1 1 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Slides by John Loucks St. Edward’s University

2 2 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. 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 Network Representation Network Representation General LP Formulation General LP Formulation n Transshipment Problem Network Representation Network Representation General LP Formulation General LP Formulation

3 3 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Transportation, Assignment, and Transshipment Problems n Network model 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. 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. n Types of network models Transportation Transportation assignment assignment Transshipment Transshipment shortest-route shortest-route maximal flow problems maximal flow problems

4 4 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Transportation Problem n Transportation problem seeks to minimize the total shipping costs of transporting goods from m origins (each with a supply s i ) to n destinations (each with a demand d j ) seeks to minimize the total shipping costs of transporting goods from m origins (each with a supply s i ) to n destinations (each with a demand d j ) when the unit shipping cost from an origin, i, to a destination, j, is c ij. when the unit shipping cost from an origin, i, to a destination, j, is c ij. n Network representation Graphical depiction of the problem Graphical depiction of the problem Consists of Consists of Nodes (dots)Nodes (dots) Arcs (lines)Arcs (lines)

5 5 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Transportation Problem n Network Representation 2 2 c 11 c 12 c 13 c 21 c 22 c 23 d1d1d1d1 d2d2d2d2 d3d3d3d3 s1s1s1s1 s2s2 SourcesDestinations

6 6 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. 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 i = number assigned to each origin i = number assigned to each origin j = number assigned to each destination j = number assigned to each destination

7 7 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Transportation Problem n Linear Programming Formulation (continued) x ij > 0 for all i and j

8 8 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. or duplicated, or posted to a publicly accessible website, in whole or in part. n LP Formulation Special Cases Total supply exceeds total demand: Total supply exceeds total demand: Total demand exceeds total supply: 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. * * Since there is not an example of this in the book we will do one next class. Assign a zero shipping cost per unit Maximum route capacity from i to j : Maximum route capacity from i to j : x ij < L i x ij < L i Remove the corresponding decision variable. Remove the corresponding decision variable. Transportation Problem No modification of LP formulation is necessary.

9 9 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. or duplicated, or posted to a publicly accessible website, in whole or in part. n LP Formulation Special Cases (continued) The objective is maximizing profit or revenue: The objective is maximizing profit or revenue: Minimum shipping guarantee from i to j : Minimum shipping guarantee from i to j : 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.

10 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Transportation Problem: Example #1 Acme Block Company has orders for 80 tons of concrete blocks at three suburban locations as follows: Northwood tons, Westwood tons, and Eastwood tons. Acme has two plants, each of which can produce 50 tons per week. Delivery cost per ton from each plant to each suburban location is shown on the next slide. How should end of week shipments be made to fill the above orders?

11 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. or duplicated, or posted to a publicly accessible website, in whole or in part. n Delivery Cost Per Ton Northwood Westwood Eastwood Northwood Westwood Eastwood Plant Plant Plant Plant Transportation Problem: Example #1

12 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. or duplicated, or posted to a publicly accessible website, in whole or in part. NorthWood WestWood EastWood Plant2 Plant Demand (t)Dest Supply (t) Orig Costs

13 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Transportation Problem: Example #1 n MIN SHIPPING COSTS SHIPPING COSTS n ST SUPPLY SUPPLY DEMAND DEMAND NON-NEGATIVE NON-NEGATIVE

14 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Transportation Problem: Example #1 n LET x ij = number of tons shipped from origin i to destination j n MIN 24x x x 13 + (Shipping cost from P1) 30x x x 23 (Shipping cost from P2) 24x x x 13 + (Shipping cost from P1) 30x x x 23 (Shipping cost from P2) n ST x 11 + x 12 + x 13 <= 50 (Supply of Plant 1) x 11 + x 12 + x 13 <= 50 (Supply of Plant 1) x 21 + x 22 + x 23 <= 50 (Supply of Plant 2) x 21 + x 22 + x 23 <= 50 (Supply of Plant 2) x 11 + x 21 = 25(Northwood Demand) x 11 + x 21 = 25(Northwood Demand) x 12 + x 22 = 45(Westwood Demand) x 12 + x 22 = 45(Westwood Demand) x 13 + x 23 = 10(Eastwood Demand) x 13 + x 23 = 10(Eastwood Demand) X ij >= 0 (Nonnegative) X ij >= 0 (Nonnegative)

