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• 1. B - 1© 2011 Pearson Education, Inc. publishing as Prentice HallB Linear ProgrammingPowerPoint presentation to accompanyHeizer and RenderOperations Management, 10ePrinciples of Operations Management, 8ePowerPoint slides by Jeff Heyl
• 2. B - 2© 2011 Pearson Education, Inc. publishing as Prentice HallOutline Why Use Linear Programming? Requirements of a LinearProgramming Problem Formulating Linear ProgrammingProblems Shader Electronics Example
• 3. B - 3© 2011 Pearson Education, Inc. publishing as Prentice HallOutline – Continued Graphical Solution to a LinearProgramming Problem Graphical Representation ofConstraints Iso-Profit Line Solution Method Corner-Point Solution Method
• 4. B - 4© 2011 Pearson Education, Inc. publishing as Prentice HallOutline – Continued Sensitivity Analysis Sensitivity Report Changes in the Resources of theRight-Hand-Side Values Changes in the Objective FunctionCoefficient Solving Minimization Problems
• 5. B - 5© 2011 Pearson Education, Inc. publishing as Prentice HallOutline – Continued Linear Programming Applications Production-Mix Example Diet Problem Example Labor Scheduling Example The Simplex Method of LP
• 6. B - 6© 2011 Pearson Education, Inc. publishing as Prentice HallLearning ObjectivesWhen you complete this module youshould be able to:1. Formulate linear programmingmodels, including an objectivefunction and constraints2. Graphically solve an LP problem withthe iso-profit line method3. Graphically solve an LP problem withthe corner-point method
• 7. B - 7© 2011 Pearson Education, Inc. publishing as Prentice HallLearning ObjectivesWhen you complete this module youshould be able to:4. Interpret sensitivity analysis andshadow prices5. Construct and solve a minimizationproblem6. Formulate production-mix, diet, andlabor scheduling problems
• 8. B - 8© 2011 Pearson Education, Inc. publishing as Prentice HallWhy Use Linear Programming? A mathematical technique tohelp plan and make decisionsrelative to the trade-offsnecessary to allocate resources Will find the minimum ormaximum value of the objective Guarantees the optimal solutionto the model formulated
• 9. B - 9© 2011 Pearson Education, Inc. publishing as Prentice HallLP Applications1. Scheduling school buses to minimizetotal distance traveled2. Allocating police patrol units to highcrime areas in order to minimizeresponse time to 911 calls3. Scheduling tellers at banks so thatneeds are met during each hour of theday while minimizing the total cost oflabor
• 10. B - 10© 2011 Pearson Education, Inc. publishing as Prentice HallLP Applications4. Selecting the product mix in a factoryto make best use of machine- andlabor-hours available while maximizingthe firm’s profit5. Picking blends of raw materials in feedmills to produce finished feedcombinations at minimum costs6. Determining the distribution systemthat will minimize total shipping cost
• 11. B - 11© 2011 Pearson Education, Inc. publishing as Prentice HallLP Applications7. Developing a production schedule thatwill satisfy future demands for a firm’sproduct and at the same time minimizetotal production and inventory costs8. Allocating space for a tenant mix in anew shopping mallso as to maximizerevenues to theleasing company
