Linear Programming Simplex

8/10/2019 Linear Programming Simplex

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Linear Programming: The Simplex Method CHAPTER 9

TRUE/FALSE

9.1 A basic feasible solution is a solution to a linear programming problem that corresponds to acorner point of the feasible region.

ANSWER: TRUE

9.! A surplus "ariable is added to a constraint in order to create an e#ualit$% and represents a#uantit$ of unused resource.

ANSWER: &A'SE

9.( A surplus "ariable is added to an = constraint in order to utili)e the simple* algorithm.

ANSWER: &A'SE

9.+ ,f all of a resource represented b$ a slac- "ariable is used% that slac- "ariable ill not be in theproduction mi* column of a linear programming simple* tableau.

ANSWER: TRUE

9./ E"en if an '0 problem in"ol"es man$ "ariables% an optimal solution ill ala$s be found at acorner point of the ndimensional pol$hedron forming the feasible region.

ANSWER: TRUE

9.2 A correctl$ formulated linear program% hen sol"ed ith the simple* algorithm% ill ala$s $ield asingle optimal solution.

ANSWER: &A'SE

9.3 Unli-e the Sol"er algorithm% Simple* ill onl$ produce a single solution e"en if multiple solutionse*ist.

ANSWER: &A'SE

9.4 Surplus "ariables% li-e slac- "ariables% carr$ a )ero cost.

ANSWER: TRUE

9.9 The constraint /516 25!(7% hen con"erted to an 8 constraint for use in the simple* algorithm%

ill be / 516 2 5!S 6 A 8 (7.

ANSWER: TRUE

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9.17 The constraint / 516 2 5!(7% hen con"erted to an 8 constraint for use in the simple*

algorithm% ill be / 516 2 5!S 8 (7.

ANSWER: &A'SE

9.11 The constraint / 516 2 5!8 (7% hen con"erted to an 8 constraint for use in the simple*algorithm% ill be / 516 2 5!6 8 (7.

ANSWER: &A'SE

9.1! 'inear programming has fe applications in the real orld due to the assumption of certaint$ inthe data and relationships of a problem.

ANSWER: &A'SE

9.1( T$picall$% real orld applications of linear programming are sol"ed ith a computer program thatutili)es the simple* algorithm.

ANSWER: TRUE

9.1+ The basic process of the simple* algorithm is to find solutions to a set of simultaneous e#uationshere e ha"e more "ariables than e#uations.

ANSWER: TRUE

9.1/ ,t is possible for an e#uation in the simple* table to ha"e both a slac- and a surplus "ariable at thesame time.

ANSWER: &A'SE

9.12 ,n the simple* table% a coefficient of )ero in a constraint implies that the "ariable ith the )erocoefficient has no influence on the solution for that iteration.

ANSWER: TRUE

9.13 The simple* method considers both feasible and infeasible solutions.

ANSWER: TRUE

9.14 The simple* method finds a solution b$ sol"ing for the intersection of to constraints.

ANSWER: TRUE

9.19 When the optimal solution is found% all slac- and surplus "ariables ha"e a "alue of )ero.

ANSWER: &A'SE

9.!7 The numbers in the bod$ of the simple* table ma$ be thought of as the solutions.

ANSWER: &A'SE9.!1 ,f there are se"en lessthanore#ual constraints in a problem% the simple* table contains se"en

slac- "ariables.

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ANSWER: TRUE

9.!! &or a ma*imi)ation problem% the ;"alues in the bod$ of the simple* table represent the grossprofit gi"en up b$ adding one unit of this "ariable into the current solution.

ANSWER: TRUE

9.!( ,n a ma*imi)ation problem% the must benonnegati"e.

ANSWER: TRUE

9.(1 Artificial "ariables can be used in both ma*imi)ation and minimi)ation problems.

ANSWER: TRUE

9.(! Artificial "ariables are most often used in minimi)ation problems.

ANSWER: TRUE

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9.(( ,n an$ linear programming problem% if a "ariable is to enter the solution% it must ha"e a positi"ecoefficient in the


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=b> if 1 unit of 51is added to the solution% 5!ill decrease b$ !.=c> if 1 unit of 5!is added to the solution% 51ill increase b$ !.=d> if 1 unit of 51is added to the solution% 5!ill increase b$ !.=e> none of the abo"e

ANSWER: c

9./2 Which of the folloing is not true of artificial "ariables:

=a> The$ ha"e no meaning in a ph$sical sensenothing more than a computational tool.

