Using QM for Windows. Using QM for Windows. Using QM for Windows LEARNING OBJECTIVES. Solving Flair Furniture s LP Problem
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1 LEARNING OBJECTIVES Vlu%on nd pricing (November 5, 2013) Lecture 11 Liner Progrmming (prt 2) 10/8/16, 2:46 AM Olivier J. de Jong, LL.M., MM., MBA, CFD, CFFA, AA Solving Flir Furniture s LP Problem Pge 1 of 1 Most orgniztions hve ccess to softwre to solve big LP problems There re differences between softwre implementtions, the pproch is bsiclly the sme With experience with computerized LP lgorithms, it is esy to djust to minor chnges 1. Understnd the bsic ssumptions nd properties of liner progrmming (LP). 2. Grphiclly solve ny LP problem tht hs only two vribles by both the corner point nd isoprofit line methods. 3. Understnd specil issues in LP such s infesibility, unboundedness, redundncy, nd lterntive optiml solutions. 4. Understnd the role of sensitivity nlysis. 5. Use Excel spredsheets to solve LP problems. 73 Using QM for Windows 72 Using QM for Windows Select the Liner Progrmming module Specify the number of constrints (nonnegtivity is ssumed) Specify the number of decision vribles Specify whether the objective is to be mximized or minimized For Flir Furniture there re two constrints, two decision vribles, nd the objective is to mximize profit 74 Using QM for Windows PROGRAM 7.1A QM for Windows Liner Progrmming Computer Input Screen PROGRAM 7.1B QM for Windows Dt Input The equtions will utomticlly pper s you enter the coefficients in the other columns. Type over X1 nd X2 to chnge the nmes of the Vribles. Click here to chnge the type of constrint. Click Solve fter entering the dt. Input the coefficients in the pproprite columns
2 PROGRAM 7.1C QM for Windows Output nd Grph Using QM for Windows Using Excel s Solver The Solver tool in Excel cn be used to find solutions to LP problems Integer progrmming problems Noninteger progrmming problems Solver is limited to 200 vribles nd, in some situtions, 100 constrints Recll the model for Flir Furniture is Mximize profit = $70T + $50C Subject to 4T + 3C 240 2T + 1C 100 To use Solver, it is necessry to enter dt nd formuls 1. Enter problem dt Vrible nmes, coefficients for the objective function nd constrints, RHS vlues for ech constrint 2. Designte specific cells for the vlues of the decision vribles 3. Write formul to clculte the vlue of the objective function 4. Write formul to compute the left-hnd sides of ech of the constrints PROGRAM 7.2A Excel Dt Input PROGRAM 7.2B Formuls These cells re selected to contin the vlues of the decision vribles. Solver will enter the optiml solution here, but you my enter numbers here lso. The signs for the constrints re entered here for reference only
3 PROGRAM 7.2C Excel Spredsheet PROGRAM 7.2D Strting Solver Becuse there is 1 in ech of these cells, the LHS vlues cn be clculted very esily to see if mistke hs been mde. You cn chnge these vlues to see how the profit nd resource utiliztion chnge PROGRAM 7.2E Solver Prmeters Dilog Box PROGRAM 7.2F Solver Add Constrint Dilog Box Enter the ddress for the LHS of the constrints. These my be entered one t time or ll together if they re of the sme type (e.g., ll < or ll >). Enter the ddress for the RHS of the constrints. Click OK when finished. Click button to select the type of constrint reltionship PROGRAM 7.2G Solver Results Dilog Box PROGRAM 7.2H Solution Additionl informtion is vilble. The optiml solution is T = 30, C = 40, profit = 4,100. The hours used re given here. Check this to be sure tht solution ws found
4 PROGRAM 7.3A Excel QM in Excel 2013 Using Excel QM PROGRAM 7.3B Excel QM Input Dt Using Excel QM After entering the problem, click the Dt tb nd select Solver from the Dt ribbon. When the window for Solver opens, simply click Solve s ll the necessry inputs hve been entered by Excel QM. Instructions to ccess Solver re here. Enter the dt in the pproprite cells. Do not chnge ny other cells in the spredsheet Using Excel QM Solving Minimiztion Problems PROGRAM 7.3C Excel QM Output Solution is shown here. Mny LP problems involve minimizing n objective such s cost Minimiztion problems cn be solved grphiclly Set up the fesible solution region Use either the corner point method or n isocost line pproch Find the vlues of the decision vribles (e.g., nd ) tht yield the minimum cost The is considering buying two different brnds of turkey feed nd blending them to provide good, low-cost diet for its turkeys TABLE 7.5 dt INGREDIENT COMPOSITION OF EACH POUND OF FEED (OZ.) BRAND 1 FEED BRAND 2 FEED MINIMUM MONTHLY REQUIREMENT PER TURKEY (OZ.) A B C Cost per pound 2 cents 3 cents Let = number of pounds of brnd 1 feed purchsed = number of pounds of brnd 2 feed purchsed Minimize cost (in cents) = subject to: ounces (ingredient A constrint) ounces (ingredient B constrint) ounces (ingredient C constrint) 0 (nonnegtivity constrint) 0 (nonnegtivity constrint)
