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Slides by John Loucks St. Edward’s University

Slides by John Loucks St. Edward’s University. Chapter 6, Part A Distribution and Network Models. Transportation Problem Network Representation General LP Formulation Transshipment Problem Network Representation General LP Formulation.

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Slides by John Loucks St. Edward’s University

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  1. Slides by John Loucks St. Edward’s University

  2. Chapter 6, Part ADistribution and Network Models • Transportation Problem • Network Representation • General LP Formulation • Transshipment Problem • Network Representation • General LP Formulation

  3. Transportation, Assignment, and Transshipment Problems • 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. • Types of network models • Transportation • assignment • Transshipment • shortest-route • maximal flow problems

  4. Transportation Problem • 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. • Network representation • Graphical depiction of the problem • Consists of • Nodes (dots) • Arcs (lines)

  5. Transportation Problem • Network Representation 1 d1 c11 1 c12 s1 c13 2 d2 c21 c22 2 s2 c23 3 d3 Sources Destinations

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

  7. Transportation Problem • Linear Programming Formulation (continued) xij> 0 for all i and j

  8. Transportation Problem • LP Formulation Special Cases • Total supply exceeds total demand: • 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: xij<Li Remove the corresponding decision variable. No modification of LP formulation is necessary.

  9. Transportation Problem • LP Formulation Special Cases (continued) • The objective is maximizing profit or revenue: • 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. Solve as a maximization problem.

  10. Transportation Problem: Example #1 Acme Block Company has orders for 80 tons of concrete blocks at three suburban locations as follows: Northwood -- 25 tons, Westwood -- 45 tons, and Eastwood -- 10 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. Transportation Problem: Example #1 • Delivery Cost Per Ton NorthwoodWestwoodEastwood Plant 1 24 30 40 Plant 2 30 40 42

  12. Supply (t) Orig. Dest. Demand (t) Costs North Wood 25 24 Plant 1 50 40 West Wood 30 45 40 30 Plant 2 50 42 East Wood 10

  13. Transportation Problem: Example #1 • MIN • SHIPPING COSTS • ST • SUPPLY • DEMAND • NON-NEGATIVE

  14. Transportation Problem: Example #1 • LET xij = number of tons shipped from origin i to destination j • MIN • 24x11 + 30x12 + 40x13 + (Shipping cost from P1)30x21 + 40x22 + 42x23 (Shipping cost from P2) • ST • x11+ x12+ x13 <= 50 (Supply of Plant 1) • x21+ x22+ x23 <= 50 (Supply of Plant 2) • x11 + x21 = 25 (Northwood Demand) • x12+ x22= 45 (Westwood Demand) • x13+ x23= 10 (Eastwood Demand) • Xij >= 0 (Nonnegative)

  15. A B C D E F G 10 X11 X12 X13 X21 X22 X23 11 Dec.Var.Values 5 45 0 20 0 10 12 Minimized Total Shipping Cost 2490 13 14 Constraints LHS RHS 15 Plant 1 Capacity 50 <= 50 16 Plant 2 Capacity 30 <= 50 17 Northwood Demand 25 = 25 18 Westwood Demand 45 = 45 19 Eastwood Demand 10 = 10 Transportation Problem: Example #1 • Partial Spreadsheet Showing Optimal Solution

  16. Transportation Problem: Example #1 • Optimal Solution FromToAmountCost Plant 1 Northwood 5 120 Plant 1 Westwood 45 1,350 Plant 2 Northwood 20 600 Plant 2 Eastwood 10 420 Total Cost = $2,490

  17. Transportation Problem: Example #1 • Partial Sensitivity Report (first half) • 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. Constraints Constraints Final Shadow Constraint Allowable Allowable Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease Cell Name Value Price R.H. Side Increase Decrease $E$17 P2.Cap 30.0 0.0 50 1E+30 20 $E$17 P2.Cap 30.0 0.0 50 1E+30 20 $E$18 N.Dem 25.0 30.0 25 20 20 $E$18 N.Dem 25.0 30.0 25 20 20 $E$19 W.Dem 45.0 36.0 45 5 20 $E$19 W.Dem 45.0 36.0 45 5 20 $E$20 E.Dem 10.0 42.0 10 20 10 $E$20 E.Dem 10.0 42.0 10 20 10 $E$16 P1.Cap 50.0 -6.0 50 20 5 $E$16 P1.Cap 50.0 -6.0 50 20 5 Transportation Problem: Example #1 • Partial Sensitivity Report (second half) • How much unused capacity is there at plant 1? plant 2?

  19. Transshipment Problem • Transshipment problems • transportation problems in which a shipment may move through intermediate 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.

  20. Transshipment Problem • Network Representation c36 3 c13 c37 1 6 s1 d1 c14 c46 c15 4 Demand c47 Supply c23 c56 c24 7 2 d2 s2 c25 5 c57 Destinations Sources Intermediate Nodes

  21. 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 nodei dj= demand at destination nodej 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. Transshipment Problem • Linear Programming Formulation (continued) xij> 0 for all i and j continued

  23. 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.

  24. 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.