=b> ,n all linear programs% the$ appear in the ob;ecti"e function ith a "er$ lo cost =>.

=c> The$ are usuall$ used ith constraints.

=d> The$ are usuall$ used ith 8 constraints.=e> none of the abo"e

ANSWER: b

9./3 ,f% in the optimal tableau of a linear programming problem% an artificial "ariable is present in thesolution mi*% this implies

=a> infeasibilit$.=b> unboundedness.=c> degenerac$.=d> alternate optimal solutions.=e> a finite optimal solution.

ANSWER: a

9./4 ,f% in the final optimal simple* tableau% the feasibilit$.=b> unboundedness.=c> degenerac$.=d> alternate optimal solutions.=e> none of the abo"e

ANSWER: d

9./9 Which of the folloing is not true about slac- "ariables in a simple* tableauB

=a> The$ are used to con"ert constraint ine#ualities to e#uations.

=b> The$ represent unused resources.=c> The$ re#uire the addition of an artificial "ariable.=d> The$ ma$ represent machine time% labor hours% or arehouse space.

=e> The$ $ield no profit.

ANSWER: c9.27 Casic "ariables in the simple* method of linear programming are

=a> "ariables in the solution mi*.=b> "ariables not in the solution mi*.=c> the real "ariables in the initial solution.=d> the slac- "ariables in the optimum solution.

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=e> ala$s the slac-% surplus% and artificial "ariables.

ANSWER: a

9.21 The folloing is not true for the net profit ro of the simple* method for linear programming:

=a> ,t contains shado prices.

=b> The "alues under the slac- columns ma$ be "ieed as the potential increase in profit if onemore unit of that resource as made a"ailable.

=c> An optimum solution has been reached hen there are no positi"e numbers.=d> ,t ma$ indicate the e*istence of more than one optimal solution.=e> ,nitiall$% the "alues under the slac- "alues columns ill be positi"e numbers.

ANSWER: e

9.2! ,n sol"ing a linear programming minimi)ation problem using the simple* method%

=a> e"er$ time an artificial "ariable is added% a surplus "ariable must also be added.=b> e"er$ time an artificial "ariable is added% a surplus "ariable must be subtracted.=c> e"er$ time a surplus "ariable is added% an artificial "ariable must be added.=d> e"er$ time a surplus "ariable is added% an artificial "ariable must be subtracted.=e> none of the abo"e

ANSWER: e

9.2( ,n using the simple* method for minimi)ation linear programming problems%

=a> an optimal solution is reached hen the the pi"ot column is selected b$ the smallest positi"e number in the the pi"ot column is selected b$ the most negati"e number in the the pi"ot column is selected b$ the smallest negati"e number in the ;ro.=e> none of the abo"e

ANSWER: c

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9.2+ The folloing are constraints hich ha"e had slac- "ariables added. 516 ! 5!6 S18 1!! 516 ( 5!6 S!8 (7

,f 518 ! and 5!8 +% hat are the "alues for S1and S!B

=a> S18 1!% S!8 (7=b> S18 2% S!8 !+

=c> S18 !% S!8 1+=d> S18 7% S!8 7=e> none of the abo"e

ANSWER: c

9.2/ The folloing are constraints hich ha"e had slac- "ariables added.! 516 + 5!6 S18 !72 516 + 5!6 S!8 (2

,f 518 ! and 5!8 !% hat are the "alues for S1and S!B

=a> S18 !7% S!8 (2=b> S18 12% S!8 (!=c> S18 4% S!8 12=d> S18 7% S!8 7=e> none of the abo"e

ANSWER: e

9.22 A sol"ed '0 problem indicated that the optimal solution as 51817 and 5! 8!7. Dne of the

constraints as + 516! 5!47. This constraint has

=a> surplus greater than )ero.=b> slac- greater than )ero.=c> surplus e#ual to )ero.

=d> slac- e#ual to )ero.=e> none of the abo"e

ANSWER: d

9.23


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=a> ( slac-% ( surplus% and ( artificial=b> 1 slac-% 1 surplus% and 1 artificial=c> 1 slac-% + surplus% and + artificial=d> 1 slac-% 1 surplus% and 1 artificial=e> none of the abo"e

ANSWER: d

9.24 ( slac-% 2 surplus% and 2 artificial=b> ! slac-% 1 surplus% and ! artificial=c> 1 slac-% ! surplus% and ! artificial=d> 1 slac-% ! surplus% and 1 artificial=e> none of the abo"e

ANSWER: b

9.29 Sensiti"it$ anal$sis in linear programming

=a> centers around applications of computer routines to sol"e '0 problems.=b> concerns studies of the e*tra information associated ith the dual '0.