• 12. B - 12© 2011 Pearson Education, Inc. publishing as Prentice HallRequirements of anLP Problem1. LP problems seek to maximize orminimize some quantity (usuallyprofit or cost) expressed as anobjective function2. The presence of restrictions, orconstraints, limits the degree towhich we can pursue ourobjective
• 13. B - 13© 2011 Pearson Education, Inc. publishing as Prentice HallRequirements of anLP Problem3. There must be alternative coursesof action to choose from4. The objective and constraints inlinear programming problemsmust be expressed in terms oflinear equations or inequalities
• 14. B - 14© 2011 Pearson Education, Inc. publishing as Prentice HallFormulating LP ProblemsThe product-mix problem at Shader Electronics Two products1. Shader x-pod, a portable musicplayer2. Shader BlueBerry, an internet-connected color telephone Determine the mix of products that willproduce the maximum profit
• 15. B - 15© 2011 Pearson Education, Inc. publishing as Prentice HallFormulating LP Problemsx-pods BlueBerrys Available HoursDepartment (X1) (X2) This WeekHours Requiredto Produce 1 UnitElectronic 4 3 240Assembly 2 1 100Profit per unit \$7 \$5Decision Variables:X1 = number of x-pods to be producedX2 = number of BlueBerrys to be producedTable B.1
• 16. B - 16© 2011 Pearson Education, Inc. publishing as Prentice HallFormulating LP ProblemsObjective Function:Maximize Profit = \$7X1 + \$5X2There are three types of constraints Upper limits where the amount used is ≤the amount of a resource Lower limits where the amount used is ≥the amount of the resource Equalities where the amount used is =the amount of the resource
• 17. B - 17© 2011 Pearson Education, Inc. publishing as Prentice HallFormulating LP ProblemsSecond Constraint:2X1 + 1X2 ≤ 100 (hours of assembly time)Assemblytime availableAssemblytime used is ≤First Constraint:4X1 + 3X2 ≤ 240 (hours of electronic time)Electronictime availableElectronictime used is ≤
• 18. B - 18© 2011 Pearson Education, Inc. publishing as Prentice HallGraphical Solution Can be used when there are twodecision variables1. Plot the constraint equations at theirlimits by converting each equationto an equality2. Identify the feasible solution space3. Create an iso-profit line based onthe objective function4. Move this line outwards until theoptimal point is identified
• 19. B - 19© 2011 Pearson Education, Inc. publishing as Prentice HallGraphical Solution100 ––80 ––60 ––40 ––20 –––| | | | | | | | | | |0 20 40 60 80 100NumberofBlueBerrysNumber of x-podsX1X2Assembly (Constraint B)Electronics (Constraint A)FeasibleregionFigure B.3
• 20. B - 20© 2011 Pearson Education, Inc. publishing as Prentice HallGraphical Solution100 ––80 ––60 ––40 ––20 –––| | | | | | | | | | |0 20 40 60 80 100NumberofBlueBerrysNumber of x-podsX1X2Assembly (Constraint B)Electronics (Constraint A)FeasibleregionFigure B.3Iso-Profit Line Solution MethodChoose a possible value for theobjective function\$210 = 7X1 + 5X2Solve for the axis intercepts of the functionand plot the lineX2 = 42 X1 = 30
• 21. B - 21© 2011 Pearson Education, Inc. publishing as Prentice HallGraphical Solution100 ––80 ––60 ––40 ––20 –––| | | | | | | | | | |0 20 40 60 80 100NumberofBlueBerrysNumber of x-podsX1X2Figure B.4(0, 42)(30, 0)\$210 = \$7X1 + \$5X2