15 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. or duplicated, or posted to a publicly accessible website, in whole or in part. n Partial Spreadsheet Showing Optimal Solution Transportation Problem: Example #1

16 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. or duplicated, or posted to a publicly accessible website, in whole or in part. n Optimal Solution From To Amount Cost From To Amount Cost Plant 1 Northwood Plant 1 Westwood 45 1,350 Plant 1 Westwood 45 1,350 Plant 2 Northwood Plant 2 Northwood Plant 2 Eastwood Plant 2 Eastwood Total Cost = $2,490 Total Cost = $2,490 Transportation Problem: Example #1

17 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. or duplicated, or posted to a publicly accessible website, in whole or in part. n Partial Sensitivity Report (first half) Transportation Problem: Example #1 -How much would the shipping cost from plant 1 to dest. 1 have to increase before a new optimal solution is met -What about from plant 1 to dest 3?

18 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. or duplicated, or posted to a publicly accessible website, in whole or in part. n Partial Sensitivity Report (second half) Transportation Problem: Example #1 Constraints FinalShadowConstraintAllowable CellNameValuePriceR.H. SideIncreaseDecrease $E$17P2.Cap E+3020 $E$18N.Dem $E$19W.Dem $E$20E.Dem $E$16P1.Cap Constraints FinalShadowConstraintAllowable CellNameValuePriceR.H. SideIncreaseDecrease $E$17P2.Cap E+3020 $E$18N.Dem $E$19W.Dem $E$20E.Dem $E$16P1.Cap How much unused capacity is there at plant 1? plant 2?

19 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Transshipment Problem n Transshipment problems transportation problems in which a shipment may move through intermediate nodes before reaching a particular destination node. transportation problems in which a shipment may move through intermediate nodes before reaching a particular destination node. n Transshipment problems can be converted to larger transportation problems and solved by a special transportation program. n Transshipment problems can also be solved by general purpose linear programming codes. n The network representation for a transshipment problem with two sources, three intermediate nodes, and two destinations is shown on the next slide.

20 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Transshipment Problem n Network Representation c 13 c 14 c 23 c 24 c 25 c 15 s1s1s1s1 c 36 c 37 c 46 c 47 c 56 c 57 d1d1d1d1 d2d2d2d2 Intermediate Nodes Sources Destinations s2s2s2s2 Demand Supply

21 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Transshipment Problem n Linear Programming Formulation Using the notation: Using the notation: x ij = number of units shipped from node i to node j x ij = number of units shipped from node i to node j c ij = cost per unit of shipping from node i to node j c ij = cost per unit of shipping from node i to node j s i = supply at origin node i s i = supply at origin node i d j = demand at destination node j d j = demand at destination node j i = number assigned to each origin i = number assigned to each origin j = number assigned to each destination * In this case, each transshipment node is both an origin and a destination!

22 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Transshipment Problem x ij > 0 for all i and j n Linear Programming Formulation (continued) continued

23 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Transshipment Problem n LP Formulation Special Cases Total supply not equal to total demand Total supply not equal to total demand Maximization objective function Maximization objective function Route capacities or route minimums Route capacities or route minimums Unacceptable routes Unacceptable routes The LP model modifications required here are identical to those required for the special cases in the transportation problem.