=c> concerns changes in the data used to build the '0 model and the effects on the optimal solution.=d> all of the abo"e

ANSWER: c

9.37 ,f one changes the contribution rates in the ob;ecti"e function of an '0 problem%

=a> the feasible region ill change.=b> the slope of the isoprofit or isocost line ill change.=c> the optimal solution to the '0 ill no longer be optimal.=d> all of the abo"e=e> none of the abo"e

ANSWER: b

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9.31 ,f one changes a nonbasic ob;ecti"e function coefficient% the optimal solution of a ma*imi)ationproblem ill remain optimal if

=a> the increase in the coefficient does not e*ceed the "alue of the ;associated ith that nonbasic"ariable.

=b> the increase in the coefficient does not e*ceed the "alues of the ;Fs of e"er$ basic "ariable.=c> the decrease in the coefficient does not e*ceed the "alue of the ;associated ith the nonbasic

"ariable.=d> the ne none of the abo"e

ANSWER: a

9.3! presents the marginal profits of each additional unit of a resource.=b> can ala$s be deri"ed b$ e*amining the ;ro of the primal?s optimal simple* tableau.=c> is better than the solution to the primal.=d> all of the abo"e

ANSWER: a

9.3+ Shado prices

=a> can be deri"ed from the coefficients of the slac- "ariables in represent the "alue of one additional unit of a resource.=c> are found in the solution to the dual '0.=d> all of the abo"e=e> none of the abo"e

ANSWER: d

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9.3/ Sensiti"it$ anal$ses are used to e*amine the effects of changes in

=a> contribution rates for each "ariable.=b> technological coefficients.=c> a"ailable resources.=d> all of the abo"e

ANSWER: d

9.32 Sensiti"it$ anal$sis ma$ be used to

=a> e*periment ith possible future changes in the firm that ma$ affect profits.=b> determine hether a corner point of the feasible region actuall$ $ields optimal profit.=c> replace the simple* method.=d> reduce the number of "ariables in a comple* '0 problem.=e> sol"e '0 problems that re#uire solutions onl$ in hole numbers.

ANSWER: a

9.33 A shado price is

=a> the coefficient of a constraint.=b> the "alue o"er standard cost of one additional unit of a resource that becomes a"ailable.=c> the coefficients that are deleted hen transposing a matri* from a primal to a dual.=d> an$ negati"e "alue in the ;ro.=e> the smallest result obtained hen basic "ariable "alues in a column are di"ided into their

respecti"e ro #uantities.

ANSWER: b

9.34 Right handside ranging

=a> tells us the number of units of a constraint that ma$ be added or subtracted ithout changingthe profit =ob;ecti"e function "alue>.

=b> is a method to find the range o"er hich shado prices remain constant.=c> re#uires computation of the dual before being accomplished.=d> all of the abo"e

ANSWER: b

9.39 &or e"er$ primal

=a> that is a ma*imi)ation problem% the dual is a minimi)ation problem.=b> the right handside #uantities become the dual?s ob;ecti"e function coefficients.=c> constraint ine#ualit$ signs are re"ersed in the dual.=d> the transpose of the constraint coefficients become the dual?s constraint coefficients.=e> all of the abo"e

ANSWER: e

9.47 Sol"ing linear programming problems b$ computer

=a> is practical onl$ for small programs.

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=b> can be done onl$ on G for Windos.=c> can sol"e enormous programs.=d> is useful% but doesn?t include sensiti"it$ anal$sis.=e> re#uires e*pert programmers.

ANSWER: c

9.41 The dual of a linear programming problem

=a> ala$s e*ists.=b> ma$ be easier to sol"e.=c> ma$ contain economic information useful to management.=d> hile e#ui"alent% is deri"ed through an alternati"e procedure.=e> all of the abo"e

ANSWER: e

9.4! ,t ould not change.=b> ,t ould increase b$ !/.=c> ,t ould decrease b$ !/.=d> ,t ould increase b$ +7.=e> ,t ould decrease b$ +7.

