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Quantitative Analysis for Management Chapter 9 Solutions

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  • Chapter 9To accompany Quantitative Analysis for Management, Eleventh Edition, Global Editionby Render, Stair, and Hanna Power Point slides created by Brian PetersonTransportation and Assignment Models

  • Learning ObjectivesStructure LP problems using the transportation, transshipment and assignment models.Use the northwest corner and stepping-stone methods.Solve facility location and other application problems with transportation models.Solve assignment problems with the Hungarian (matrix reduction) method.After completing this chapter, students will be able to:

  • Chapter Outline 9.1Introduction 9.2The Transportation Problem 9.3The Assignment Problem 9.4The Transshipment Problem 9.5The Transportation Algorithm 9.6Special Situations with the Transportation Algorithm 9.7Facility Location Analysis 9.8 The Assignment Algorithm 9.9 Special Situations with the Assignment Algorithm

  • IntroductionIn this chapter we will explore three special linear programming models:The transportation problem.The assignment problem.The transshipment problem.These problems are members of a category of LP techniques called network flow problems.

  • The Transportation ProblemThe transportation problem deals with the distribution of goods from several points of supply (sources) to a number of points of demand (destinations).Usually we are given the capacity of goods at each source and the requirements at each destination.Typically the objective is to minimize total transportation and production costs.

  • The Transportation ProblemThe Executive Furniture Corporation manufactures office desks at three locations: Des Moines, Evansville, and Fort Lauderdale.The firm distributes the desks through regional warehouses located in Boston, Albuquerque, and Cleveland.

  • The Transportation ProblemNetwork Representation of a Transportation Problem, with Costs, Demands and SuppliesFigure 9.1Executive Furniture Company$5$4$3$8$4$3$9$7$5SupplyDemand

  • Linear Programming for the Transportation ExampleLet Xij = number of units shipped from source i to destination j,Where:i = 1, 2, 3, with 1 = Des Moines, 2 = Evansville, and 3 = Fort Lauderdalej = 1, 2, 3, with 1 = Albuquerque, 2 = Boston, and 3 = Cleveland.

  • Linear Programming for the Transportation ExampleMinimize total cost = 5X11 + 4X12 + 3X13 +8X21 + 4X22 + 3X23 + 9X31 +7X32 + 5X33Subject to:X11 + X12 + X13 100 (Des Moines supply)X21 + X22 + X23 300 (Evansville supply)X31 + X32 + X33 300 (Fort Lauderdale supply)X11 + X21 + X31 = 300 (Albuquerque demand)X12 + X22 + X32 = 200 (Boston demand)X13 + X23 + X33 = 200 (Cleveland demand)Xij 0 for all i and j.

  • Executive Furniture Corporation Solution in Excel 2010Program 9.1

  • A General LP Model for Transportation ProblemsLet:Xij = number of units shipped from source i to destination j.cij = cost of one unit from source i to destination j.si = supply at source i.dj = demand at destination j.

  • A General LP Model for Transportation ProblemsMinimize cost = Subject to:

    i = 1, 2,, m.

    j = 1, 2, , n.

    xij 0for all i and j.

  • The Assignment ProblemThis type of problem determines the most efficient assignment of people to particular tasks, etc.Objective is typically to minimize total cost or total task time.

  • Linear Program for Assignment ExampleThe Fix-it Shop has just received three new repair projects that must be repaired quickly: a radio, a toaster oven, and a coffee table.Three workers with different talents are able to do the jobs.The owner estimates the cost in wages if the workers are assigned to each of the three jobs.Objective: minimize total cost.

  • Example of an Assignment Problem in a Transportation Network FormatFigure 9.2

  • Linear Program for Assignment ExampleLet:Xij = 1 if person i is assigned to project j, or 0 otherwise.Where:i = 1,2,3 with 1 = Adams, 2 = Brown, and 3 = Cooperj = 1,2,3, with 1 = Project 1, 2 = Project 2, and 3 = Project 3.

  • Linear Program for Assignment ExampleMinimize total cost = 11X11 + 14X12 + 6X13 + 8X21 + 10X22 + 11X23 + 9X31 + 12X32 + 7X33Subject to:X11 + X12 + X13 1X21 + X22 + X23 1X31 + X32 + X33 1X11 + X21 + X31 = 1X12 + X22 + X32 = 1X13 + X23 + X33 = 1Xij = 0 or 1 for all i and j

  • Fix-it Shop Solution in Excel 2010Program 9.2

  • Linear Program for Assignment ExampleX13 = 1, so Adams is assigned to project 3.X22 = 1, so Brown is assigned to project 2.X31 = 1, so Cooper is assigned to project 3.Total cost of the repairs is $25.

  • Transshipment ApplicationsWhen the items are being moved from a source to a destination through an intermediate point (a transshipment point), the problem is called a transshipment problem.

    Distribution CentersFrosty Machines manufactures snow blowers in Toronto and Detroit.These are shipped to regional distribution centers in Chicago and Buffalo.From there they are shipped to supply houses in New York, Philadelphia, and St Louis.Shipping costs vary by location and destination.Snow blowers cannot be shipped directly from the factories to the supply houses.

