Faaliyetler İş Araştırma ve İktisat LP Sensitivity Analysis (Questions) , " MAN 203 CEMlLKUZEY -.2. LP Sensitivity Analysis and Interpretation of Solution "" •. _,' ...... _'c~,....'"'-_,=-~. __ . ""'" ," 1 ",1 \ _ •.. ----_ ... _ .. - - - . PROBLEMS 1. In a linear programming problem. the binding constraints for the optJ.aJ solution are a. b. 5X + 3Y .:': 30 2X + 5Y .:': 20 Fill in the blanks in the following sentence: As long as the slope of the objective function stays between and ___ , the current optimal solution point will remain optimal. Which of these objective functions wiJllead to th e same o~ solution? I) 2X+ IY 2) 7X+8Y 3) 80X+60Y 4) 25J1:+35Y 2. The optimal solution of the linear programming problem is allhe intasa:tion of constraints I and 2. a. b. c. Max 2x! + X2 S.t. 4x, + ' Xl .:5. 400 4x, + 3x, .:': 600 lx, + 2x, " 300 X I , X2 ~ 0 Over what range can the coefficient of XI vary before the cwwad solution is no longer optimal? Over what range can the coefficient of X2 vary before the c...a:d solution is no longer optimal? Compute the dual prices for the three constrajnts. TOPIC: Graphical sensitivity analysis a. b. c. d. e. The binding constraints for this problem are the first and se""" Min XI + 2X2 s.t. x,+ x, <- 300 2xI + x, <- 400 2x) + 5.X2 ~ 750 XI, X2 ::: 0 Keeping C2 fixed at 2, over what range can CI vary before then: is a change in the optimal solution point? Keeping Cl fixed at ] J over what Pange can C2 vary before then: is a change in the optimal solution point? If the objective function becomes Min 1.5xl + 2X2. what will lie the optimal values of x .. X2. and the objective function? lfthe objective function becomes Min 7xI + 6X2, what conslniots will be binding? Find the dual price for each constraint jn the original preble,-. • 4~ - "" Use the following Management Scientist output to answer the questio.s.. MAX 3IXI+l5X2+l2X3 S.T I) lXI +5X2+2X3>90 2) 6XI+7X2+8X3<150 3) 5XI+ 3X2+ 3X3<120 OPTIMAL SOLUTION Objective Function Value = 763.333 Variable Value Reduced Cost XI 13.333 0.000 X2 10.000 0.000 X3 0.000 10.889 Constraint Slack/Su!]1lus Dual Price 0.000 -0.778 2 0.000 5.556 3 23.333 0.000 OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value XI 30.000 31.000 X2 No Lower Limit 35.000 X3 No Lower Limit 32.000 RIGHT HAND SIDE RANGES a. b. Constraint 2 3 Lower Limit 77.647 126.000 96.667 Current Value 90.000 150.000 120.000 Give the solution to the problem. Which constraillts are binding? Ulmer Limil NoUpperL'" 36.161 42.889 Upper Limit 107.143 163 .125 No Upper LiIoiiI c. What would happen if the coefficient ofXt increased by 3? d. What would happen iflhe right-hand side of constraint 1 incn:z;ed by 10? • 5. Use th e following Management Scienti st ou tput to answer the questimls.. M1N 4X I +5 X2+6X3 S.T. I ) X I + X2+ X3 <85 2) 3X I +4X2+2X3>280 -. _, . . 3:l4X ' B§*" 2±.4>J¥.ll2'3J! O',"o=-", . ,.~~.~="_",. ,..,.....,". ···~."",",".C= _ __ ""~'=~'·' . .. ·.'c.'