ANSWER: b

9.4(


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=a> ,t ould not change.=b> ,t ould increase b$ !/.=c> ,t ould decrease b$ !/.=d> ,t ould increase b$ +7.=e> ,t ould decrease b$ +7.

ANSWER: a

9.4+ A primal linear programming problem has four "ariables and three constraints. The dual of thisill ha"e

=a> four "ariables and three constraints.=b> three "ariables and four constraints.=c> four "ariables and se"en constraints.=d> se"en "ariables and four constraints.=e> none of the abo"e

ANSWER: b

TAC'E 91The folloing is a partial simple* tableau for a ma*imi)ation problem after one iteration


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9.42 According to Table 91% hich is a simple* tableau after at least one iteration% hich ro ould beselected as the pi"ot roB

=a> first=b> second=c> third=d> none of the abo"e

ANSWER: b

9.43 According to Table 91% hich is a simple* tableau after at least one iteration% hich "ariableshould enter the solution at the ne*t iterationB

=a> 51=b> 5!=c> 5(=d> S1=e> none of the abo"e

ANSWER: a

9.44 According to Table 91% hich is a simple* tableau after at least one iteration% ho man$ units ofthe "ariable entering the solution ne*t ill be in the basis in the ne*t tableauB

=a> 1./=b> 2.23=c> 17=d> 2=e> none of the abo"e

ANSWER: d

TAC'E 9!Simple* Tableau : 1


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=a> no feasible solution=b> multiple optimal solutions=c> degenerac$=d> unbounded solution=e> none of the abo"e

ANSWER: C

9.97 According to Table 9!% the 7./77 in the 51column means that

=a> to produce 1 unit of 51% 7./ units of 5!must be gi"en up.=b> to produce 1 unit of 5!% 7./ units of 51must be gi"en up.=c> if 1 unit of 51is produced% profits on 5!ill decrease b$ 7./77.=d> if 1 unit of 51is produced% profits on 5!ill increase b$ 7./77.=e> none of the abo"e

ANSWER: a

9.91 According to Table 9!% hich is the final simple* tableau for a problem ith to "ariables andto constraints% hat can be said about the optimal solution and the constraintsB

=a> There is slac- remaining in the first constraint.=b> There is slac- remaining in the second constraint.=c> There is slac- remaining in both constraints.=d> There is no slac- remaining in either constraint.=e> none of the abo"e

ANSWER: b

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TAC'E 9(

to produce 1 unit of 51% 7.223 units of 5!must be gi"en up.=b> to produce 1 unit of 5!% 7.223 units of 51must be gi"en up.=c> if 1 unit of 51is produced% profits on 5!ill decrease b$ 7.223.=d> if 1 unit of 51is produced% profits on 5!ill increase b$ 7.223.=e> none of the abo"e

ANSWER: a

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9.9+ According to Table 9(% hich is the final simple* tableau for a problem ith to "ariables andto constraints% hat can be said about the optimal solution and the constraintsB

=a> There is slac- remaining in the first constraint.=b> There is slac- remaining in the second constraint.=c> There is slac- remaining in both constraints.=d> There is no slac- remaining in either constraint.

=e> none of the abo"e

ANSWER: b

9.9/ According to Table 9(% hich is the final simple* tableau for a problem ith to "ariables andto constraints% hat is the ma*imum possible profit =ob;ecti"e function "alue> for this problemB

=a> !7=b> +7=c> 477=d> !2.223=e> none of the abo"e

ANSWER: c

9.92 According to Table 9(% hich is the final simple* tableau for a linear programming problem=ma*imi)ation>% hat ould happen to profit if the 51column ere selected as the pi"ot columnand another iteration of the simple* algorithm ere performedB

=a> Total profits ould increase.=b> Total profits ould decrease.=c> An infeasible solution ould be found.=d> Another optimal solution ould be found.=e> none of the abo"e

ANSWER: b

Table 9+a*imi)e profit !7516 (75!6 125(Sub;ect to: 4516 +5 64 Resource A

+516 1!5!6 +5( 96 Resource C

All "ariables 0

+12

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=b> +13=c> +12./=d> +1/./=e> none of the abo"e

ANSWER: c

9.94 According to Table 9+% all of the resources are being used. ,f the amount of resource C erechanged from 92 to 93% then the ma*imum possible total profit ould be

=a> +12=b> +13=c> +19=d> +!7=e> none of the abo"e

ANSWER: d

9.99 According to Table 9+% it is currentl$ profitable to produce some units of 51and the current profitper unit of 51is !7. What is the loest "alue that this could be to allo this "ariable to remain inthe basisB

=a> 4=b> 12=c> !7=d> (7=e> none of the abo"e

ANSWER: b

TAC'E 9/a*imi)e profit + 516 4 5!6 ( 5(

Sub;ect to: / 516 17 5!97 resource A+ 516 1! 5!6 ! 5(27 resource C

All "ariables 7

The final simple* tableau is pro"ided belo.

