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Readings

Readings. Readings. Chapter 6 Distribution and Network Models. Overview. Overview. Overview. Overview. Tool Summary Define decision variable x ij = units moving from origin i to destination j. Write origin constraints (with < or =):

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Readings

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  1. Readings • Readings • Chapter 6 • Distribution and Network Models

  2. Overview • Overview

  3. Overview

  4. Overview Tool Summary • Define decision variable xij = units moving from origin i to destination j. • Write origin constraints (with < or =): • Write destination constraints (with < or =): • Write transshipment constraints (with < or =):

  5. Overview Tool Summary • Identify implicit assumptions needed to complete a formulation, such as all agents having an equal value of time.

  6. Transshipment • Transshipment

  7. Transshipment Overview Transshipment Problems are Transportation Problems extended so that a shipment may move through intermediate nodes (transshipment nodes) before reaching a particular destination node.

  8. Transshipment This is the network representation for a transshipment problem with two sources, three intermediate nodes, and two destinations: 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

  9. Transshipment 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 node j xij> 0 for all i and j

  10. Transshipment Problem Variations • 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.

  11. Transshipment Question: 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 up to 75 units to its customers. 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 Zeron N 1 5 8 Zeron S 3 4 4

  12. Transshipment Formulate and solve a transshipment linear programming problem for Zeron Industries.

  13. Transshipment Answer: Since demands by the three customer firms (Zrox, Hewes, Rockrite) are fixed, revenue for Zeron Industries is fixed, and so profit maximization is the same as cost minimization. There is data on transportation costs, but there is no data on the cost of ordering from suppliers. Nevertheless, if the unit cost from each supplier is the same, say P per unit, then the cost of ordering supply equal to the fixed demand of 50, 60, and 40 is fixed at 150P. Hence, to minimize cost, we just have to minimize transportation cost. To that end, it may help to draw a network model:

  14. Transshipment 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

  15. Transshipment • Next, Define decision variables: 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) • Problem Features: • There will be 1 variable for each manufacturer-supplier pair and each supplier-customer pair, so 10 variables all together. • There will be 1 constraint for each manufacturer, 1 for each supplier, and 1 for each customer, so 7 constraints all together.

  16. Transshipment • Define objective function: Minimize total shipping costs. Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 8x37+ 3x45 + 4x46 + 4x47 • Constrain amount out of Arnold: x13 + x14< 75 • Constrain amount out of Supershelf: x23 + x24< 75 • Constrain amount through Zeron N: x13 + x23 - x35 - x36 - x37 = 0 • Constrain amount through Zeron S: x14 + x24 - x45 - x46 - x47 = 0 • Constrain amount into Zrox: x35 + x45 = 50 • Constrain amount into Hewes: x36 + x46 = 60 • Constrain amount into Rockrite: x37 + x47 = 40

  17. Transshipment Indicies: i = 1 (Arnold), 2 (Supershelf) j = 3 (Zeron N), 4 (Zeron S) k = 5 (Zrox), 6 (Hewes), 7 (Rockrite) Minimized shipping costs Out of Arnold through Zeron N Out of Supershelf through Zeron S Through Zeron N into Zrox Through Zeron S into Hewes

  18. Transshipment Indicies: i = 1 (Arnold), 2 (Supershelf) j = 3 (Zeron N), 4 (Zeron S) k = 5 (Zrox), 6 (Hewes), 7 (Rockrite) 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

  19. Transshipment with Transshipment Origins • Transshipment with Transshipment Origins

  20. Transshipment with Transshipment Origins Overview Transshipment Problems with Transshipment Origins seek 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), where goods from one origin may move through other origins (transshipment nodes) before reaching a particular destination node.

  21. Transshipment with Transshipment Origins Question: Index cities i = 1 (Newbury Park), i = 2 (Thousand Oaks), i = 3 (Westlake Hills), i = 4 (Agoura Hills), i = 5 (Calabasas). Suppose you run rental car lots in each city. Newbury Park has a surplus of 3 cars (it has 3 more cars than it needs), Westlake Hills has a surplus of 2 cars (it has 2 more cars than it needs), and Calabasas has a deficit of 5 cars (it needs 5 more cars than it has).

