Krajewski TIF Supplement E

8/11/2019 Krajewski TIF Supplement E

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Supplement E Linear Programming

Supplement

E Linear Programming

TRUE/FALSE

1. Linear programming is useful for allocating scarce resources among competing demands.

Answer: True

Reference: Introduction

Difficulty: Easy

Keywords: linear, programming, product, mix

2. A constraint is a limitation that restricts the permissible choices.

Answer: True

Reference: Basic oncepts

Difficulty: !oderate

Keywords: constraint, limit

". #ecision $ariables are represented in both the ob%ecti$e function and the constraints &hile

formulating a linear program.

Answer: True


Difficulty: !oderateKeywords: constraint, decision, $ariable, ob%ecti$e

'. A parameter is a region that represents all permissible combinations of the decision $ariables in a

linear programming model.

Answer: (alse



Keywords: parameter, decision, $ariable, feasible, region

). In linear programming, each parameter is assumed to be *no&n &ith certainty.

Answer: True



Keywords: certainty, assumption

196




+. The ob%ecti$e function2

Maximize Z = 3x + 4y is appropriate.

Answer: (alse



Keywords: linearity, assumption, proportionality

. -ne assumption of linear programming is that a decision ma*er cannot use negati$e uantities of the

decision $ariables.

Answer: True



Keywords: nonnegati$ity, decision, $ariable

/. -nly corner points should be considered for the optimal solution to a linear programming problem.

Answer: True

Reference: 0raphic Analysis

Difficulty: !oderateKeywords: corner, point, optimal

. The graphical method is a practical method for sol$ing product mix problems of any sie, pro$ided

the decision ma*er has sufficient uantities of graph paper.

Answer: (alse



Keywords: graphical, method

13. A binding constraint is the amount by &hich the left4hand side falls short of the right4hand side.

Answer: (alse

Reference: 0raphic AnalysisDifficulty: !oderate

Keywords: binding, constraint

11. A binding constraint has slac* but does not ha$e surplus.

Answer: (alse



Keywords: binding, slac*, surplus

12. The simplex method is an interacti$e algebraic procedure for sol$ing linear programming problems.

Answer: True

Reference: omputer 5olutionsDifficulty: !oderate

Keywords: simplex, method

197




MULTIPLE CHICE

1". A manager is interested in using linear programming to analye production for the ensuing &ee*. 5he*no&s that it &ill ta*e exactly 1.) hours to run a batch of product A and that this batch &ill consumet&o tons of sugar. This is an example of the linear programming assumption of6a. linearity. b. certainty.c. continuous $ariables.d. &hole numbers.

Answer: bReference: Basic onceptsDifficulty: !oderateKeywords: certainty, assumption

1'. 7hich of the follo&ing statements regarding linear programming is 8-T true9a. A parameter is also *no&n as a decision $ariable. b. Linearity assumes proportionality and additi$ity.c. The product4mix problem is a one4period type of aggregate planning problem.d. -ne reasonable seuence for formulating a model is defining the decision $ariables, &riting out the

ob%ecti$e function, and &riting out the constraints.Answer: aReference: Basic onceptsDifficulty: !oderateKeywords: parameter, decision, $ariable

1). 7hich of the follo&ing statements regarding linear programming is 8-T true9a. A linear programming problem can ha$e more than one optimal solution. b. !ost real4&orld linear programming problems are sol$ed on a computer.c. If a binding constraint &ere relaxed, the optimal solution &ouldn:t change.

d. A surplus $ariable is added to a ; constraint to con$ert it to an euality.Answer: cReference: Basic onceptsDifficulty: !oderateKeywords: solution, surplus, $ariable

1+. (or the line that has the euation ' X 1 < / X 2 = //, an axis intercept is6a. >3, 22?. b. >+, 3?.c. >+, 22?.d. >3, 11?.

Answer: d

Reference: 0raphic AnalysisDifficulty: !oderateKeywords: axis, intercept

19!




