Quantitative Analysis Question Paper

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    Chapter 3Linear Programming: Sensitivity Analysis and Interpretation of Solution

    MULTIPL C!"IC

    1. To solve a linear programming problem with thousands of variables and constraintsa. a personal computer can be used.b. a mainframe computer is required.

    c. the problem must be partitioned into subparts.d. unique software would need to be developed.

    ANS: A PTS: 1 TOP: omputer solution

    !. A negative dual price for a constraint in a minimi"ation problem meansa. as the right#hand side increases$ the ob%ective function value will increase.b. as the right#hand side decreases$ the ob%ective function value will increase.c. as the right#hand side increases$ the ob%ective function value will decrease.d. as the right#hand side decreases$ the ob%ective function value will decrease.

    ANS: A PTS: 1 TOP: &ual price

    '. (f a decision variable is not positive in the optimal solution$ its reduced cost isa. what its ob%ective function value would need to be before it could become positive.b. the amount its ob%ective function value would need to improve before it could become

    positive.c. "ero.d. its dual price.

    ANS: ) PTS: 1 TOP: *educed cost

    +. A constraint with a positive slac, valuea. will have a positive dual price.b. will have a negative dual price.c. will have a dual price of "ero.d. has no restrictions for its dual price.

    ANS: PTS: 1 TOP: Slac, and dual price

    -. The amount b which an ob%ective function coefficient can change before a different set of values forthe decision variables becomes optimal is thea. optimal solution.b. dual solution.c. range of optimalit.d. range of feasibilit.

    ANS: PTS: 1 TOP: *ange of optimalit

    /. The range of feasibilit measuresa. the right#hand#side values for which the ob%ective function value will not change.b. the right#hand#side values for which the values of the decision variables will not change.c. the right#hand#side values for which the dual prices will not change.d# each of these choices are true.

    ANS: PTS: 1 TOP: *ange of feasibilit

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    0. The 12 *ule comparesa. proposed changes to allowed changes.b. new values to original values.c. ob%ective function changes to right#hand side changes.d. dual prices to reduced costs.

    ANS: A PTS: 1 TOP: Simultaneous changes

    3. An ob%ective function reflects the relevant cost of labor hours used in production rather than treating

    them as a sun, cost. The correct interpretation of the dual price associated with the labor hoursconstraint isa. the ma4imum premium 5sa for overtime6 over the normal price that the compan would

    be willing to pa.b. the upper limit on the total hourl wage the compan would pa.c. the reduction in hours that could be sustained before the solution would change.d. the number of hours b which the right#hand side can change before there is a change in

    the solution point.

    ANS: A PTS: 1 TOP: &ual price

    7. A section of output from The 8anagement Scientist is shown here.

    9ariable ower imit urrent 9alue ;pper imit

    1 / 1 1!

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    c. reduced cost.d. relevant cost.

    ANS: PTS: 1 TOP: (nterpretation of computer output

    1!. The dual price measures$ per unit increase in the right hand side of the constraint$a. the increase in the value of the optimal solution.b. the decrease in the value of the optimal solution.c. the improvement in the value of the optimal solution.

    d. the change in the value of the optimal solution.

    ANS: & PTS: 1 TOP: (nterpretation of computer output

    1'. Sensitivit analsis information in computer output is based on the assumption ofa. no coefficient changes.b. one coefficient changes.c. two coefficients change.d. all coefficients change.

    ANS: ) PTS: 1 TOP: Simultaneous changes

    1+.

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    '.

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    1+. The amount of a sun, cost will var depending on the values of the decision variables.

    ANS: @ PTS: 1 TOP: autionar note on the interpretation of dual prices

    1-. (f the optimal value of a decision variable is "ero and its reduced cost is "ero$ this indicates thatalternative optimal solutions e4ist.

    ANS: T PTS: 1 TOP: (nterpretation of computer output

    1/. An change to the ob%ective function coefficient of a variable that is positive in the optimal solutionwill change the optimal solution.