~~"'""··--"'·· ., .. ..... . Objective Function Value = 400.000 Variable Value Reduced Cost X I 0.000 1.500 X2 80.000 0.000 X3 0.000 1.000 Constraint Slack/Sumlus Dual Price I 5.000 0.000 2 40.000 0.000 3 0.000 ·1.250 OBJECTIVE COEFF1CJENT RANGES Variable Lower Limit Current Value UlmerLimil XI 2.500 4.000 No Upper LiooiiI X2 0.000 5.000 6.000 X3 5.000 6.000 NoUpperL ..... RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value UrmerLimil 80.000 85.000 No Upper Li8iI 2 No Lower Limit 280.000 320.000 3 280.000 320.000 340.000 a. What is the optimal solution, and what is the value of the pJUlil:contribution? h. Which constraints are binding? c . What are the dual prices for each resource? Interpret. d. Compute and interpret the ranges of optimality. e. Compute and interpret the ranges of feasibility . • 6_ a) which constraints are binding? b) What are the dual prices for each resource? lnterpret. c) Compute and interpret the ranges of optimality. d) Com put e and interpret the ranges of feasibility. -'- 'Microsoft:En-el=7;():SeD.5itjvity:~Rtport~=' - '~""'··:"'~~.!"~~~·!-· . ---.."" ........ -.>~ ~.-=-, ............ -~~ ~....,.,.,., . - .. - . Worksbeet: IxJs7base.u.ISbeetl Changing Cells Final Reduced Objectm Allowable Allowable Cell Name Value Cost Codficidll: Iocr-ease Decrease $BSI2 Variables Variable 1 8.461538462 0 S 7 3.4 SCS12 Variables Variable 2 4.615384615 0 .. 8.5 2.333333333 Constraints Final Shadow CODstraillll Allowable AUowable Cell Name Value Price RH. Side Increase Decrease SBSI5 constraint 1 Used 40 0.538461538 • 110 7 SB$16 constraint 2 Used 30 1.307692308 30 30 4.666666667 SBSI7 constraint 3 Used 13.07692308 0 12 1.076923077 IE+30 7_ The following linear programming problem has been solved by The Man~t Scientist. Use the output to answer the questions. 0 L~ARPROGRA~GPROBLEM MAX 25XI+30X2+15X3 S.T. I) 4XJ+5X2+8X3<1200 2) 9X 1+ 15X2+ 3X3<1500 OPTIMAL SOLUTION Objective Function Value = 4700.000 Variable Value Reduced Cost XI 140.000 0.000 X2 0.000 10.000 X3 80.000 0.000 COftsffaint- Slaekf&ttrolus Dual Price 0.000 1.000 2 0.000 2.333 • < OBJECT1VE COEFFJClENT RANGES Variable Xl X2 Lower Limit Current Value 19.286 25.000 No Lower Limit 30.000 Upper Limit 45.000 40.000 "~·o-',,~~"·='~~X3 . .,..<-· ~ ... __ .. .8T 31il, __ =.""l"S,OQO~ .===.IhQ~ "~-··~~·"" .. """""..~,~ - ... ' RJGHT HAND SIDE RANGES Constraint 2 Lower Limit 666.667 450.000 Current Value 1200.000 J 500.000 a. Give the complete optimal solution. b. Which constraints are binding? Upper Limil 4000.00II 2700.00II c . What is the dual price for the second constraint? What intapretation does this have? d. Over what range can the objective function coefficient of~Y3I)' before a new solution point becomes optimal? e. By how much can the amount of resource 2 decrease befOJe1hc dual price will change? f. What would happen if the first constraint's right-hand side iKreased by 700 and the second's decreased by 350? 8). LINDO output is given for the following linear programming problem.. MJN 12 Xl + 10.X2 + 9 X3 SUBJECT TO 2) 5Xl +8X2+5X3>= 60 3) 8 Xl + 10 X2 + 5 X3 >= 80 END LP OPTIMUM FOUND AT STEP OBJECTIVE FUNCTION VALUE 1) 80.000000 VARJABLE VALUE REDUCED COST Xl .000000 4.000000 X2 8.000000 .000000 X3 .000000 4.000000 ROW SLACK OR SURPLUS DUALPRlCE 2) 4.000000 .000000 3) .000000 -1.000000 • NO. lTERA TIONS= RANGES IN WHlCH THE BAS1S JS UNCHANGED: OB). COEFF1CIENT RANGES VARIABLE XI X2 X3 ROW 2 3 CURRENT COEFFICIENT 12.000000 10.000000 9.000000 • CURRENT RHS 60.000000 80.000000 ALLOWABLE INCREASE INFINlTY 5.000000 INFINITY ALLOWABLE INCREASE 4.000000 INFINlTY a. What is the solution to the problem? b. Which constraints are binding? c. Interpret the reduced cost for x)_ d. Interpret the dual price for constraint 2. ALLOWA~ DECREASE 4.000000 10.000000 4.000000 ALLOWA~ DECREASE INFlNllY 5.000000 e. What would happen if the cost of x, dropped to 10 and the COSlofx, increased to 12? 