,t ould increase b$ !.=c> ,t ould decrease b$ 1.

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=d> ,t ould decrease b$ !.=e> ,t ould not change.

ANSWER: e

9.171 According to Table 9/% the optimal solution to a linear programming problem has been found.What ould happen to the ma*imum possible total profit if the amount of resource A ere

decreased from 97 units to 27 unitsB

=a> Total profits ould decrease.=b> Total profits ould increase.=c> Total profits ould not change.=d> Unable to tell from the information pro"ided.=e> none of the abo"e

ANSWER: c

9.17! According to Table 9/% the optimal solution to a linear programming problem has been found. ,fthe amount of resource C ere increased b$ 1 unit =from 27 to 21> the ma*imum possible totalprofit ould

=a> increase b$ 1.=b> increase b$ 1./7.=c> decrease b$ 1.=d> decrease b$ 1./7.=e> not change.

ANSWER: b

9.17( According to Table 9/% the optimal solution to a linear programming problem has been found.While the current optimal solution calls for )ero units of 51to be produced% if the profit on 51ereraised% it ould be possible to produce 51and still generate a profit of 97. o much ould the

profit on 51ha"e to be raised for this to occurB

=a> !=b> +=c> 4=d> 17=e> none of the abo"e

ANSWER: a

Table 92

a*imi)e 8 1! 516 (7 5!Sub;ect to: (! 516 14 5!277 Resource A

!+ 516 +4 5!477 Resource C

&inal Dptimal Solution: 8 /77

Iariable Solution Shado0rice

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51 7 (5! 12.223 7S1 (77 7S! 7 7.2!/

Db;ecti"e 518 1!% 5!8 (7% S18 7.2!/% S!8 7=e> none of the abo"e

ANSWER: a

9.17/ According to Table 92% hich is a summari)ed solution output from simple* anal$sis% if theamount of resource A ere decreased so that there ere onl$ //7 units a"ailable instead of 277%hat ould happen to total profitsB

=a> The$ ould decrease.=b> The$ ould increase.=c> The$ ould not change.=d> Unable to determine from the gi"en information.=e> none of the abo"e

ANSWER: c

9.172 ,n appl$ing the simple* solution procedure to a minimi)ation problem to determine hich "ariableenters the solution mi*%

=a> pic- the one ith the most negati"e pic- the one ith the positi"e pic- the one ith the largest pic- the one ith the smallest ;.=e> pic- the smallest nonnegati"e number formed b$ di"iding each amount in the #uantit$ columnb$ the appropriate column at the e*iting "ariable.

ANSWER: a

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9.173 Which of the folloing is true of slac- "ariablesB

=a> The$ ha"e no meaning in a ph$sical sensenothing more than a computational tool.

=b> ,n all linear programs% the$ appear in the ob;ecti"e function ith a "er$ lo cost =>.

=c> The$ are usuall$ used ith constraints.

=d> The$ are usuall$ used ith 8 constraints.

=e> The$ are usuall$ used ith constraints.

ANSWER: e

9.174 ,f% in the optimal tableau of a linear programming problem% there is an artificial "ariable present inthe solution mi*% this implies

=a> infeasibilit$.=b> unboundedness.=c> degenerac$.=d> alternate optimal solutions.=e> a finite optimal solution.

ANSWER: a

9.179 ,f% in the final optimal simple* tableau% the feasibilit$.=b> unboundedness.=c> degenerac$.=d> alternate optimal solutions.=e> none of the abo"e

ANSWER: e

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9.117 Which of the folloing is true about surplus "ariables in a simple* tableauB

=a> The$ are used to con"ert constraint ine#ualities to e#uations.

=b> The$ represent unused resources.=c> The$ re#uire the addition of an artificial "ariable.=d> The$ ma$ represent machine time% labor hours% or arehouse space.=e> The$ $ield a positi"e profit.

