines on the network that connect nodes are called arcs Copyright ©2015 Pearson Education, Inc.9 – 5

The Transportation Problem Deals with the distribution of goods from several points of supply (sources) to a number of points of demand (destinations) –Usually given the capacity of goods at each source and the requirements at each destination –Typically objective is to minimize total transportation and production costs Copyright ©2015 Pearson Education, Inc.9 – 6

Linear Program for Transportation Executive Furniture Corporation transportation problem –Minimize transportation cost –Meet demand –Not exceed supply Copyright ©2015 Pearson Education, Inc.9 – 7

Linear Program for Transportation Let X ij = number of units shipped from source i to destination j Where i =1, 2, 3, with 1 = Des Moines, 2 = Evansville, and 3 = Fort Lauderdale j =1, 2, 3, with 1 = Albuquerque, 2 = Boston, and 3 = Cleveland Copyright ©2015 Pearson Education, Inc.9 – 8

Linear Program for Transportation Minimize total cost =5X X X X X X X 31 +7X X 33 Subject to: X 11 + X 12 + X 13 ≤ 100 (Des Moines supply) X 21 + X 22 + X 23 ≤ 300 (Evansville supply) X 31 + X 32 + X 33 ≤ 300 (Fort Lauderdale supply) X 11 + X 21 + X 31 = 300 (Albuquerque demand) X 12 + X 22 + X 32 = 200 (Boston demand) X 13 + X 23 + X 33 = 200 (Cleveland demand) X ij ≥ 0 for all i and j Copyright ©2015 Pearson Education, Inc.9 – 9

Linear Program for Transportation Optimal solution 100 units from Des Moines to Albuquerque 200 units from Evansville to Boston 100 units from Evansville to Cleveland 200 units from Ft. Lauderdale to Albuquerque 100 units from Ft. Lauderdale to Cleveland Total cost = $3,900 Copyright ©2015 Pearson Education, Inc.9 – 10

Linear Program for Transportation Copyright ©2015 Pearson Education, Inc.9 – 11 FIGURE 9.1 – Network Representation of a Transportation Problem Supply $5 $4 $3 $8 $4 $3 $9 $7 $ Demand Source Des Moines (Source 1) Evansville (Source 2) Fort Lauderdale (Source 3) Albuquerque (Destination 1) Boston (Destination 2) Cleveland (Destination 3) Destination

A General LP Model for Transportation Problems Let X ij =number of units shipped from source i to destination j c ij =cost of one unit from source i to destination j s i =supply at source i d j =demand at destination j Copyright ©2015 Pearson Education, Inc.9 – 12

Facility Location Analysis Transportation method especially useful New location is major financial importance Several alternative locations evaluated Subjective factors are considered Final decision also involves minimizing total shipping and production costs Alternative facility locations analyzed within the framework of one overall distribution system Copyright ©2015 Pearson Education, Inc.9 – 13

Facility Location Analysis Hardgrave Machine Company produces computer components in Cincinnati, Salt Lake City, and Pittsburgh Four warehouses in Detroit, Dallas, New York, and Los Angeles Two new plant sites being considered – Seattle and Birmingham Which of the new locations will yield the lowest cost for the firm in combination with the existing plants and warehouses? Copyright ©2015 Pearson Education, Inc.9 – 14

Facility Location Analysis Copyright ©2015 Pearson Education, Inc.9 – 15 WAREHOUSE MONTHLY DEMAND (UNITS) PRODUCTION PLANT MONTHLY SUPPLY COST TO PRODUCE ONE UNIT ($) Detroit10,000Cincinnati15,00048 Dallas12,000Salt Lake City6,00050 New York15,000Pittsburgh14,00052 Los Angeles9,00035,000 46,000 Supply needed from a new plant = 46,000 – 35,000 = 11,000 units per month TABLE 9.1 – Hardgrave’s Demand and Supply Data ESTIMATED PRODUCTION COST PER UNIT AT PROPOSED PLANTS Seattle$53 Birmingham$49

Facility Location Analysis Copyright ©2015 Pearson Education, Inc.9 – 16 TO FROMDETROITDALLASNEW YORK LOS ANGELES CINCINNATI$25 $55 $40$60 SALT LAKE CITY PITTSBURGH SEATTLE BIRMINGHAM TABLE 9.2 – Hardgrave’s Shipping Costs Solve two transportation problems – one for each combination

Facility Location Analysis X ij =number of units shipped from source i to destination j Where i =1, 2, 3, 4 with 1 = Cincinnati, 2 = Salt Lake City, 3 = Pittsburgh, and 4 = Seattle j =1, 2, 3, 4 with 1 = Detroit, 2 = Dallas, 3 = New York, and 4 = Los Angeles Copyright ©2015 Pearson Education, Inc.9 – 17