  • Network Representation of Transshipment ExampleFigure 9.3

  • Transshipment ApplicationsFrosty Machines Transshipment DataFrosty would like to minimize the transportation costs associated with shipping snow blowers to meet the demands at the supply centers given the supplies available.Table 9.1

    TOFROMCHICAGOBUFFALONEW YORK CITYPHILADELPHIAST LOUISSUPPLYToronto$4$7800Detroit$5$7700Chicago$6$4$5Buffalo$2$3$4Demand450350300

  • Transshipment ApplicationsA description of the problem would be to minimize cost subject to:The number of units shipped from Toronto is not more than 800.The number of units shipped from Detroit is not more than 700.The number of units shipped to New York is 450.The number of units shipped to Philadelphia is 350.The number of units shipped to St Louis is 300.The number of units shipped out of Chicago is equal to the number of units shipped into Chicago.The number of units shipped out of Buffalo is equal to the number of units shipped into Buffalo.

  • Transshipment ApplicationsThe decision variables should represent the number of units shipped from each source to the transshipment points and from there to the final destinations.X13 = the number of units shipped from Toronto to ChicagoX14 = the number of units shipped from Toronto to BuffaloX23 = the number of units shipped from Detroit to ChicagoX24 = the number of units shipped from Detroit to BuffaloX35 = the number of units shipped from Chicago to New YorkX36 = the number of units shipped from Chicago to PhiladelphiaX37 = the number of units shipped from Chicago to St LouisX45 = the number of units shipped from Buffalo to New YorkX46 = the number of units shipped from Buffalo to PhiladelphiaX47 = the number of units shipped from Buffalo to St Louis

    Annie Puciloski - X24 Key: "Chicago" s/b "Buffalo"

  • Transshipment ApplicationsThe linear program is:Minimize cost = 4X13 + 7X14 + 5X23 + 7X24 + 6X35 + 4X36 + 5X37 + 2X45 + 3X46 + 4X47 subject toX13 + X14 800(supply at Toronto)X23 + X24 700(supply at Detroit)X35 + X45= 450(demand at New York)X36 + X46= 350(demand at Philadelphia)X37 + X47= 300(demand at St Louis)X13 + X23= X35 + X36 + X37(shipping through Chicago)X14 + X24= X45 + X46 + X47(shipping through Buffalo)Xij 0for all i and j (nonnegativity)

  • Solution to Frosty Machines Transshipment ProblemProgram 9.3

  • The Transportation AlgorithmThis is an iterative procedure in which a solution to a transportation problem is found and evaluated using a special procedure to determine whether the solution is optimal.When the solution is optimal, the process stops.If not, then a new solution is generated.

  • Transportation Table for Executive Furniture CorporationTable 9.2Des Moines capacity constraintCell representing a source-to-destination (Evansville to Cleveland) shipping assignment that could be madeTotal supply and demandCleveland warehouse demandCost of shipping 1 unit from Fort Lauderdale factory to Boston warehouse

    TOFROMWAREHOUSE AT ALBUQUERQUEWAREHOUSE AT BOSTONWAREHOUSE AT CLEVELANDFACTORY CAPACITYDES MOINES FACTORY$5$4$3100

    EVANSVILLE FACTORY$8$4$3300

    FORT LAUDERDALE FACTORY$9$7$5300

    WAREHOUSE REQUIREMENTS300200200700

  • Developing an Initial Solution: Northwest Corner RuleOnce we have arranged the data in a table, we must establish an initial feasible solution.One systematic approach is known as the northwest corner rule.Start in the upper left-hand cell and allocate units to shipping routes as follows:Exhaust the supply (factory capacity) of each row before moving down to the next row.Exhaust the demand (warehouse) requirements of each column before moving to the right to the next column. Check that all supply and demand requirements are met.This problem takes five steps to make the initial shipping assignments.

  • Developing an Initial Solution: Northwest Corner RuleBeginning in the upper left hand corner, we assign 100 units from Des Moines to Albuquerque. This exhaust the supply from Des Moines but leaves Albuquerque 200 desks short. We move to the second row in the same column.

    TOFROMALBUQUERQUE (A)BOSTON (B) CLEVELAND (C)FACTORY CAPACITYDES MOINES (D)100$5$4$3100

    EVANSVILLE (E)$8$4$3300

    FORT LAUDERDALE (F)$9$7$5300

    WAREHOUSE REQUIREMENTS300200200700

  • Developing an Initial Solution: Northwest Corner RuleAssign 200 units from Evansville to Albuquerque. This meets Albuquerques demand. Evansville has 100 units remaining so we move to the right to the next column of the second row.

    TOFROMALBUQUERQUE (A)BOSTON (B) CLEVELAND (C)FACTORY CAPACITYDES MOINES (D)100$5$4$3

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Quantitative Analysis for Management Chapter 9 Solutions

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