• 22. B - 22© 2011 Pearson Education, Inc. publishing as Prentice HallGraphical Solution100 ––80 ––60 ––40 ––20 –––| | | | | | | | | | |0 20 40 60 80 100NumberofBlueBerrysNumber of x-podsX1X2Figure B.5\$210 = \$7X1 + \$5X2\$420 = \$7X1 + \$5X2\$350 = \$7X1 + \$5X2\$280 = \$7X1 + \$5X2
• 23. B - 23© 2011 Pearson Education, Inc. publishing as Prentice HallGraphical Solution100 ––80 ––60 ––40 ––20 –––| | | | | | | | | | |0 20 40 60 80 100NumberofBlueBerrysNumber of x-podsX1X2Figure B.6\$410 = \$7X1 + \$5X2Maximum profit lineOptimal solution point(X1 = 30, X2 = 40)
• 24. B - 24© 2011 Pearson Education, Inc. publishing as Prentice Hall100 ––80 ––60 ––40 ––20 –––| | | | | | | | | | |0 20 40 60 80 100NumberofBlueBerrysNumber of x-podsX1X2Corner-Point MethodFigure B.71234
• 25. B - 25© 2011 Pearson Education, Inc. publishing as Prentice HallCorner-Point Method The optimal value will always be at acorner point Find the objective function value at eachcorner point and choose the one with thehighest profitPoint 1 : (X1 = 0, X2 = 0) Profit \$7(0) + \$5(0) = \$0Point 2 : (X1 = 0, X2 = 80) Profit \$7(0) + \$5(80) = \$400Point 4 : (X1 = 50, X2 = 0) Profit \$7(50) + \$5(0) = \$350
• 26. B - 26© 2011 Pearson Education, Inc. publishing as Prentice HallCorner-Point Method The optimal value will always be at acorner point Find the objective function value at eachcorner point and choose the one with thehighest profitPoint 1 : (X1 = 0, X2 = 0) Profit \$7(0) + \$5(0) = \$0Point 2 : (X1 = 0, X2 = 80) Profit \$7(0) + \$5(80) = \$400Point 4 : (X1 = 50, X2 = 0) Profit \$7(50) + \$5(0) = \$350Solve for the intersection of two constraints2X1 + 1X2 ≤ 100 (assembly time)4X1 + 3X2 ≤ 240 (electronics time)4X1 + 3X2 = 240- 4X1 - 2X2 = -200+ 1X2 = 404X1 + 3(40) = 2404X1 + 120 = 240X1 = 30
• 27. B - 27© 2011 Pearson Education, Inc. publishing as Prentice HallCorner-Point Method The optimal value will always be at acorner point Find the objective function value at eachcorner point and choose the one with thehighest profitPoint 1 : (X1 = 0, X2 = 0) Profit \$7(0) + \$5(0) = \$0Point 2 : (X1 = 0, X2 = 80) Profit \$7(0) + \$5(80) = \$400Point 4 : (X1 = 50, X2 = 0) Profit \$7(50) + \$5(0) = \$350Point 3 : (X1 = 30, X2 = 40) Profit \$7(30) + \$5(40) = \$410
• 28. B - 28© 2011 Pearson Education, Inc. publishing as Prentice HallSensitivity Analysis How sensitive the results are toparameter changes Change in the value of coefficients Change in a right-hand-side value ofa constraint Trial-and-error approach Analytic postoptimality method
• 29. B - 29© 2011 Pearson Education, Inc. publishing as Prentice HallSensitivity ReportProgram B.1
• 30. B - 30© 2011 Pearson Education, Inc. publishing as Prentice HallChanges in Resources The right-hand-side values ofconstraint equations may changeas resource availability changes The shadow price of a constraint isthe change in the value of theobjective function resulting from aone-unit change in the right-hand-side value of the constraint
• 31. B - 31© 2011 Pearson Education, Inc. publishing as Prentice HallChanges in Resources Shadow prices are often explainedas answering the question “Howmuch would you pay for oneadditional unit of a resource?” Shadow prices are only valid over aparticular range of changes in right-hand-side values Sensitivity reports provide theupper and lower limits of this range