5 FIGURE 7.10 Fesible Region Pounds of Brnd Ingredient C Constrint Fesible Region Ingredient B Constrint Solve for the vlues of the three corner points Point is the intersection of ingredient constrints C nd B = 48 = 3 Substituting 3 in the first eqution, we find = 12 Solving for point b we find = 8.4 nd = 4.8 Solving for point c we find = 18 nd = 0 5 b Ingredient A Constrint 0 c Pounds of Brnd Substituting these vlues bck into the objective function we find Cost = Cost t point = 2(3) + 3(12) = 42 Cost t point b = 2(8.4) + 3(4.8) = 31.2 Cost t point c = 2(18) + 3(0) = 36 Solving using n isocost line Move the isocost line towrd the lower left The lst point touched in the fesible region will be the optiml solution The lowest cost solution is to purchse 8.4 pounds of brnd 1 feed nd 4.8 pounds of brnd 2 feed for totl cost of 31.2 cents per turkey FIGURE 7.11 Grphicl Solution Using the Isocost Approch PROGRAM 7.4 Solution in QM for Windows Fesible Region Pounds of Brnd = = Isocost Line Direction of Decresing Cost ( = 8.4, = 4.8) Pounds of Brnd
6 PROGRAM 7.5A Excel 2013 Solution PROGRAM 7.5B Excel 2013 Formuls Formuls re written to find the vlues in column D Four Specil Cses in LP Four specil cses nd difficulties rise t times when using the grphicl pproch 1. No fesible solution 2. Unboundedness 3. Redundncy 4. Alternte Optiml Solutions No Fesible Solution No solution to the problem tht stisfies ll the constrint equtions No fesible solution region exists A common occurrence in the rel world Generlly one or more constrints re relxed until solution is found Consider the following three constrints No Fesible Solution FIGURE 7.12 A problem with no fesible solution Region Stisfying Third Constrint Unboundedness Sometimes liner progrm will not hve finite solution In mximiztion problem One or more solution vribles, nd the profit, cn be mde infinitely lrge without violting ny constrints In grphicl solution, the fesible region will be open ended Usully mens the problem hs been formulted improperly Region Stisfying First Two Constrints
7 10 5 Unboundedness FIGURE 7.13 A Fesible Region Tht Is Unbounded to the Right Mximize profit = $3 + $5 subject to , Fesible Region Redundncy A redundnt constrint is one tht does not ffect the fesible solution region One or more constrints my be binding This is very common occurrence in the rel world Cuses no prticulr problems, but eliminting redundnt constrints simplifies the model Mximize profit = $1 + $2 subject to , FIGURE 7.14 Problem with Redundnt Constrint Fesible Region Redundncy Mximize profit = $1 + $2 subject to , 0 Redundnt Constrint Eduction, Inc. Copyright 2015 Person 7 39 Alternte Optiml Solutions Occsionlly two or more optiml solutions my exist Grphiclly this occurs when the objective function s isoprofit or isocost line runs perfectly prllel to one of the constrints Allows mngement gret flexibility in deciding which combintion to select s the profit is the sme t ech lternte solution Mximize profit = $3 + $2 subject to , Alternte Optiml Solutions FIGURE 7.15 Exmple of Alternte Optiml Solutions 8 7 A Mximize profit = $3 + $2 subject to , 0 Optiml Solution Consists of All Combintions of nd Along the AB Segment Isoprofit Line for $8 2 B Isoprofit Line for $12 1 Fesible Overlys Line Segment AB Region Person Eduction, Inc. Copyright Sensitivity Anlysis Optiml solutions to LP problems thus fr hve been found under deterministic ssumptions We ssume complete certinty in the dt nd reltionships of problem Rel world conditions re dynmic Anlyze how sensitive deterministic solution is to chnges in the ssumptions of the model This is clled sensitivity nlysis, postoptimlity nlysis, prmetric progrmming, or optimlity nlysis 7 42
8 Sensitivity Anlysis Involves series of wht-if? questions concerning constrints, vrible coefficients, nd the objective function Tril-nd-error method Vlues re chnged nd the entire model is resolved Preferred wy is to use n nlytic postoptimlity nlysis After problem hs been solved, we determine rnge of chnges in problem prmeters tht will not ffect the optiml solution or chnge the vribles in the solution High Note Sound Compny The compny mnufctures qulity spekers nd stereo receivers Products require certin mount of skilled rtisnship which is in limited supply Product mix LP model Mximize profit = $50 + $120 subject to (hours of electricins time vilble) (hours of udio technicins time vilble), High Note Sound Compny FIGURE 7.16 The High Note Sound Compny Grphicl Solution (Receivers) = (0, 20) 10 b = (16, 12) Optiml Solution t Point = 0 Spekers = 20 Receivers Profits = $2,400 Isoprofit Line: $2,400 = c = (20, 0) (Spekers) 7 45 High Note Sound Compny Electricin hours used re = 2(0) + 4(20) = 80 All hours re utilized so slck = 0 Additionl units of binding constrint will generlly increse profits Technicin hours used re = 3(0) + 1(20) = 20 Avilble hours = 60 so slck = 