    ANS: @ PTS: 1 TOP: *ange of optimalit

    10. *elevant costs should be reflected in the ob%ective function$ but sun, costs should not.

    ANS: T PTS: 1 TOP: autionar note on the interpretation of dual prices

    13. (f the range of feasibilit for b1is between 1/ and '0$ then if b1B !! the optimal solution will notchange from the original optimal solution.

    ANS: @ PTS: 1 TOP: *ight#hand sides

    17. The 1 percent rule can be applied to changes in both ob%ective function coefficients and right#handsides at the same time.

    ANS: @ PTS: 1 TOP: Simultaneous changes

    !. (f the dual price for the right#hand side of a constraint is "ero$ there is no upper limit on its range of

    feasibilit.

    ANS: T PTS: 1 TOP: *ight#hand sides

    S!"$T A'S($

    1. &escribe each of the sections of output that come from The 8anagement Scientist and how ou woulduse each.

    ANS:Answer not provided.

    PTS: 1 TOP: (nterpretation of computer output

    !. C4plain the connection between reduced costs and the range of optimalit$ and between dual prices

    and the range of feasibilit.

    ANS:Answer not provided.

    PTS: 1 TOP: (nterpretation of computer output

    '. C4plain the two interpretations of dual prices based on the accounting assumptions made in calculatingthe ob%ective function coefficients.

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    ANS:Answer not provided.

    PTS: 1 TOP: &ual price

    +. Dow can the interpretation of dual prices help provide an economic %ustification for new technolog=

    ANS:

    Answer not provided.

    PTS: 1 TOP: &ual price

    -. Dow is sensitivit analsis used in linear programming= Eiven an e4ample of what tpe of questionsthat can be answered.

    ANS:Answer not provided.

    PTS: 1 TOP: Sensitivit analsis

    /. Dow would sensitivit analsis of a linear program be underta,en if one wishes to considersimultaneous changes for both the right#hand#side values and ob%ective function.

    ANS:Answer not provided.

    PTS: 1 TOP: Simultaneous sensitivit analsis

    P$")LM

    1. (n a linear programming problem$ the binding constraints for the optimal solution are

    -F G 'H '

    !F G -H !

    a. @ill in the blan,s in the following sentence:As long as the slope of the ob%ective function stas between IIIIIII and IIIIIII$ thecurrent optimal solution point will remain optimal.

    b.

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    s.t. +41G 14!+

    +41G '4!/

    141G !4!'

    41$ 4!

    a. Over what range can the coefficient of 41var before the current solution is no longeroptimal=

    b. Over what range can the coefficient of 4!var before the current solution is no longeroptimal=

    c. ompute the dual prices for the three constraints.

    ANS:

    a. 1.'' c1+

    b. .- c!1.-

    c. &ual prices are .!-$ .!-$

    PTS: 1 TOP: Eraphical sensitivit analsis

    '. The binding constraints for this problem are the first and second.

    8in 41G !4!

    s.t. 41G 4!'

    !41G 4!+

    !41G -4!0-

    41 $ 4!

    a. Jeeping c!fi4ed at !$ over what range can c1var before there is a change in the optimalsolution point=

    b. Jeeping c1fi4ed at 1$ over what range can c!var before there is a change in the optimalsolution point=c. (f the ob%ective function becomes 8in 1.-41G !4!$ what will be the optimal values of 41$

    4!$ and the ob%ective function=d. (f the ob%ective function becomes 8in 041G /4!$ what constraints will be binding=e. @ind the dual price for each constraint in the original problem.

    ANS:

    a. .3 c1!

    b. 1 c!!.-

    c. 41B !-$ 4!B -$ " B +0-

    d. onstraints 1 and ! will be binding.e. &ual prices are .''$ $ .'' 5The first and third values are negative.6

    PTS: 1 TOP: Eraphical sensitivit analsis

    +. C4cel?s Solver tool has been used in the spreadsheet below to solve a linear programming problem

    with a ma4imi"ation ob%ective function and all constraints.