Facility Location Analysis Minimize total cost =73X X X X X X X X X X X X X X X X 44 Subject to: X 11 + X 21 + X 31 + X 41 = 10,000 Detroit demand X 12 + X 22 + X 32 + X 42 = 12,000 Dallas demand X 13 + X 23 + X 33 + X 43 = 15,000 New York demand X 14 + X 24 + X 34 + X 44 = 9,000 Los Angeles demand X 11 + X 12 + X 13 + X 14 ≤ 15,000 Cincinnati supply X 21 + X 22 + X 23 + X 24 ≤ 6,000 Salt Lake City supply X 31 + X 32 + X 33 + X 34 ≤ 14,000 Pittsburgh supply X 41 + X 42 + X 43 + X 44 ≤ 11,000 Seattle supply All variables X ij ≥ 0 Copyright ©2015 Pearson Education, Inc.9 – 18

Minimize total cost =73X X X X X X X X X X X X X X X X 44 Subject to: X 11 + X 21 + X 31 + X 41 = 10,000 Detroit demand X 12 + X 22 + X 32 + X 42 = 12,000 Dallas demand X 13 + X 23 + X 33 + X 43 = 15,000 New York demand X 14 + X 24 + X 34 + X 44 = 9,000 Los Angeles demand X 11 + X 12 + X 13 + X 14 ≤ 15,000 Cincinnati supply X 21 + X 22 + X 23 + X 24 ≤ 6,000 Salt Lake City supply X 31 + X 32 + X 33 + X 34 ≤ 14,000 Pittsburgh supply X 41 + X 42 + X 43 + X 44 ≤ 11,000 Seattle supply All variables X ij ≥ 0 Facility Location Analysis Copyright ©2015 Pearson Education, Inc.9 – 19 The total cost for the Seattle alternative = $3,704,000

Minimize total cost =73X X X X X X X X X X X X X X X X 44 Subject to: X 11 + X 21 + X 31 + X 41 = 10,000 Detroit demand X 12 + X 22 + X 32 + X 42 = 12,000 Dallas demand X 13 + X 23 + X 33 + X 43 = 15,000 New York demand X 14 + X 24 + X 34 + X 44 = 9,000 Los Angeles demand X 11 + X 12 + X 13 + X 14 ≤ 15,000 Cincinnati supply X 21 + X 22 + X 23 + X 24 ≤ 6,000 Salt Lake City supply X 31 + X 32 + X 33 + X 34 ≤ 14,000 Pittsburgh supply X 41 + X 42 + X 43 + X 44 ≤ 11,000 Seattle supply All variables X ij ≥ 0 Facility Location Analysis Copyright ©2015 Pearson Education, Inc.9 – 20 The total cost for the Seattle alternative = $3,704,000 Reformulating the problem for the Birmingham alternative and solving, the total cost = $3,741,000

The Assignment Problem This class of problem determines the most efficient assignment of people or equipment to particular tasks Objective is typically to minimize total cost or total task time Copyright ©2015 Pearson Education, Inc.9 – 21

Linear Program for Assignment Example The Fix-it Shop has just received three new repair projects that must be repaired quickly 1.A radio 2.A toaster oven 3.A coffee table Three workers with different talents are able to do the jobs Owner estimates wage cost for workers on projects Objective – minimize total cost Copyright ©2015 Pearson Education, Inc.9 – 22

Linear Program for Assignment Example Copyright ©2015 Pearson Education, Inc.9 – 23 FIGURE 9.2 – Assignment Problem in a Transportation Network Format Supply $11 $14 $6 $8 $10 $11 $9 $12 $ Demand Person Adams (Source 1) Brown (Source 2) Cooper (Source 3) Project 1 (Destination 1) Project 2 (Destination 2) Project 3 (Destination 3) Project

Linear Program for Assignment Example Copyright ©2015 Pearson Education, Inc.9 – 24 Let where i =1, 2, 3, with 1 = Adams, 2 = Brown, and 3 = Cooper j =1, 2, 3, with 1 = Project 1, 2 = Project 2, and 3 = Project 3

Linear Program for Assignment Example Copyright ©2015 Pearson Education, Inc.9 – 25 Minimize total cost =11X X X X X X X X X 33 subject to X 11 + X 12 + X 13 = 1 X 21 + X 22 + X 23 = 1 X 31 + X 32 + X 33 = 1 X 11 + X 21 + X 31 = 1 X 12 + X 22 + X 32 = 1 X 13 + X 23 + X 33 = 1 X ij = 0 or 1 for all i and j