ANSWER: c

9.111 Nonbasic "ariables in the simple* method of linear programming are

=a> "ariables in the solution mi*.=b> "ariables not in the solution mi*.=c> the real "ariables in the initial solution.=d> the slac- "ariables in the optimum solution.=e> ala$s the slac-% surplus% and artificial "ariables.

ANSWER: b

9.11! ,f% in sol"ing a linear programming minimi)ation problem using the simple* method% a slac-"ariable is gi"en a non)ero coefficient in the ob;ecti"e function% this implies

=a> there is a cost for using an amount of the resource greater than that hich is a"ailable.=b> there is no cost for using the resource.=c> there is a cost for failing to use the entire amount of the resource.=d> the resource can be substituted for another resource.=e> none of the abo"e

ANSWER: c

9.11( A sol"ed '0 problem indicated that the optimal solution as 518 /% 5!8 17% A1 8 +7. The first

constraint as: +516 !5!47. This solution is:

=a> indeterminant.=b> degenerate.=c> infeasible.=d> unbounded.=e> none of the abo"e

ANSWER: c

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9.11+ ,f one changes the coefficient of a constraint in an '0 problem%

=a> the feasible region ill change.=b> the slope of the isoprofit or isocost line ill change.=c> the "alue of the ob;ecti"e function ill increase.=d> the feasible region ma$ change.=e> none of the abo"e

ANSWER: d

9.11/ Sensiti"it$ anal$sis cannot be used to e*amine the effects of:

=a> changes in the contribution rates for each "ariable.=b> changes in the technological coefficients.=c> changes in the a"ailable resources.=d> the addition or deletion of a constraint.=e> none of the abo"e

ANSWER: d

9.112 ,n the initial basic solution% the "alues of the "ariables in the basic solution are

=a> ala$s )ero.=b> ala$s negati"e.=c> ne"er )ero.=d> ala$s nonnegati"e.=e> none of the abo"e

ANSWER: d

9.113 The range of insignificance is

=a> a range of "alues here the "ariables can be said to differ insignificantl$ from )ero.=b> the range of "ariables here the "alue of the artificial "ariables can be said to differ

insignificantl$ from )ero.=c> the range of "alues o"er hich the coefficient of a basic "ariable can "ar$ ithout causing a

change in the optimal solution mi*.=d> the range of "alues o"er hich the coefficient of a nonbasic "ariable can "ar$ ithout causing

a change in the optimal mi*.=e> none of the abo"e

ANSWER: d

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9.114 The range of optimalit$ is

=a> the range of "alues o"er hich the coefficient of the slac- "ariables can "ar$ ithout causing achange in the optimal mi*.

=b> the range of "alues o"er hich the coefficient of a basic "ariable can "ar$ ithout causing achange in the optimal solution mi*.

=c> the range o"er hich the coefficients of a surplus "ariable can "ar$ ithout causing a change

in the optimal solution mi*.=d> the range of "alues o"er hich the coefficient of a nonbasic "ariable can "ar$ ithout causing

a change in the optimal mi*.=e> none of the abo"e

ANSWER: b

9.119 Shado prices represent

=a> the "alue of one additional unit of a basic "ariable.=b> the "alue of one less unit of a basic "ariable.=c> the "alue of one additional unit of a specific resource.=d> the "alue of one less unit of a basic "ariable.=e> none of the abo"e

ANSWER: c

9.1!7 An artificial "ariable has no ph$sical interpretation but

=a> is added to each constraint to facilitate the Simple* process.

=b> is added to each constraint to facilitate the Simple* process.

=c> is added to each or 8 constraint to facilitate the Simple* process.

=d> is merel$ another manner of introducing a negati"e slac-.=e> none of the abo"e

ANSWER: b

9.1!1 A slac- "ariable

=a> is added to each constraint to facilitate the Simple* process.

=b> is added to each constraint to facilitate the Simple* process.

=c> is added to each or 8 constraint to facilitate the Simple* process.