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    Input Se*tion

    "+,e*tive &un*tion Coeffi*ients

    F H

    + /

    Constraints Avail#

    K1 ' - /

    K! ' ! +3K' 1 1 !

    "utput Se*tion

    -aria+les 1'.'''''' +

    Profit -'.'''''' !+ 00.''''''

    Constraint Usage Sla*.

    K1 / 1.037C#11

    K! +3 #!./7C#11

    K' 10.'''''' !.//////0

    a. Eive the original linear programming problem.b. Eive the complete optimal solution.

    ANS:

    a. 8a4 +F G /H

    s.t. 'F G -H /

    'F G !H +3

    1F G 1H ! F $ H

    b. The complete optimal solution is F B 1'.'''$ H B +$ L B 0'.'''$ S1B $ S!B $ S'B !.//0

    PTS: 1 TOP: Spreadsheet solution of Ps

    -. C4cel?s Solver tool has been used in the spreadsheet below to solve a linear programming problem

    with a minimi"ation ob%ective function and all constraints.

    Input Se*tion

    "+,e*tive &un*tion Coeffi*ients

    F H

    - +

    Constraints $e/0d

    K1 + ' /

    K! ! - -

    K' 7 3 1++

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    "utput Se*tion

    -aria+les 7./ 0.!

    Profit +3 !3.3 0/.3

    Constraint Usage Sla*.

    K1 / 1.'-C#11

    K! --.! #-.!

    K' 1++ #!./!C#11

    a. Eive the original linear programming problem.b. Eive the complete optimal solution.

    ANS:

    a. 8in -F G +H

    s.t. +F G 'H /

    !F G -H -

    7F G 3H 1++

    F $ H

    b. The complete optimal solution is F B 7./$ H B 0.!$ L B 0/.3$ S1B $ S!B -.!$ S'B

    PTS: 1 TOP: Spreadsheet solution of Ps

    /. ;se the spreadsheet and Solver sensitivit report to answer these questions.a.

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    2 Profit B Total

    23

    24 *esources ;sed Sl,Surp

    25 onstraint 1

    26 onstraint !

    27 onstraint '

    28

    29

    Sensitivity $eport

    Changing Cells

    &inal $edu*ed "+,e*tive Allo;a+le Allo;a+le

    Cell 'ame -alue Cost Coeffi*ient In*rease 1e*rease

    M)M1! 9ariables 9ariable 1 3.+/1-'3+/! - 0 '.+

    MM1! 9ariables 9ariable ! +./1-'3+/1- + 3.- !.'''''''''

    Constraints

    &inal Shado; Constraint Allo;a+le Allo;a+le

    Cell 'ame -alue Pri*e $#!# Side In*rease 1e*rease

    M)M1- constraint 1 ;sed + .-'3+/1-'3 + 11 0M)M1/ constraint ! ;sed ' 1.'0/7!'3 ' ' +.////////0

    M)M10 constraint ' ;sed 1'.0/7!'3 1! 1.0/7!'00 1CG'

    ANS:

    a. B)3)11b. B311c. B)1!G1!d. B)+)11G+11e. B)-)11G-11

    f. B)/)11G/11g. 3.+/h. +./1i. es%. no

    PTS: 1 TOP: Spreadsheet solution of Ps

    0. ;se the following 8anagement Scientist output to answer the questions.

    (NCA* P*OE*A88(NE P*O)C8

    8AF '1F1G'-F!G'!F'

    S.T.16 'F1G-F!G!F'7!6 /F1G0F!G3F'1-'6 -F1G'F!G'F'1!

    OPT(8A SO;T(ON

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    Ob%ective @unction 9alue B 0/'.'''

    9ariable 9alue *educed ostF1 1'.''' .F! 1. .F' . 1.337

    onstraint Slac,Surplus &ual Price

    1 . .003

    ! . -.--/' !'.''' .