Linear Program for Assignment Example Copyright ©2015 Pearson Education, Inc.9 – 26 Minimize total cost =11X X X X X X X X X 33 subject to X 11 + X 12 + X 13 ≤ 1 X 21 + X 22 + X 23 ≤ 1 X 31 + X 32 + X 33 ≤ 1 X 11 + X 21 + X 31 = 1 X 12 + X 22 + X 32 = 1 X 13 + X 23 + X 33 = 1 X ij = 0 or 1 for all i and j Solution X 13 = 1, Adams assigned to Project 3 X 22 = 1, Brown assigned to Project 2 X 31 = 1, Cooper is assigned to Project 1 Total cost = $25

Shortest-Route Problem Find the shortest distance from one location to another Can be modeled as –A linear programming problem with 0-1 variables –Using specialized algorithm Copyright ©2015 Pearson Education, Inc.9 – 27

Shortest-Route Problem Ray Design transports beds, chairs, and other furniture items from the factory to the warehouse –Travel through several cities –No direct interstate highways Find the route with the shortest distance Copyright ©2015 Pearson Education, Inc.9 – 28 Plant Warehouse FIGURE 9.5 – Roads from Ray’s Plant to Warehouse

Shortest-Route Problem Variables X ij = 1 if arc from node i to node j is selected and X ij = 0 otherwise where i = 1, 2, 3, 4, 5 j = 2, 3, 4, 5, 6 Constraints specify the number of units going into a node must equal the number of units going out of the node Copyright ©2015 Pearson Education, Inc.9 – 29

Shortest-Route Problem Minimize distance =100X X X X X X X X X X X X X X 56 subject to Copyright ©2015 Pearson Education, Inc.9 – 30 X 12 + X 13 =1Node 1 X 12 + X 32 – X 23 – X 24 – X 25 =0Node 2 X 13 + X 23 – X 32 – X 35 =0Node 3 X 24 + X 54 – X 42 – X 45 – X 46 =0Node 4 X 25 + X 35 + X 45 – X 52 – X 53 – X 54 – X 56 =0Node 5 X 46 + X 56 =1Node 6 All variables=0or 1

Minimal-Spanning Tree Problem Connecting all points of a network together while minimizing the total distance of the connections Linear programming can be used but is complex Minimal-spanning tree technique is quite easy Copyright ©2015 Pearson Education, Inc.9 – 31

Minimal-Spanning Tree Problem Steps for the Minimal-Spanning Tree Technique 1.Select any node in the network. 2.Connect this node to the nearest node that minimizes the total distance. 3.Considering all of the nodes that are now connected, find and connect the nearest node that is not connected. If there is a tie for the nearest node, select one arbitrarily. A tie suggests there may be more than one optimal solution. 4.Repeat the third step until all nodes are connected. Copyright ©2015 Pearson Education, Inc.9 – 32

Minimal-Spanning Tree Problem Lauderdale Construction –Housing project in Panama City Beach –Determine the least expensive way to provide water and power to each house Copyright ©2015 Pearson Education, Inc.9 – Gulf FIGURE 9.6 – Network for Lauderdale Construction

Minimal-Spanning Tree Problem Step 1 –Arbitrarily select node 1 Step 2 –Connect node 1 to node 3 (nearest) Copyright ©2015 Pearson Education, Inc.9 – Gulf FIGURE 9.7 – First Iteration

Minimal-Spanning Tree Problem Step 3 –Connect next nearest unconnected node, node 4 Continue for other unconnected nodes Copyright ©2015 Pearson Education, Inc.9 – FIGURE 9.8 – Second and Third Iterations (a) Second Iteration(b) Third Iteration

Minimal-Spanning Tree Problem Step 4 –Repeat the process Copyright ©2015 Pearson Education, Inc.9 – FIGURE 9.9 – Last Four Iterations (a) Fourth Iteration(b) Fifth Iteration

Step 4 –Repeat the process Minimal-Spanning Tree Problem Copyright ©2015 Pearson Education, Inc.9 – FIGURE 9.9 – Last Four Iterations (c) Sixth Iteration(d) Seventh Iteration

Minimal-Spanning Tree Problem Copyright ©2015 Pearson Education, Inc.9 – 38 TABLE 9.4 – Summary of Steps in Lauderdale Construction Minimal-Spanning Tree Problem STEP CONNECTED NODES UNCONNECTED NODES CLOSE UNCONNECTED NODES ARC SELECTED ARC LENGTH TOTAL DISTANCE 112, 3, 4, 5, 6, 7, 831–322 21, 32, 4, 5, 6, 7, 843–424 31, 3, 42, 5, 6, 7, 82 or 63–237 41, 2, 3, 45, 6, 7, 85 or 62– , 2, 3, 4, 56, 7, 863– , 2, 3, 4, 5, 67, 886– , 2, 3, 4, 5, 6, 8778–7216