9. The LP problem whose output follows determines how many necklaa::s. bracelets, rings, and earrings a jewelry store should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint I measures display space in units, constraint 2 . mcasures time to set up the display in minutes. Constraints 3 and 4 are marketing restrictions. LINEAR PROGRAMMING PROBLEM MAX 100XI+120X2+ 150X3+125X4 S.T. I) XI+2X2+2X3+2X4
50 OPTIMAL SOLUTION Objective Function Value = Variable XI X2 X3 X4 Constraint • 2 3 4 Value 8.000 0.000 17.000 33.000 Slack/Sumlus 0.000 63.000 0.000 0.000 7475.000 Reduced Cost 0.000 5.000 0.000 0.000 Dual Price 75.000 0.000 25.000 ·25.000 OBJECTIVE COEFFICIENT RANGES • Variable Lower Lim it Currenl Valu e U~Qer Limit Xl 87.500 100.000 No Upper Limi. X2 No Lower Limil 120.000 125.000 X3 125.000 150.000 162.500 X4 120.000 125.000 150.000 RlGHT HAND SIDE RANGES Constraint Lower Limit Current Value URRer Limit 100.000 108.000 123.750 2 57.000 120.000 No Upper Limi. 3 8.000 25. 000 58.000 4 41.500 50.000 54.000 Use the output 10 answer the questions. a. How many necklaces should be stocked? b. Now many bracelets should be stocked ? c. How many rings should be stocked? d. How many earrings should be stocked? e. How much space will be left unused? f. How much tim e will be used? g. By how much will the second marketing restriction be exceeded? h. What is the profit? I. To what value can the profit on necklaces drop before the solalion would change? J By how much can the profit on rings increase before the solution would change? k. By how much can the amount of space decrease before there is a change in the profit? I. You are offered the chance to obtain more space. The offer is for J5 units and (he Iota I price is 1500. What should you do? 10. The decision variables represent the amounts of ingredients 1,2.:and 3 to put into a blend. The objective function represents profit. The first three constraints measure the usage and availability of resources A, B, and C. The fourth constraint is a minimum R>quirement for ingredient 3. Use th e output to answer these questions. a. How much of ingredient I will be put into the blend? b. How much of ingredient 2 will be put into the blend? c. How much of ingredient 3 will be put into the blend? d. How much resource A is used? e. How much resource B will be left unused? f. What will the profit be? g. What will happen to the solution if the profit from ingredient 2 drops to 4? h. What will happen to the solution ifthe profit from ingredient 3 increases by J? I. What will happen to the solution if the amount of resoun:e C increases by 2? J. What will happen to the solution if the minimum requirement for ingredient 3 increases to IS? L~ARPROGRA~GPROBLEM MA X 4XI+6X2+7X3 S.T. I) 3X J+2 X2+5X3<120 2) IX J+ 3X2+ 3X3<80 3) 5X J+ 5X2+8X3 < 1 60 4) +I X3> 10 Objective Function Value = 166.000 Variable Value Reduced Cost Xl 0.000 2.000 X2 16.000 0.000 X3 10.000 0.000 Constraint Slack/Sumlus Dual Price 38.000 0.000 2 2.000 0.000 3 0.000 1.200 4 0.000 -2.600 OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Um~er Lindt Xl No Lower Limit 4.000 6.000 X2 4.375 6.000 NoUpper ..... iJ X3 No Lower Limit 7.000 9.600 RlGHT HAND SIDE RANGES Constraint Lower Limit Current Value Uu(!er Li .... 