=d> is added to each 8 constraint to facilitate the Simple* process.=e> none of the abo"e

ANSWER: a

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9.1!! Using the Simple* method% e -no e ha"e an optimal solution hen

=a> all slac- "ariables ha"e a )ero "alue.=b> all basic "ariables are negati"e.=c> hen all the real "ariables ha"e a non)ero "alue.=d> hen all the artificial "ariables ha"e a positi"e "alue.=e> none of the abo"e

ANSWER: e

9.1!( The substitution rates

=a> are the coefficients in the #uantit$ column.=b> tell us the number of units of a basic "ariable hich must be remo"ed from the solution inorder for another "ariable to enter the basis.=c> tell the amount of one resource hich can be substituted for another.=d> tell us the amount of a resource hich must be used to gain another unit of profit.=e> none of the abo"e

ANSWER: b

PROBLEMS

9.1!+ a*imi)e 8 ( 516 ! 5!5(6 7 S16 7 S!6 7 S(Sub;ect to 516 5!6 ! 5(6 S18 17

! 515!6 5(6 S!8 !7

( 516 5!6 S(8 1/

51% 5!% 5( 7

!24


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=b>


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a*imi)e 45 6 17L

Sub;ect to: /5 6 (L (+

!5 6 (L 8 !+

L (

5% L 7

ANSWER: a*imi)e 45 6 17L 6 7 S16 7 S! A1 A!Sub;ect to: /5 6 (L 6 S1 8 (+

!5 6 (L 6 A1 8 !+

L S!6 A!8 (

9.1!4 Sol"e the folloing linear programming problem using the simple* method.

a*imi)e ( 516 /5!Sub;ect to: + 516 ( 5! +4

516 ! 5! !7

51% 5! 7

ANSWER:


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Table 9.3a*imi)e /7 51 6 !7 5! 6 (7 5(

Sub;ect to ! 51 6 5! 6 5( 97 =Resource A>

51 6 ! 5! 6 5( /7 =Resource C>

51 6 5! 6 5( 47 =Resource

51% 5!% 5( 7

and the corresponding final simple* tableau.

What is the range of optimalit$ for the contribution rate of the "ariable 51B=b> anagement has decided to decrease the price of product 51. The profit contribution illdecrease from /7 to +7 for each unit of 51produced. Will the optimal produce mi* changeB=c> What is the range of insignificance of the contribution rate of the "ariable 5!B=d> o much ould $ou be illing to pa$ for one more unit of Resource AB E*plain.=e> o much ould $ou be illing to pa$ for an additional unit of Resource (7 No. This is ithin the range of optimalit$.=c>

=d> Up to !7 =the shado price of Resource A>=e> Nothing. Shado price is 7 =there are e*cess units a"ailable>.

9.1(1 According to Table 93% the final simple* tableau for a linear programming problem is pro"ided.Ji"en this table% calculate the right handside range for Resource C.

ANSWER:

Range of Resource C is from +/ units to 47 units.

9.1(! According to Table 93% the final simple* tableau for a linear programming problem is pro"ided.Ji"en this table:=a> What is the dual to the abo"e ma*imi)ation problemB=b> What is the optimal "alue of the dual programB=c> What is the optimal solution to the dual programB

!3!


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ANSWER:

=a> inimi)e 97U16 /7U!6 47U(Sub;ect to: ! U16 U!6 U(/7

U16 !U!6 U(!7

U16 U!6 U((7

U1%U!% U( 7

=b> !(77

=c> U18 !7% U!8 17% U(8 7.

9.1(( Upon retirement% r. Mlas started to ma-e to t$pes of childrenFs ooden to$s in his shop.Wuns $ield a "ariable profit of 4 each% and Toos ha"e a contribution margin of 9 apiece. E"enthough his electric sa o"erheats% he can ma-e 3 Wuns or 1+ Toos each da$. Since he doesn?t ha"ee#uipment for dr$ing the lac#uer finish he puts on the to$s% the dr$ing operation limits him to 12Wuns or 4 Toos per da$. The final tableau for r. Mlas? problem ould be:

3H(=b> 17H(

9.1(+ The folloing is a final simple* tableau for a ma*imi)ation problem. Use this to anser the#uestions belo.


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=a> What is the range of optimalit$ for the contribution rate of "ariable 5 1B=b> What is the range of optimalit$ for the contribution rate of "ariable 5!B=c> o much can the right handside of the second constraint be reduced ithout affecting the

ma*imum profitB=d> o much can the right handside of the first constraint be reduced ithout affecting the

ma*imum profitB

ANSWER:

=a> 3

=c> 37 units=d> 47 units

9.1(/ The folloing is a final simple* tableau for a ma*imi)ation problem. Use this to anser the#uestions belo.