• 32. B - 32© 2011 Pearson Education, Inc. publishing as Prentice HallSensitivity Analysis–100 ––80 ––60 ––40 ––20 –––| | | | | | | | | | |0 20 40 60 80 100 X1X2Figure B.8 (a)Changed assembly constraint from2X1 + 1X2 = 100to 2X1 + 1X2 = 110Electronics constraintis unchangedCorner point 3 is still optimal, butvalues at this point are now X1 = 45,X2 = 20, with a profit = \$4151234
• 33. B - 33© 2011 Pearson Education, Inc. publishing as Prentice HallSensitivity Analysis–100 ––80 ––60 ––40 ––20 –––| | | | | | | | | | |0 20 40 60 80 100 X1X2Figure B.8 (b)Changed assembly constraint from2X1 + 1X2 = 100to 2X1 + 1X2 = 90Electronics constraintis unchangedCorner point 3 is still optimal, butvalues at this point are now X1 = 15,X2 = 60, with a profit = \$4051234
• 34. B - 34© 2011 Pearson Education, Inc. publishing as Prentice HallChanges in theObjective Function A change in the coefficients in theobjective function may cause adifferent corner point to become theoptimal solution The sensitivity report shows howmuch objective functioncoefficients may change withoutchanging the optimal solution point
• 35. B - 35© 2011 Pearson Education, Inc. publishing as Prentice HallSolving MinimizationProblems Formulated and solved in much thesame way as maximizationproblems In the graphical approach an iso-cost line is used The objective is to move the iso-cost line inwards until it reachesthe lowest cost corner point
• 36. B - 36© 2011 Pearson Education, Inc. publishing as Prentice HallMinimization ExampleX1 = number of tons of black-and-white picturechemical producedX2 = number of tons of color picture chemicalproducedMinimize total cost = 2,500X1 + 3,000X2Subject to:X1 ≥ 30 tons of black-and-white chemicalX2 ≥ 20 tons of color chemicalX1 + X2 ≥ 60 tons totalX1, X2 ≥ \$0 nonnegativity requirements
• 37. B - 37© 2011 Pearson Education, Inc. publishing as Prentice HallMinimization ExampleTable B.960 –50 –40 –30 –20 –10 ––| | | | | | |0 10 20 30 40 50 60X1X2FeasibleregionX1 = 30X2 = 20X1 + X2 = 60ba
• 38. B - 38© 2011 Pearson Education, Inc. publishing as Prentice HallMinimization ExampleTotal cost at a = 2,500X1 + 3,000X2= 2,500 (40) + 3,000(20)= \$160,000Total cost at b = 2,500X1 + 3,000X2= 2,500 (30) + 3,000(30)= \$165,000Lowest total cost is at point a
• 39. B - 39© 2011 Pearson Education, Inc. publishing as Prentice HallLP ApplicationsProduction-Mix ExampleDepartmentProduct Wiring Drilling Assembly Inspection Unit ProfitXJ201 .5 3 2 .5 \$ 9XM897 1.5 1 4 1.0 \$12TR29 1.5 2 1 .5 \$15BR788 1.0 3 2 .5 \$11Capacity MinimumDepartment (in hours) Product Production LevelWiring 1,500 XJ201 150Drilling 2,350 XM897 100Assembly 2,600 TR29 300Inspection 1,200 BR788 400
• 40. B - 40© 2011 Pearson Education, Inc. publishing as Prentice HallLP ApplicationsX1 = number of units of XJ201 producedX2 = number of units of XM897 producedX3 = number of units of TR29 producedX4 = number of units of BR788 producedMaximize profit = 9X1 + 12X2 + 15X3 + 11X4subject to .5X1 + 1.5X2 + 1.5X3 + 1X4 ≤ 1,500 hours of wiring3X1 + 1X2 + 2X3 + 3X4 ≤ 2,350 hours of drilling2X1 + 4X2 + 1X3 + 2X4 ≤ 2,600 hours of assembly.5X1 + 1X2 + .5X3 + .5X4 ≤ 1,200 hours of inspectionX1 ≥ 150 units of XJ201X2 ≥ 100 units of XM897X3 ≥ 300 units of TR29X4 ≥ 400 units of BR788