    O)QCT(9C OC@@((CNT *ANECS

    9ariable ower imit urrent 9alue ;pper imitF1 '. '1. No ;pper imitF! No ower imit '-. '/.1/0F' No ower imit '!. +!.337

    *(EDT DAN& S(&C *ANECS

    onstraint ower imit urrent 9alue ;pper imit1 00./+0 7. 10.1+'! 1!/. 1-. 1/'.1!-' 7/.//0 1!. No ;pper imit

    a. Eive the solution to the problem.b.

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    F! 3. .F' . 1.

    onstraint Slac,Surplus &ual Price1 -. .! +. .

    ' . 1.!-

    O)QCT(9C OC@@((CNT *ANECS

    9ariable ower imit urrent 9alue ;pper imitF1 !.- +. No ;pper imitF! . -. /.F' -. /. No ;pper imit

    *(EDT DAN& S(&C *ANECS

    onstraint ower imit urrent 9alue ;pper imit1 3. 3-. No ;pper imit! No ower imit !3. '!.' !3. '!. '+.

    a.

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    (NCA* P*OE*A88(NE P*O)C8

    8AF !-F1G'F!G1-F'

    S.T.16 +F1G-F!G3F'1!!6 7F1G1-F!G'F'1-

    OPT(8A SO;T(ON

    Ob%ective @unction 9alue B +0.

    9ariable 9alue *educed ostF1 1+. .F! . 1.F' 3. .

    onstraint Slac,Surplus &ual Price1 . 1.! . !.'''

    O)QCT(9C OC@@((CNT *ANECS

    9ariable ower imit urrent 9alue ;pper imitF1 17.!3/ !-. +-.F! No ower imit '. +.F' 3.''' 1-. -.

    *(EDT DAN& S(&C *ANECS

    onstraint ower imit urrent 9alue ;pper imit1 ///.//0 1!. +.

    ! +-. 1-. !0.

    a. Eive the complete optimal solution.b.

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    f. The sum of percentage changes is 0!3 G 5'-651-6 1 so the solution will not

    change.

    PTS: 1 TOP: (nterpretation of 8anagement Scientist output

    1. (N&O output is given for the following linear programming problem.

    8(N 1! F1 G 1 F! G 7 F'

    S;)QCT TO!6 - F1 G 3 F! G - F' B /'6 3 F1 G 1 F! G - F' B 3

    CN&

    P OPT(8;8 @O;N& AT STCP 1

    O)QCT(9C @;NT(ON 9A;C

    16 3.

    9A*(A)C 9A;C *C&;C& OSTF1 . +.F! 3. .F' . +.

    *O< SAJ O* S;*P;S &;A P*(C!6 +. .

    '6 . 1.

    NO. (TC*AT(ONSB 1

    *ANECS (N

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    ANS:

    a. 41B $ 4!B 3$ 4'B $ s1B +$ s!B $ " B 3b. onstraint ! is binding.c. c1would have to decrease b + or more for 41to become positive.d. (ncreasing the right#hand side b 1 will cause a negative improvement$ or increase$ of 1 in

    this minimi"ation ob%ective function.e. The sum of the percentage changes is 5!65+6 G !- 1 so the solution would not

    change.

    PTS: 1 TOP: (nterpretation of (N&O output

    11. The P problem whose output follows determines how man nec,laces$ bracelets$ rings$ and earrings a%ewelr store should stoc,. The ob%ective function measures profitR it is assumed that ever piecestoc,ed will be sold. onstraint 1 measures displa space in units$ constraint ! measures time to set upthe displa in minutes. onstraints ' and + are mar,eting restrictions.

    (NCA* P*OE*A88(NE P*O)C8

    8AF 1F1G1!F!G1-F'G1!-F+

    S.T.16 F1G!F!G!F'G!F+13!6 'F1G-F!GF+1!'6 F1GF'!-+6 F!GF'GF+-

    OPT(8A SO;T(ON

    Ob%ective @unction 9alue B 0+0-.