20 = 40 Additionl units of nonbinding constrint will only increse slck 7 46 Chnges in the Objective Function Coefficient Contribution rtes in the objective functions fluctute The fesible solution region remins exctly the sme The slope of the isoprofit or isocost line chnges Modest increses or decreses in objective function coefficients my not chnge the current optiml corner point Know how much n objective function coefficient cn chnge before the optiml solution would be t different corner point 7 47 Chnges in the Objective Function Coefficient FIGURE 7.17 Chnges in the Receiver Contribution Coefficients Profit Line for (Psses through Point b) b Old Profit Line for (Psses through Point ) Profit Line for (Psses through Point ) c
9 QM for Windows QM for Windows PROGRAM 7.6A Input to QM for Windows High Note Sound PROGRAM 7.6B High Note Sound Sensitivity Anlysis Excel Solver Excel Solver PROGRAM 7.7A Excel Spredsheet for High Note Sound PROGRAM 7.7B Excel 2013 Solution nd Solver Results The By Chnging Vrible Cells in the Solver Dilog Box re B4:C4. The Set Objective cell in the Solver Dilog Box is D5. The constrints dded into Solver will be D8:D9 <=F8:F PROGRAM 7.7C Excel 2013 Sensitivity Report Excel Solver The nmes presented in the Sensitivity Report combine the text in column A nd the text bove the dt, unless the cells hve been nmed using the Nme Mnger from the Formuls tb. The profit on spekers my chnge by these mounts nd the current corner point will remin optiml. The resources used re here. The RHS cn chnge by these mounts, nd the shdow price will still be relevnt. Chnges in the Technologicl Coefficients Chnges in the technologicl coefficients often reflect chnges in the stte of technology If the mount of resources needed to produce product chnges, coefficients in the constrint equtions will chnge Objective function does not chnge My produce significnt chnge in the shpe of the fesible region My cuse chnge in the optiml solution
10 Stereo Receivers Chnges in the Technologicl Coefficients FIGURE 7.18 Chnge in the Technologicl Coefficients () Originl Problem Optiml Solution d b c CD Plyers (b) Chnge in Circled Coefficient Still Optiml e f (c) Chnge in Circled Coefficient Optiml Solution g c Chnges in Resources or Right-Hnd-Side Vlues Right-hnd-side vlues of the constrints often represent resources vilble to the firm Additionl resources my led to higher totl profit Sensitivity nlysis bout resources helps nswer questions bout How much should be pid for dditionl resources How much more of resource would be useful Chnges in Resources or Right-Hnd-Side Vlues Chnging the RHS will chnge the fesible region, unless the constrint is redundnt Often chnges the optiml solution The dul price or dul vlue The mount of chnge in the objective function vlue tht results from unit chnge in one of the resources The dul price for constrint is the improvement in the objective function vlue tht results from one-unit increse in the right-hnd side of the constrint Chnges in Resources or Right-Hnd-Side Vlues The mount of possible increse in the RHS is limited If the RHS is incresed beyond the upper bound, then the objective function would no longer increse by the dul price There would be excess (slck) resources or the objective function my chnge by n mount different from the dul price The dul price is relevnt only within limits FIGURE 7.19 Chnges in the Electricins Time Resource FIGURE 7.19 Chnges in the Electricins Time Resource () (b) Constrint Representing 60 Hours of Audio Technicin s Time Resource Constrint Representing 60 Hours of Audio Technicin s Time Resource 25 b Chnged Constrint Representing 100 Hours of Electricin s Time Resource 25 Chnged Constrint Representing 60 Hours of Electricin s Time Resource b c c
11 Chnges in the Electricins Time Resource FIGURE 7.19 (c) Chnged Constrint Representing 240 Hours of Electricin s Time Resource QM for Windows PROGRAM 7.6B High Note Sound Sensitivity Anlysis 25 Constrint Representing 60 Hours of Audio Technicin s Time Resource Excel Solver Remember PROGRAM 7.7C Excel 2013 Sensitivity Report The nmes presented in the Sensitivity Report combine the text in column A nd the text bove the dt, unless the cells hve been nmed using the Nme Mnger from the Formuls tb. The profit on spekers my chnge by these mounts nd the current corner point will remin optiml. The resources used re here. The RHS cn chnge by these mounts, nd the shdow price will still be relevnt Chpter 3: Where Prices Come From: The Interction of Demnd nd Supply Red Quntittive Methods-module guide. Any questions plese e-mil: olivier.edu@gmil.com nd mke notes s you do so, in whtever wy works best for you in terms of remembering informtion (your performnce on this course is only ssessed by exm). Copyright 2010 Person Eduction, Inc. Economics R. Glenn Hubbrd, Anthony Ptrick O Brien, 3e. 64 of 46
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