• 41. B - 41© 2011 Pearson Education, Inc. publishing as Prentice HallLP ApplicationsDiet Problem ExampleA 3 oz 2 oz 4 ozB 2 oz 3 oz 1 ozC 1 oz 0 oz 2 ozD 6 oz 8 oz 4 ozFeedProduct Stock X Stock Y Stock Z
• 42. B - 42© 2011 Pearson Education, Inc. publishing as Prentice HallLP ApplicationsX1 = number of pounds of stock X purchased per cow each monthX2 = number of pounds of stock Y purchased per cow each monthX3 = number of pounds of stock Z purchased per cow each monthMinimize cost = .02X1 + .04X2 + .025X3Ingredient A requirement: 3X1 + 2X2 + 4X3 ≥ 64Ingredient B requirement: 2X1 + 3X2 + 1X3 ≥ 80Ingredient C requirement: 1X1 + 0X2 + 2X3 ≥ 16Ingredient D requirement: 6X1 + 8X2 + 4X3 ≥ 128Stock Z limitation: X3 ≤ 80X1, X2, X3 ≥ 0Cheapest solution is to purchase 40 pounds of grain Xat a cost of \$0.80 per cow
• 43. B - 43© 2011 Pearson Education, Inc. publishing as Prentice HallLP ApplicationsLabor Scheduling ExampleF = Full-time tellersP1 = Part-time tellers starting at 9 AM (leaving at 1 PM)P2 = Part-time tellers starting at 10 AM (leaving at 2 PM)P3 = Part-time tellers starting at 11 AM (leaving at 3 PM)P4 = Part-time tellers starting at noon (leaving at 4 PM)P5 = Part-time tellers starting at 1 PM (leaving at 5 PM)Time Number of Time Number ofPeriod Tellers Required Period Tellers Required9 AM - 10 AM 10 1 PM - 2 PM 1810 AM - 11 AM 12 2 PM - 3 PM 1711 AM - Noon 14 3 PM - 4 PM 15Noon - 1 PM 16 4 PM - 5 PM 10
• 44. B - 44© 2011 Pearson Education, Inc. publishing as Prentice HallLP Applications= \$75F + \$24(P1 + P2 + P3 + P4 + P5)Minimize total dailymanpower costF + P1 ≥ 10 (9 AM - 10 AM needs)F + P1 + P2 ≥ 12 (10 AM - 11 AM needs)1/2 F + P1 + P2 + P3 ≥ 14 (11 AM - 11 AM needs)1/2 F + P1 + P2 + P3 + P4 ≥ 16 (noon - 1 PM needs)F + P2 + P3 + P4 + P5 ≥ 18 (1 PM - 2 PM needs)F + P3 + P4 + P5 ≥ 17 (2 PM - 3 PM needs)F + P4 + P5 ≥ 15 (3 PM - 7 PM needs)F + P5 ≥ 10 (4 PM - 5 PM needs)F ≤ 124(P1 + P2 + P3 + P4 + P5) ≤ .50(10 + 12 + 14 + 16 + 18 + 17 + 15 + 10)
• 45. B - 45© 2011 Pearson Education, Inc. publishing as Prentice HallLP Applications= \$75F + \$24(P1 + P2 + P3 + P4 + P5)Minimize total dailymanpower costF + P1 ≥ 10 (9 AM - 10 AM needs)F + P1 + P2 ≥ 12 (10 AM - 11 AM needs)1/2 F + P1 + P2 + P3 ≥ 14 (11 AM - 11 AM needs)1/2 F + P1 + P2 + P3 + P4 ≥ 16 (noon - 1 PM needs)F + P2 + P3 + P4 + P5 ≥ 18 (1 PM - 2 PM needs)F + P3 + P4 + P5 ≥ 17 (2 PM - 3 PM needs)F + P4 + P5 ≥ 15 (3 PM - 7 PM needs)F + P5 ≥ 10 (4 PM - 5 PM needs)F ≤ 124(P1 + P2 + P3 + P4 + P5) ≤ .50(112)F, P1, P2, P3, P4, P5 ≥ 0
• 46. B - 46© 2011 Pearson Education, Inc. publishing as Prentice HallLP ApplicationsThere are two alternate optimal solutions to thisproblem but both will cost \$1,086 per dayF = 10 F = 10P1 = 0 P1 = 6P2 = 7 P2 = 1P3 = 2 P3 = 2P4 = 2 P4 = 2P5 = 3 P5 = 3First SecondSolution Solution
• 47. B - 47© 2011 Pearson Education, Inc. publishing as Prentice HallThe Simplex Method Real world problems are toocomplex to be solved using thegraphical method The simplex method is an algorithmfor solving more complex problems Developed by George Dantzig in thelate 1940s Most computer-based LP packagesuse the simplex method
• 48. B - 48© 2011 Pearson Education, Inc. publishing as Prentice HallAll rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying,recording, or otherwise, without the prior written permission of the publisher.Printed in the United States of America.