    9ariable 9alue *educed ostF1 3. .F! . -.F' 10. .F+ ''. .

    onstraint Slac,Surplus &ual Price1 . 0-.! /'. .' . !-.

    + . !-.

    O)QCT(9C OC@@((CNT *ANECS

    9ariable ower imit urrent 9alue ;pper imitF1 30.- 1. No ;pper imitF! No ower imit 1!. 1!-.F' 1!-. 1-. 1/!.-F+ 1!. 1!-. 1-.

    *(EDT DAN& S(&C *ANECS

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    onstraint ower imit urrent 9alue ;pper imit1 1. 13. 1!'.0-! -0. 1!. No ;pper imit' 3. !-. -3.+ +1.- -. -+.

    ;se the output to answer the questions.

    a. Dow man nec,laces should be stoc,ed=

    b. Now man bracelets should be stoc,ed=c. Dow man rings should be stoc,ed=d. Dow man earrings should be stoc,ed=e. Dow much space will be left unused=f. Dow much time will be used=g. ) how much will the second mar,eting restriction be e4ceeded=h.

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    1-=

    (NCA* P*OE*A88(NE P*O)C8

    8AF +F1G/F!G0F'

    S.T.16 'F1G!F!G-F'1!!6 1F1G'F!G'F'3

    '6 -F1G-F!G3F'1/+6 G1F'1

    OPT(8A SO;T(ON

    Ob%ective @unction 9alue B 1//.

    9ariable 9alue *educed ostF1 . !.F! 1/. .F' 1. .

    onstraint Slac,Surplus &ual Price1 '3. .! !. .' . 1.!

    + . !./

    O)QCT(9C OC@@((CNT *ANECS

    9ariable ower imit urrent 9alue ;pper imitF1 No ower imit +. /.F! +.'0- /. No ;pper imitF' No ower imit 0. 7./

    *(EDT DAN& S(&C *ANECS

    onstraint ower imit urrent 9alue ;pper imit1 3!. 1!. No ;pper imit! 03. 3. No ;pper imit' 3. 1/. 1/'.'''+ 3.337 1. !.

    ANS:

    a. b. 1/c. 1d. ++e. !f. 1//g. rerunh. L B 10/i. L B 1/3.+

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    %. L B 1-'

    PTS: 1 TOP: (nterpretation of 8anagement Scientist output

    1'. The P model and (N&O output represent a problem whose solution will tell a specialt retailer howman of four different stles of umbrellas to stoc, in order to ma4imi"e profit. (t is assumed that everone stoc,ed will be sold. The variables measure the number of women?s$ golf$ men?s$ and foldingumbrellas$ respectivel. The constraints measure storage space in units$ special displa rac,s$ demand$and a mar,eting restriction$ respectivel.

    8AF + F1 G / F! G - F' G '.- F+

    S;)QCT TO!6 ! F1 G ' F! G ' F' G F+ B 1!'6 1.- F1 G ! F! B -++6 ! F! G F' G F+ B 0!-6 F! G F' B 1!

    CN&

    O)QCT(9C @;NT(ON 9A;C

    16 '13.

    9A*(A)C 9A;C *C&;C& OSTF1 1!. .F! . .-F' 1!. .F+ /. .

    *O< SAJ O* S;*P;S &;A P*(C!6 . !.'6 '/. .

    +6 . 1.-

    -6 . !.-

    *ANECS (N

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    ;se the output to answer the questions.

    a. Dow man women?s umbrellas should be stoc,ed=b. Dow man golf umbrellas should be stoc,ed=c. Dow man men?s umbrellas should be stoc,ed=d. Dow man folding umbrellas should be stoc,ed=e. Dow much space is left unused=f. Dow man rac,s are used=

    g. ) how much is the mar,eting restriction e4ceeded=h.

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    9A*(A)C 9A;C *C&;C& OSTF1 IIIIIIII 1.'1!-F! IIIIIIII IIIIIIII F' !0.- IIIIIIII

    *O< SAJ O* S;*P;S &;A P*(C

    !6 IIIIIIII .1!-

    '6 IIIIIIII .031!-

    NO. (TC*AT(ONSB !