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9.1(2 Write the dual of the folloing problem.

a*imi)e 45 6 17L


!5 6 (L !+

L (

5% L 7

ANSWER: inimi)e (+U16 !+U!6 (U(Sub;ect to: /U16 !U! 4

(U16 (U!6 U(17

All "ariables 7

9.1(3 The folloing is a final simple* tableau for a ma*imi)ation problem. All the "ariables are eitherdecision "ariables or slac- "ariables. There are no surplus "ariables in this problem.


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a*imi)e 45 6 17L


!5 6 (L 8 !+

L (

5% L 7

ANSWER:

a*imi)e 45 6 17L 6 1S1 6 1S( A! A(

Sub;ect to: /5 6 (L 6 1S1 6 7S( 6 7A! 6 7A( 8 (+!5 6 (L 6 7S1 6 7S( 6 1A! 6 7A( 8 !+ L 6 7S1 1S( 6 7A! 6 1A( 8 (

5% L 7

9.1+7 Write the dual of the folloing linear program:

a*imi)e ( 516 /5!Sub;ect to: + 516 ( 5! +4

516 ! 5! !7

51% 5! 7

ANSWER:

inimi)e +4U16 !7U!Sub;ect to: +U16 1U! (

(U16 !U! /

9.1+1 Write the dual of the folloing linear program:

inimi)e 8 1! 516 (7 5!Sub;ect to: (! 516 14 5!277

!+ 516 +4 5!477

ANSWER:

a*imi)e 277U16 477U!

Sub;ect to: (!U16 !+U!1!

14U16 +4U! (7

9.1+! Write the dual for the folloing linear program:

a*imi)e + 516 4 5!6 ( 5(Sub;ect to: / 516 17 5! 97

+ 516 1! 5!6 ! 5(177

( 516 3 5!6 / 5(27

!32


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ANSWER:

inimi)e 97U16 177U!6 27U(

Sub;ect to: /U16 +U!6 (U( +

17U16 1!U!6 3U(4

7U16 !U!6 /U( (

9.1+( Ji"en the folloing dual linear programming problem% rite the primal program:

a*imi)e /7U16 179U!6 !/U(

Sub;ect to: 2U16 +U!6 !U( 9

4U16 1(U! 6 11U(3

!U16 (U! 6 7U( (

ANSWER:

a*imi)e 9*16 3*!6 (*(Sub;ect to: 2*16 4*!6 !*(/7

+*16 1(*!6 (*(179

!*16 11*!6 7*(!/

9.1++ Ji"en the folloing dual linear program% rite the primal program:

a*imi)e (U16 !U!

Sub;ect to:

3U1 !U!7

/U16 U!17

U163U!1!

ANSWER:

inimi)e 7*16 17*!6 1!*(

Sub;ect to: 3*16 /*!6 1*( (

!*16 1*!6 1*!!

9.1+/ Write the dual of the folloing linear program.

a*imi)e +7 516 (7 5!6 27 5(

Sub;ect to 516 5!6 5(971! 516 4 5!6 17 5(1/77

51 !7

51 !7

5(177

51% 5!% 5(7

ANSWER:

!33


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inimi)e 97U16 1/77U!6 !7U(6 177U+Sub;ect to: 1U16 1!U!6 1U(6 7U+ +7

1U16 4U!6 7U(6 7U+ > (7

1U16 17U!6 7U(6 1U+ 27

SHORT ANSWER/ESSAY

9.1+2 E*plain hat a slac- "ariable represents in a constraint.

ANSWER: the amount of unused resource for that constraint

9.1+3 E*plain hat a surplus "ariable represents in a constraint.

ANSWER: the amount of o"erused resource for that constraint

9.1+4 E*plain hat an artificial "ariable represents in a constraint.

ANSWER: Nothingit onl$ acts as a tool to help generate an initial feasible solution.

9.1+9 What does the


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9.1// Ji"en the folloing linear program to be sol"ed manuall$ b$ use of the simple* algorithm% e*plainhat $ou might do before setting up the initial tableauB

a*imi)e + 516 5!Sub;ect to: 9 516 177 5! 977

4 516 /7 5! +77

177 516 177 5! 1777

177 516 +77 5! 477

(77 516 +77 5!1!77

51% 5!7

ANSWER: con"ert the present formulation to its dual representation of onl$ to constraints

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Linear Programming Simplex

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