82.000 120.000 No Upper Linoit 2 78.000 80.000 No Upper Limit 3 80.000 160.000 163.333 4 8.889 10.000 20.0llO I I. The LP model and LINDO output represent a problem whose solutio. Will tell a specially retailer how many of four different styles ofumbreHas to stock in order to maximilE profit. ]1 is assumed that every one stocked will be sold. The variables measure the number ofwomen"s. ... r. men's, and folding umbrellas. respectively_ The constraints measure storage space in units, special6spJay racks, demand. and a marketing restriction, respectively. MAX 4 Xl + 6 X2 + 5 X3 + 3.5 X4 • SUBJECT TO 2) 2 X I + 3 X2 + 3 X3 + X4 <= 120 3) 1.5 X I + 2 X2 <= 54 4) 2 X2 + X3 + X4 <= 72 5) X2+X3 >= 12 END , • OBJECTIVE FUNCTION VALUE I) 318.00000 VARIABLE VALUE REDUCED COST ~~=''''''' ... _"":l(L,~ _ _ ~",,,....!2)lllQQllo.. .• -.OQQQOO_· __ ·"!,,,,,,,,,~c:-'Cc~~ _____ '~,=,,~,.,_._·,_ .. _~."' __ ~_"_ X2 .000000 X3 12.000000 X4 60.000000 .500000 .000000 .000000 ROW 2) 3) 4) 5) SLACK OR SURPLUS .000000 36.000000 .000000 .000000 DUAL PRICE 2.000000 .000000 L500000 ·2.500000 RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ. COEFFICIENT RANGES VARIABLE CURRENT ALLOWABLE ALLOWABLE COEFFICIENT INCREASE DECREASE XI 4.000000 1.000000 2.500000 X2 6.000000 .500000 INFINITY X3 5.000000 2.500000 .500000 X4 3.500000 INFINITY 500000 RlGHTHAND SIDE RANGES CURRENT ALLOWABLE ALLOWABLE ROW RHS INCREASE DECREASE 2 120.000000 48.000000 24.000008 3 54.000000 INFINITY 36.000008 4 12.000000 24.000000 48.000000 5 12.000000 12.000000 12.000000 Use the output to answer the questions. a. How many women's umbrellas should be stocked? b. How many golf umbrellas should be stocked? c. How many men's umbrellas should be stocked? d. How many folding umbrellas should be stocked? e. How much space is left unused? f. How many racks are used? g. By how much is the marketing restriction exceeded? h. What is the total profit? • I. By how much can the profit on women's umhrenas increascbefore the solution would change? j. To what value can the profit on golf umbrellas increase hef'()Kthe solution would change? k. By how much can the amount of space increase before then: is a change in the dual price? I. You are offered an advertisement that should increase the daaand constraint from 72 to 86 for a total cost of $20. Wou ld you say yes or no? 12_ HOMEWORK: A large sporting goods store is placing an order for bit:ycJes with its supplier. Four models can be ordered: the adult Open Trail, the adult Cityscape. the girl's Sea Sprite, and the boy's Trail Blazer. ]1 is assumed that every oike ordered will be sold, and their profits, respectively, are 30, 25, 22, and 20. The LP model should maximize profit. There are several conditions that the store needs to worry about. _ .' (Lne of th~e :1 S.Rac£.tl11l'lliLth':..lQ.\'.gUQ[y.~.J1.aJ!u1t~~ill!<,~~1!l;~Q~~~~,Jll!u.~Jill
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