1 / 49

Review

Review. Review. We will spend up to 30 minutes reviewing Exam 1 K now how your answers were graded. Know how to correct your mistakes. Your final exam is cumulative, and may contain similar questions. Readings. Readings. Chapter 6 Distribution and Network Models. Overview. Overview.

brit
Download Presentation

Review

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Review • Review We will spend up to 30 minutes reviewing Exam 1 • Know how your answers were graded. • Know how to correct your mistakes. Your final exam is cumulative, and may contain similar questions.

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

  3. Overview • Overview

  4. Overview

  5. Overview Tool Summary • Write the objective of maximizing a minimum as a linear program. • For example, maximize min {2x, 3y} as maximize M subject to 2x > M and 3y > M. • Define decision variable xij = units moving from origin i to destination j. • Write origin constraints (with < or =): • Write destination constraints (with < or =):

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

  7. Network Models • Network Models

  8. Network Models Network Models are nodes, arcs, and functions (costs, supplies, demands, etc.) associated with the arcs and nodes. Transportation, assignment, transshipment, and shortest-route problems are examples.

  9. Network Models • Each of the four network models (transportation, assignment, transshipment, and shortest-route problems) can be formulated as linear programs and solved by general-purpose linear programming codes. • For each of the four models, if the right-hand side of the linear programming formulations are all integers, the optimal solution will be integer values for the decision variables. • There are many computer packages (including The Management Scientist) that contain convenient separate computer codes for these models, which take advantage of their network structure. But do not use such codes on exams because they lack the flexibility of the general-purpose linear programming codes.

  10. Transportation • Transportation

  11. Transportation Overview Transportation Problems are Resource Allocation Problems when outputs are fixed, and when outputs and inputs occur at different locations. Transportation Problems thus help determine the transportation of goods from m origins (each with a supply si) to n destinations (each with a demand dj) to minimize cost.

  12. Transportation Here is the network representation for a transportation problem with two sources and three destinations. 1 d1 c11 1 c12 s1 c13 2 d2 c21 c22 2 s2 c23 3 d3 Sources Destinations

  13. Transportation • 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 • Linear programming formulation (supply inequality, demand equality). = xij> 0 for all i and j

  14. Transportation Possible 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.

  15. Transportation Question: Acme Block Company has orders for 80 tons of concrete blocks at three suburban locations: Northwood -- 25 tons, Westwood -- 45 tons, and Eastwood -- 10 tons. Acme has two plants, each of which can produce 50 tons per week. Delivery costs per ton from each plant to each suburban location are thus: NorthwoodWestwoodEastwood Plant 1 24 30 40 Plant 2 30 40 42 Formulate then solve the linear program that determines how shipments should be made to fill the orders above.

  16. Transportation • Answer: Linear programming formulation (supply inequality, demand equality). • Variables: Xij = Tons shipped from Plant i to Destination j • Objective: • Min 24 X11 + 30 X12 + 40 X13 + 30 X21 + 40 X22 + 42 X23 • Supply Constraints: • X11 + X12 + X13 < 50 • X21 + X22 + X23 < 50 • Demand Constraints: • X11 + X21 = 25 • X12 + X22 = 45 • X13 + X23 = 10

  17. Transportation

  18. Transportation • Define sources: Source 1 = Plant 1, Source 2 = Plant 2. • Define destinations: 1 = Northwood, 2 = Westwood, 3 = Eastwood. • Define costs: c11= 24 c12= 30 c13= 40 c21= 30 c22= 40 c23= 42 2+3=5 2x3 = 6 • Define 2 supplies: s1= 50, s2= 50. • Define 3 demands: d1= 25, d2= 45, d3= 10. • Define variables: Xij = number of units • shipped from Source i to Destination j. Cost c13= 40 Supply s1 = 50 Demand d2 = 45

  19. Transportation Variable names: Xij= number of units shipped from Plant i to Destination j. Destination 1 = Northwood; 2 = Westwood; 3 = Eastwood Optimal shipments: 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

  20. Transportation

  21. Transportation Variable names: Origin i = Plant i Destination 1 = Northwood Destination 2 = Westwood Destination 3 = Eastwood Cost from Plant 1 to Northwood Optimal shipments: 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

  22. Transportation with Modes of Transport • Transportation with Modes of Transport

  23. Transportation with Modes of Transport Overview Transportation Problems with Modes of Transport re-interpret some of the different “origins” in a basic transportation problem to include not only location but modes of transportation. For example, instead of “San Diego” as an origin specifying location, we have “San Diego by Truck” as an origin specifying both location and mode of transportation.