24 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. or duplicated, or posted to a publicly accessible website, in whole or in part. 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. Transshipment Problem: Example

25 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Because of long standing contracts based on past orders, unit costs from the manufacturers to the suppliers are: Zeron N Zeron S Zeron N Zeron S Arnold 5 8 Arnold 5 8 Supershelf 7 4 Supershelf 7 4 The costs to install the shelving at the various locations are: Zrox Hewes Rockrite Zrox Hewes Rockrite Thomas Thomas Washburn Washburn Transshipment Problem: Example

26 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. or duplicated, or posted to a publicly accessible website, in whole or in part. n Network Representation ARNOLD WASH BURN ZROX HEWES Arnold SuperShelf Hewes Zrox ZeronN ZeronS Rock-Rite Transshipment Problem: Example

27 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Transshipment Problem: Example n How to model transhipment nodes Flow out equal to flow in Flow out equal to flow in (Unless a transshipment node has some sort of demand) (Unless a transshipment node has some sort of demand) n Flow out = Flow in n Flow out – flow in = 0 n OR n Flow in – flow out = 0

28 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. or duplicated, or posted to a publicly accessible website, in whole or in part. n LET x ij = amount shipped from manufacturer i to supplier j x jk = amount shipped from supplier j to customer k x jk = amount shipped from supplier j to customer k where i = 1 (Arnold), 2 (Supershelf) where i = 1 (Arnold), 2 (Supershelf) j = 3 (Zeron N), 4 (Zeron S) j = 3 (Zeron N), 4 (Zeron S) k = 5 (Zrox), 6 (Hewes), 7 (Rockrite) k = 5 (Zrox), 6 (Hewes), 7 (Rockrite) Transshipment Problem: Example

n Cont’d n MIN 5 x x x x 24 + (From supply to xship) 1 x x x x x x 47 (From xship to demand) n SUBJECT TO Amount Out of Arnold: x 13 + x 14 < 75 Amount Out of Supershelf: x 23 + x 24 < 75 Amount Through Zeron N: x 13 + x 23 - x 35 - x 36 - x 37 = 0 Amount Through Zeron S: x 14 + x 24 - x 45 - x 46 - x 47 = 0 Amount Into Zrox: x 35 + x 45 = 50 Amount Into Hewes: x 36 + x 46 = 60 Amount Into Rockrite: x 37 + x 47 = 40 Non-negativity of Variables: x ij > 0, for all i and j.

Objective Function Value = Objective Function Value = Variable Value Reduced Costs Variable Value Reduced Costs X X X X X X X X X X X X X X X X X X X X Transshipment Problem: Example How much would the cost to ship from node 1 to 4 have to be reduced In order for this route to be a factor in the solution?

31 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. or duplicated, or posted to a publicly accessible website, in whole or in part. n Solution ARNOLD WASH BURN ZROX HEWES Arnold SuperShelf Hewes Zrox ZeronN ZeronS Rock-Rite Transshipment Problem: Example

32 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Shortest-Route Problem n Shortest-route problem finding the shortest path in a network from one node (or set of nodes) to another node (or set of nodes). finding the shortest path in a network from one node (or set of nodes) to another node (or set of nodes). n Can be used to find minimized cost or time as well as distance n Flow through a node is represented by “1” that mea that mea

33 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. 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 the arc from node i to node j x ij = 1 if the arc from node i to node j is on the shortest route is on the shortest route 0 otherwise 0 otherwise c ij = distance, time, or cost associated c ij = distance, time, or cost associated with the arc from node i to node j with the arc from node i to node j continued Shortest-Route Problem

34 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. or duplicated, or posted to a publicly accessible website, in whole or in part. n Linear Programming Formulation (continued) Shortest-Route Problem

35 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Susan Winslow has an important business meeting Susan Winslow has an important business meeting in Paducah this evening. She has a number of alternate routes by which she can travel from the company headquarters in Lewisburg to Paducah. The network of alternate routes and their respective travel time, ticket cost, and transport mode appear on the next two slides. If Susan earns a wage of $15 per hour, what route If Susan earns a wage of $15 per hour, what route should she take to minimize the total travel cost? Example: Shortest Route

36 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. or duplicated, or posted to a publicly accessible website, in whole or in part. 6 A B C D E F G H I J K L M Example: Shortest Route Paducah Lewisburg n Network Representation