    *ANECS (N

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    F' IIIIII '.-

    onstraint Slac,Surplus &ual Price1 . 1-.! IIIIII .' IIIIII .+ . IIIIII

    O)QCT(9C OC@@((CNT *ANECS

    9ariable ower imit urrent 9alue ;pper imitF1 -.+ 1!. No ;pper imitF! !. 7. !.F' No ower imit 0. 1.-

    *(EDT DAN& S(&C *ANECS

    onstraint ower imit urrent 9alue ;pper imit1 1+-. 1-. 1-/.//0! IIIIII IIIIII /+.' IIIIII IIIIII 3.

    + 11.!3/ 11/. 1!.

    ANS:4'B because the reduced cost is positive.41B !+ after plugging into the ob%ective functionThe second reduced cost is .s!B ! and s'B !! from plugging into the constraints.

    The fourth dual price is 1/.- from plugging into the dual ob%ective function$ which our students

    might not understand full until hapter /.The lower limit for constraint ! is ++ and for constraint ' is -3$ from the amount of slac, in eachconstraint. There are no upper limits for these constraints.

    PTS: 1 TOP: (nterpretation of solution

    1/. A large sporting goods store is placing an order for biccles with its supplier. @our models can beordered: the adult Open Trail$ the adult itscape$ the girl?s Sea Sprite$ and the bo?s Trail )la"er. (t isassumed that ever bi,e ordered will be sold$ and their profits$ respectivel$ are '$ !-$ !!$ and !. TheP model should ma4imi"e profit. There are several conditions that the store needs to worr about.One of these is space to hold the inventor. An adult?s bi,e needs two feet$ but a child?s bi,e needsonl one foot. The store has - feet of space. There are 1! hours of assembl time available. Thechild?s bi,e need + hours of assembl timeR the Open Trail needs - hours and the itscape needs /hours. The store would li,e to place an order for at least !0- bi,es.a. @ormulate a model for this problem.b. Solve our model with an computer pac,age available to ou.c. Dow man of each ,ind of bi,e should be ordered and what will the profit be=d.

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    ANS:NOTC TO (NST*;TO*: The problem is suitable for a ta,e#home or lab e4am. The student mustformulate the model$ solve the problem with a computer pac,age$ and then interpret the solution toanswer the questions.

    a. 8AF ' F1 G !- F! G !! F' G ! F+

    S;)QCT TO!6 ! F1 G ! F! G F' G F+ B -

    '6 - F1 G / F! G + F' G + F+ B 1!+6 F1 G F! G F' G F+ B !0-

    b. O)QCT(9C @;NT(ON 9A;C

    16 /3-.

    9A*(A)C 9A;C *C&;C& OSTF1 1. .F! . 1'.F' 10-. .F+ . !.

    *O< SAJ O* S;*P;S &;A P*(C!6 1!-. .'6 . 3.

    +6 . 1.

    NO. (TC*AT(ONSB !

    *ANECS (N

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    PTS: 1 TOP: @ormulation and computer solution

    10. A compan produces two products made from aluminum and copper. The table below gives the unitrequirements$ the unit production man#hours required$ the unit profit and the availabilit of theresources 5in tons6.

    Aluminum opper 8an#hours ;nit Profit

    Product 1 1 ! -Product ! 1 1 ' /Available 1 / !+

    The Management Scientistprovided the following solution output:

    Ob%ective @unction 9alue B -+.

    9A*(A)C 9A;C *C&;C& OSTF1 /. .F! +. .

    ONST*A(NT SAJS;*P;S &;A P*(C1 . '.! !. .' . 1.