  25. Transshipment Problem: Example Because of long standing contracts based on past orders, unit costs from the manufacturers to the suppliers are: Zeron NZeron S Arnold 5 8 Supershelf 7 4 The costs to install the shelving at the various locations are: ZroxHewesRockrite Thomas 1 5 8 Washburn 3 4 4

  26. Transshipment Problem: Example • Network Representation ZROX Zrox 50 1 5 Zeron N Arnold 75 ARNOLD 5 8 8 Hewes 60 HEWES 3 7 Super Shelf Zeron S 4 75 WASH BURN 4 4 Rock- Rite 40

  27. Transshipment Problem: Example • How to model transhipment nodes • Flow out equal to flow in • (Unless a transshipment node has some sort of demand) • Flow out = Flow in • Flow out – flow in = 0 • OR • Flow in – flow out = 0

  28. Transshipment Problem: Example • LET 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)

  29. Transshipment Problem: Example • Cont’d • MIN 5x13+ 8x14 + 7x23 + 4x24 + (From supply to xship) 1x35+ 5x36 + 8x37 + 3x45 + 4x46 + 4x47 (From xship to demand) • SUBJECT TOAmount Out of Arnold: x13 + x14< 75 Amount Out of Supershelf: x23 + x24< 75 Amount Through Zeron N: x13 + x23 - x35 - x36 - x37 = 0 Amount Through Zeron S: x14 + x24 - x45 - x46 - x47 = 0 Amount Into Zrox: x35 + x45 = 50 Amount Into Hewes: x36 + x46 = 60 Amount Into Rockrite: x37 + x47 = 40 Non-negativity of Variables: xij> 0, for all i and j.

  30. Transshipment Problem: Example Objective Function Value = 1150.000 VariableValueReduced Costs X13 75.000 0.000 X14 0.000 2.000 X23 0.000 4.000 X24 75.000 0.000 X35 50.000 0.000 X36 25.000 0.000 X37 0.000 3.000 X45 0.000 3.000 X46 35.000 0.000 X47 40.000 0.000 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. Transshipment Problem: Example • Solution Zrox ZROX 50 50 75 1 5 Zeron N Arnold 75 ARNOLD 5 25 8 8 Hewes 60 35 HEWES 3 4 7 Super Shelf Zeron S 40 75 WASH BURN 4 4 75 Rock- Rite 40

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

  33. Shortest-Route Problem • Linear Programming Formulation Using the notation: xij = 1 if the arc from node i to node j is on the shortest route 0 otherwise cij= distance, time, or cost associated with the arc from node i to node j continued

  34. Shortest-Route Problem • Linear Programming Formulation (continued)

  35. Example: Shortest Route 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 should she take to minimize the total travel cost?

  36. Example: Shortest Route • Network Representation F 2 5 K L A B G J 3 C 6 1 D I Paducah H Lewisburg M E 4

  37. Example: Shortest Route Transport Time Ticket RouteMode(hours)Cost A Train 4 $ 20 B Plane 1 $115 C Bus 2 $ 10 D Taxi 6 $ 90 E Train 3 1/3 $ 30 F Bus 3 $ 15 G Bus 4 2/3 $ 20 H Taxi 1 $ 15 I Train 2 1/3 $ 15 J Bus 6 1/3 $ 25 K Taxi 3 1/3 $ 50 L Train 1 1/3 $ 10 M Bus 4 2/3 $ 20

  38. Example: Shortest Route Transport Time Time Ticket Total RouteMode(hours)CostCostCost A Train 4 $60 $ 20 $ 80 B Plane 1 $15 $115 $130 C Bus 2 $30 $ 10 $ 40 D Taxi 6 $90 $ 90 $180 E Train 3 1/3 $50 $ 30 $ 80 F Bus 3 $45 $ 15 $ 60 G Bus 4 2/3 $70 $ 20 $ 90 H Taxi 1 $15 $ 15 $ 30 I Train 2 1/3 $35 $ 15 $ 50 J Bus 6 1/3 $95 $ 25 $120 K Taxi 3 1/3 $50 $ 50 $100 L Train 1 1/3 $20 $ 10 $ 30 M Bus 4 2/3 $70 $ 20 $ 90

  39. Example: Shortest Route • LP Formulation • Objective Function Min 80x12 + 40x13 + 80x14 + 130x15 + 180x16 + 60x25 + 100x26 + 30x34 + 90x35 + 120x36 + 30x43 + 50x45 + 90x46 + 60x52 + 90x53 + 50x54 + 30x56 • Node Flow-Conservation Constraints x12 + x13 + x14 + x15 + x16 = 1 (origin) – x12 + x25 + x26 – x52 = 0 (node 2) – x13 + x34 + x35 + x36 – x43 – x53 = 0 (node 3) – x14 – x34 + x43 + x45 + x46 – x54 = 0 (node 4) – x15 – x25 – x35 – x45 + x52 + x53 + x54 + x56 = 0 (node 5) x16 + x26 + x36 + x46 + x56 = 1 (destination)

  40. Example: Shortest Route • Solution Summary Minimum total cost = $150 x12 = 0 x25 = 0 x34 = 1 x43 = 0 x52 = 0 x13 = 1 x26 = 0 x35 = 0 x45 = 1 x53 = 0 x14 = 0 x36 = 0 x46 = 0 x54 = 0 x15 = 0 x56 = 1 x16 = 0

  41. End of Chapter 6, Part A

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