  24. Transportation with Modes of Transport • Question: The Navy has 9,000 pounds of material in Albany, Georgia that it wishes to ship to three installations: San Diego, Norfolk, and Pensacola. They require 4,000, 2,500, and 2,500 pounds, respectively. Government regulations require equal distribution of shipping among the three carriers. • The shipping costs per pound by truck, railroad, and airplane are: • Formulate then solve the linear program that determines shipping arrangements (mode, destination, and quantity) that minimize the total shipping cost. Destination Mode San Diego Norfolk Pensacola Truck $12 $ 6 $ 5 Railroad $20 $11 $ 9 Airplane $30 $26 $28

  25. Transportation with Modes of Transport

  26. Transportation with Modes of Transport • Define the variables. We want to determine the pounds of material, xij, to be shipped by mode i to destination j. • Variable names: • Define the objective. Minimize the total shipping cost. Min: (shipping cost per pound for each mode-destination pairing) x (number of pounds shipped by mode-destination pairing). Min: 12x11 + 6x12 + 5x13 + 20x21 + 11x22 + 9x23 + 30x31 + 26x32 + 28x33 San Diego Norfolk Pensacola Truck x11x12x13 Railroad x21x22x23 Airplane x31x32x33

  27. Transportation with Modes of Transport • Define the constraints of equal use of transportation modes: (1) x11 + x12 + x13 = 3000 (2) x21 + x22 + x23 = 3000 (3) x31 + x32 + x33 = 3000 • Define the destination material constraints: (4) x11 + x21 + x31 = 4000 (5) x12 + x22 + x32 = 2500 (6) x13 + x23 + x33 = 2500

  28. Transportation with Modes of Transport • Linear programming summary. • Variables: Xij = Pounds shipped by Mode i to Destination j • Objective: • Min 12 X11 + 6 X12 + 5 X13 • + 20 X21 + 11 X22 + 9 X23 + 30 X31 + 26 X32 + 28 X33 • Mode (Supply equality) Constraints: • X11 + X12 + X13 = 3000 • X21 + X22 + X23 = 3000 • X31 + X32 + X33 = 3000 • Destination Constraints: • X11 + X21 + X31 = 4000 • X12 + X22 + X32 = 2500 • X13 + X23 + X33 = 2500

  29. Transportation with Modes of Transport San Diego Norfolk Pensacola Truck X11 X12 X13 Railroad X21 X22 X23 Airplane X31 X32 X33 Variable names: Units to San Diego by truck • Solution Summary: • San Diego receives 1000 lbs. by truck • and 3000 lbs. by airplane. • Norfolk receives 2000 lbs. by truck • and 500 lbs. by railroad. • Pensacola receives 2500 lbs. by railroad. • The total shipping cost is $142,000.

  30. Transportation with Modes of Transport The Management Science Transportation module is not available. Remember, in that formulation, the supply constraints are inequalities. = xij> 0 for all i and j But in Example 2, the “origins” are the modes of shipping, and the supply constraint on each mode is an equality.

  31. Assignment • Assignment

  32. Assignment Overview Assignment Problems are Transportation Problems when the “goods” are workers that are transported to jobs, and each worker either does all of a job or none of it. Assignment Problems thusminimize the total cost of assigning of m workers (or agents) to m jobs (or tasks). The simplest way to model all-or-nothing in any linear program is to restrict the fraction of the job completed to be a binary (0 or 1) decision variable.

  33. Assignment An assignment problem is thus a special case of a transportation problem in which all supplies and all demands equal to 1; hence assignment problems may be solved as linear programs. And although the only sensible solution quantities are binary (0 or 1), the special form of the problem and of The Management Scientist guarantees all solutions are binary (0 or 1).