    *ANECS (N

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    compan should be willing to pa for e4tra aluminum=

    ANS:

    a. / product 1$ + product !$ Profit B M-+b. )etween M- and M0-R at M0 the profit is M-3c. NoR total 2 change is 3' 1'2 12d. &ual prices are the shadow prices for the resourcesR since there was unused copper

    5because S!B !6$ e4tra copper is worth Me. M'f. M1R this is the amount e4tra man#hours are worthg. The shadow price is the premium for aluminum ## would be willing to pa up to M1 G

    M' B M+ for e4tra aluminum

    PTS: 1 TOP: (nterpretation of solution

    13. Eiven the following linear program:

    8AF -x1G 0x!

    s.t. x1/

    !x1G 'x!17

    x1Gx!3

    x1$x!

    The graphical solution to the problem is shown below. @rom the graph we see that the optimal solutionoccurs atx1B -$x!B '$ andzB +/.

    a. alculate the range of optimalit for each ob%ective function coefficient.b. alculate the dual price for each resource.

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    ANS:

    a. *anges of optimalit: 1+' c10 and - c!1-!

    b. Summari"ing$ the dual price for the first resource is $ for the second resource is !$ and forthe third is 1

    PTS: 1 TOP: (ntroduction to sensitivit analsis

    17. onsider the following linear program:

    8AF 'x1G +x!5M Profit6

    s.t. x1G 'x!1!

    !x1Gx!3

    x1'

    x1$x!

    The Management Scientistprovided the following solution output:

    OPT(8A SO;T(ON

    Ob%ective @unction 9alue B !.

    9ariable 9alue *educed ostF1 !.+ .F! '.! .

    onstraint Slac,Surplus &ual Price1 . 1.! . 1.' ./ .

    O)QCT(9C OC@@((CNT *ANECS

    9ariable ower imit urrent 9alue ;pper imitF1 1.''' '. 3.F! 1.- +. 7.

    *(EDT DAN& S(&C *ANECS

    onstraint ower imit urrent 9alue ;pper imit1 7. 1!. !+.! +. 3. 7.

    ' !.+ '. No ;pper imit

    a.

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    ANS:

    a. x1B !.+ andx!B '.!$ andzB M!..b. Optimal solution will not change. Optimal profit will equal M!7./.c. )ecause 1 is outside the range of 1.- to 7.$ the optimal solution li,el would change.d. Sum of the change percentages is -2 G +2 B 72. Since this does not e4ceed 12

    the optimal solution would not change.

    PTS: 1 TOP: (nterpretation of solution

    !. onsider the following linear program:

    8(N /x1G 7x!5M cost6

    s.t. x1G !x!3

    1x1G 0.-x!'

    x!!

    x1$x!

    The Management Scientistprovided the following solution output:

    OPT(8A SO;T(ON

    Ob%ective @unction 9alue B !0.

    9ariable 9alue *educed ostF1 1.- .F! !. .

    onstraint Slac,Surplus &ual Price1 !.- .

    ! . ./

    ' . +.-

    O)QCT(9C OC@@((CNT *ANECS

    9ariable ower imit urrent 9alue ;pper imitF1 . /. 1!.F! +.- 7. No ;pper imit

    *(EDT DAN& S(&C *ANECS

    onstraint ower imit urrent 9alue ;pper imit1 -.- 3. No ;pper imit

    ! 1-. '. --.' . !. +.

    a.

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    d. (f simultaneousl the cost of 41was raised to M0.- and the cost ofx!was reduced to M/$would the current solution still remain optimal=

    e. (f the right#hand side of constraint ' is increased b 1$ what will be the effect on theoptimal solution=

    ANS:

    a. x1B 1.- andx!B !.$ and the ob%ective function value B !0..b. + is within this range of to 1!$ so the optimal solution will not change. Optimal total cost

    will be M!+..c. x!can fall to +.- without concern for the optimal solution changing.d. Sum of the change percentages is 71.02. This does not e4ceed 12$ so the optimal

    solution would not change.e. The right#hand side remains within the range of feasibilit$ so there is no change in the

    optimal solution. Dowever$ the ob%ective function value increases b M+.-.

    PTS: 1 TOP: (nterpretation of solution