  22. Transshipment with Transshipment Origins • Suppose you calculate the following costs per car of transporting cars between the cities: • transporting between 1 and 2 (that is, either 1 to 2, or 2 to 1) costs $2 • transporting between 1 and 3 costs $3 • transporting between 1 and 4 costs $4 • transporting between 1 and 5 costs $5 • transporting between 2 and 3 costs $2 • transporting between 2 and 4 costs $3 • transporting between 2 and 5 costs $4 • transporting between 3 and 4 costs $2 • transporting between 3 and 5 costs $3 • transporting between 4 and 5 costs $2 • How should you move cars between cities? Formulate your rental-car problem as a linear program, but you need not solve for the optimum. • Tip: Your written answer should define the decision variables, and formulate the objective and constraints.

  23. Transshipment with Transshipment Origins Answer: Define decision variables: xij = amount of cars moved from City i to City j Define objective function: Minimize total costs. Min 2(x12+x21) + 3(x13+x31) + 4(x14+x41) + 5(x15+x51) + 2(x23+x32) + 3(x24+x42) + 4(x25+x52) + 2(x34+x43) + 3(x35+x53) + 2(x45+x54) Constrain cars from City 1: x12 + x13 + x14 + x15 = 3 + x21 + x31 + x41 + x51 Constrain cars from City 2: x21 + x23 + x24 + x25 = x12 + x32 + x42 + x52 Constrain cars from City 3: x31 + x32 + x34 + x35 = 2 + x13 + x23 + x43 + x53 Constrain cars from City 4: x41 + x42 + x43 + x45 = x14 + x24 + x34 + x54 Constrain cars from City 5: x51 + x52 + x53 + x54 = -5 + x15 + x25 + x35 + x45 (Or constraints can be written with < rather than =. It does not matter since excess supply exactly cancels excess demand.)

  24. Shortest Route • Shortest Route

  25. Shortest Route Overview Shortest Route Problems are Transshipment Problems where there is one origin, one destination, one unit supplied, and one unit demanded, and where that unit is indivisible. Shortest Route Problems find the shortest path in a network from one node (or set of nodes) to another node (or set of nodes). The criterion to be minimized in the shortest-route problem is not limited to distance, but can be minimum time or cost.

  26. Shortest Route 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 xij> 1

  27. Shortest Route Question: 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?

  28. Shortest Route F 2 5 K L A B G J 3 C 6 1 D I Paducah H Lewisburg M E 4

  29. Shortest Route Route Transport Time Ticket (Arc)Mode(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

  30. Shortest Route Using the wage of $15 per hour, compute total cost. Transport Time TimeTicket 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

  31. Shortest Route • Define indices: Nodes 1 (origin), 2, …, 6 (destination) • Define decision variables: xij = 1 if the route from node i to node j is on the shortest route • Problem Features: • There is 1 decision variable for each possible route. • There is 1 constraint for each node. Cost of route from 1 to 4 • Define objective function: Minimize total transportation costs. Min 80x12 + 40x13 + 80x14 + 130x15 + 180x16 + 60x25 + 100x26 + 30x34 + 90x35 + 120x36 + 30x43 + 50x45 + 90x46 + 60x52 + 90x53 + 50x54 + 30x56

  32. Shortest Route • 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)

  33. Shortest Route Variables: xij = 1 if the arc from node i to node j is on the shortest route 0 otherwise Possible arcs: x12, x13, x14, x15, x16, x25, x26, x34, x35, x36, x43, x45, x46, x52, x53, x54, x56 Minimized transportation cost From Origin to Node 3 From Node 3 to Node 4 From Node 4 to Node 5 From Node 5 to Destination

  34. Shortest Route

  35. Shortest Route Input window Cost of Arc from Node 1 to Node 3 Output window Minimized transportation costs

  36. BA 452 Quantitative Analysis End of Lesson B.2

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