  34. Assignment Here is the network representation of an assignment problem with three workers (agents) and three jobs (tasks): c11 1 1 c12 c13 Agents Tasks c21 c22 2 2 c23 c31 c32 3 3 c33

  35. Assignment Notation: xij = 1 if agent i is assigned to task j 0 otherwise cij= cost of assigning agent i to task j s.t. xij> 0 for all i and j

  36. Assignment • Possible variations: • Number of agents exceeds the number of tasks: Extra agents simply remain unassigned. • An assignment is unacceptable: Remove the corresponding decision variable. • An agent is permitted to work t tasks:

  37. Assignment Question: Russell electrical contractors pay their subcontractors a fixed fee plus mileage for work performed. On a given day the contractor is faced with three electrical jobs associated with various projects. Given below are the distances between the subcontractors and the projects. Projects SubcontractorABC Westside 50 36 16 Federated 28 30 18 Goliath 35 32 20 Universal 25 25 14 Assume each subcontractor can perform at most one project. Formulate then solve the linear program that assigns contractors to minimize total mileage costs.

  38. Assignment Answer: 50 West. A 36 16 Subcontractors Projects 28 30 Fed. B 18 32 35 Gol. C 20 25 25 Univ. 14

  39. Assignment Project A Project B Project C Westside x11x12x13 Federated x21x22x23 Goliath x31x32x33 Universal x41x42x43 Variable names: • There will be 1 variable for each agent-task pair, so 12 variables all together. • There will be 1 constraint for each agent and for each task, so 7 constraints all together.

  40. Assignment Project A Project B Project C Westside x11x12x13 Federated x21x22x23 Goliath x31x32x33 Universal x41x42x43 Variable names: Min 50x11+36x12+16x13+28x21+30x22+18x23 +35x31+32x32+20x33+25x41+25x42+14x43 s.t. x11+x12+x13 < 1 x21+x22+x23 < 1 x31+x32+x33 < 1 x41+x42+x43 < 1 x11+x21+x31+x41 = 1 x12+x22+x32+x42 = 1 x13+x23+x33+x43 = 1 xij = 0 or 1 for all i and j Agents Tasks

  41. Assignment Agent 1 capacity: x11+x12+x13 < 1 Project A Project B Project C Westside x11x12x13 Federated x21x22x23 Goliath x31x32x33 Universal x41x42x43 Task 3 done: x13+x23+x33+x43 = 1

  42. Assignment Project A Project B Project C Westside x11x12x13 Federated x21x22x23 Goliath x31x32x33 Universal x41x42x43 Variable names: Optimal assignment: SubcontractorProjectDistance Westside C 16 Federated A 28 Goliath (unassigned) Universal B 25 Total distance = 69 miles

  43. Assignment

  44. Assignment Projects SubcontractorABC Westside 50 36 16 Federated 28 30 18 Goliath 35 32 20 Universal 25 25 14 Optimal assignment: SubcontractorProjectDistance Westside C 16 Federated A 28 Goliath (unassigned) Universal B 25 Total distance = 69 miles

  45. Assignment Question: Now change Example 3 to take into account the recent marriage of the Goliath subcontractor to your youngest daughter. That is, you have to assign Goliath one of the jobs. How should the contractors now be assigned to minimize total mileage costs?

  46. Assignment Alternative notation: WA = 0 if Westside does not get task A 1 if Westside does get task A and so on. Min 50WA+36WB+16WC+28FA+30FB+18FC +35GA+32GB+20GC+25UA+25UB+14UC s.t. WA+WB+WC< 1 FA+FB+FC < 1 GA+GB+GC = 1 UA+UB+UC < 1 WA+FA+GA+UA = 1 WB+FB+GB+UB = 1 WC+FC+GC+UC = 1 Agents Tasks

  47. Assignment Goliath gets a task: GA+GB+GC = 1 Task A gets done:WA+FA+GA+UA=1

  48. Assignment Optimal assignment: SubcontractorProjectDistance Westside C 16 • Federated (unassigned) Goliath B 32 Universal A 25 Total distance = 73 miles

  49. BA 452 Quantitative Analysis End of Lesson B.1

More Related