37 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Shortest Route Transport Time Ticket Transport Time Ticket Route Mode (hours) Cost A Train 4 $ 20 A Train 4 $ 20 B Plane 1 $115 B Plane 1 $115 C Bus 2 $ 10 C Bus 2 $ 10 D Taxi 6 $ 90 D Taxi 6 $ 90 E Train 3  $ 30 E Train 3  $ 30 F Bus 3 $ 15 F Bus 3 $ 15 G Bus 4  $ 20 G Bus 4  $ 20 H Taxi 1 $ 15 H Taxi 1 $ 15 I Train 2  $ 15 I Train 2  $ 15 J Bus 6  $ 25 J Bus 6  $ 25 K Taxi 3  $ 50 K Taxi 3  $ 50 L Train 1  $ 10 L Train 1  $ 10 M Bus 4  $ 20 M Bus 4  $ 20

38 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Shortest Route Transport Time Time Ticket Total Transport Time Time Ticket Total Route Mode (hours) Cost Cost Cost A Train 4 $60 $ 20 $ 80 A Train 4 $60 $ 20 $ 80 B Plane 1 $15 $115 $130 B Plane 1 $15 $115 $130 C Bus 2 $30 $ 10 $ 40 C Bus 2 $30 $ 10 $ 40 D Taxi 6 $90 $ 90 $180 D Taxi 6 $90 $ 90 $180 E Train 3  $50 $ 30 $ 80 E Train 3  $50 $ 30 $ 80 F Bus 3 $45 $ 15 $ 60 F Bus 3 $45 $ 15 $ 60 G Bus 4  $70 $ 20 $ 90 G Bus 4  $70 $ 20 $ 90 H Taxi 1 $15 $ 15 $ 30 H Taxi 1 $15 $ 15 $ 30 I Train 2  $35 $ 15 $ 50 I Train 2  $35 $ 15 $ 50 J Bus 6  $95 $ 25 $120 J Bus 6  $95 $ 25 $120 K Taxi 3  $50 $ 50 $100 K Taxi 3  $50 $ 50 $100 L Train 1  $20 $ 10 $ 30 L Train 1  $20 $ 10 $ 30 M Bus 4  $70 $ 20 $ 90 M Bus 4  $70 $ 20 $ 90

39 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Shortest Route n LP Formulation Objective Function Objective Function Min 80 x x x x x x 25 Min 80 x x x x x x x x x x x x x x x x x x x x x x x x x x x x 56 Node Flow-Conservation Constraints Node Flow-Conservation Constraints x 12 + x 13 + x 14 + x 15 + x 16 = 1 (origin) x 12 + x 13 + x 14 + x 15 + x 16 = 1 (origin) – x 12 + x 25 + x 26 – x 52 = 0 (node 2) – x 12 + x 25 + x 26 – x 52 = 0 (node 2) – x 13 + x 34 + x 35 + x 36 – x 43 – x 53 = 0 (node 3) – x 13 + x 34 + x 35 + x 36 – x 43 – x 53 = 0 (node 3) – x 14 – x 34 + x 43 + x 45 + x 46 – x 54 = 0 (node 4) – x 14 – x 34 + x 43 + x 45 + x 46 – x 54 = 0 (node 4) – x 15 – x 25 – x 35 – x 45 + x 52 + x 53 + x 54 + x 56 = 0 (node 5) – x 15 – x 25 – x 35 – x 45 + x 52 + x 53 + x 54 + x 56 = 0 (node 5) x 16 + x 26 + x 36 + x 46 + x 56 = 1 (destination) x 16 + x 26 + x 36 + x 46 + x 56 = 1 (destination)

40 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. or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Shortest Route n Solution Summary Minimum total cost = $150 x 12 = 0 x 25 = 0 x 34 = 1 x 43 = 0 x 52 = 0 x 13 = 1 x 26 = 0 x 35 = 0 x 45 = 1 x 53 = 0 x 13 = 1 x 26 = 0 x 35 = 0 x 45 = 1 x 53 = 0 x 14 = 0 x 36 = 0 x 46 = 0 x 54 = 0 x 14 = 0 x 36 = 0 x 46 = 0 x 54 = 0 x 15 = 0 x 56 = 1 x 15 = 0 x 56 = 1 x 16 = 0 x 16 = 0

41 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. or duplicated, or posted to a publicly accessible website, in whole or in part. End of Chapter 6, Part A