1. onsider a corner point to a linear programming problem, &hich lies at the intersection of thefollo&ing t&o constraints6

+ X 1 < 1) X 2 @ "32 X 1 < X 2 @ )3

7hich of the follo&ing statements about the corner point is true9

a. X 1 @ 21 b. X 1 ; 2)c. X 1 @ 13d. X 1 ; 1

Answer: aReference: 0raphic AnalysisDifficulty: !oderateKeywords: corner, point

1/. A manager is interested in deciding production uantities for products A, B, and . e has anin$entory of 23 tons each of ra& materials 1, 2, ", and ' that are used in the production of products A,B, and . e can further assume that he can sell all of &hat he ma*es. 7hich of the follo&ing

statements is correct9a. The manager has four decision $ariables. b. The manager has three constraints.c. The manager has three decision $ariables.d. The manager can sol$e this problem graphically.

Answer: cReference: 0raphic AnalysisDifficulty: !oderateKeywords: decision, $ariable

1. A site manager has three day laborers a$ailable for eight hours each and a burning desire to maximiehis return on their &ages. The site manager uses linear programming to assign them to t&o tas*s and

notes that he has enough &or* to occupy 21 labor hours. The linear program that the site manager hasconstructed has6a. slac*. b. surplus.c. a positi$e shado& price for labor.d. no feasible solution.

Answer: dReference: 0raphic AnalysisDifficulty: !oderateKeywords: shado&, price

199




23. 5uppose that the optimal $alues of the decision $ariables to a t&o4$ariable linear programming problem remain the same as long as the slope of the ob%ecti$e function lies bet&een the slopes of thefollo&ing t&o constraints6

2 X 1 < " X 2 @ 2+2 X 1 < 2 X 2 @ 23

The current ob%ecti$e function is6/ X 1 < X 2 = Z

7hich of the follo&ing statements about the range of optimality on c1 is TCE9a. 3 @ c1 @ 2 b. 2 @ c1 @ +c. + @ c1 @ d. @ c1 @ 12

Answer: cReference: 5ensiti$ity AnalysisDifficulty: ardKeywords: range, optimality

21. Dou are faced &ith a linear programming ob%ecti$e function of6!ax = F23G < F"3D

and constraints of6"G < 'D = 2' >onstraint A?)G H D = 1/ >onstraint B?

Dou disco$er that the shado& price for onstraint A is .) and the shado& price for onstraint B is 3.7hich of these statements is TCE9a. Dou can change uantities of G and D at no cost for onstraint B. b. (or e$ery additional unit of the ob%ecti$e function you create, you lose 3 units of B.c. (or e$ery additional unit of the ob%ecti$e function you create, the price of A rises by F.)3.d. The most you &ould &ant to pay for an additional unit of A &ould be F.)3.

Answer: d

Reference: 5ensiti$ity AnalysisDifficulty: ardKeywords: shado&, price

22. 7hile glancing o$er the sensiti$ity report, you note that the stitching labor has a shado& price of F13and a lo&er limit of 2' hours &ith an upper limit of "+ hours. If your original right hand $alue forstitching labor &as "3 hours, you *no& that6a. the next &or*er that offers to &or* an extra / hours should recei$e at least F/3. b. you can send someone home + hours early and still pay them the F+3 they &ould ha$e earned &hile

on the cloc*.c. you &ould be &illing pay up to F+3 for someone to &or* another + hours.d. you &ould lose F/3 if one of your &or*ers missed an entire / hour shift.

Answer: cReference: 5ensiti$ity AnalysisDifficulty: !oderateKeywords: shado&, price

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FILL I$ THE %LA$&

2". is useful for allocating scarce resources among competing demands.

Answer: Linear programming

Reference: IntroductionDifficulty: Easy

Keywords: linear, programming

2'. The is an expression in linear programming models that states mathematically &hat is

being maximied or minimied.

Answer: ob%ecti$e function



Keywords: ob%ecti$e, function

2). represent choices the decision ma*er can control.

Answer: #ecision $ariablesReference: Basic oncepts


Keywords: decision, $ariables

2+. are the limitations that restrict the permissible choices for the decision $ariables.

Answer: onstraints



Keywords: constraint

2. The represents all permissible combinations of the decision $ariables in a linear

programming model.Answer: feasible region



Keywords: feasible, region

2/. A>n? is a $alue that the decision ma*er cannot control and that does not change &hen

the solution is implemented.

Answer: parameter



Keywords: parameter, $alue

2. Each coefficient or gi$en constant is *no&n by the decision ma*er &ith .

Answer: certainty


Difficulty: Easy

Keywords: parameter, certainty, assumption

"#1




"3. If merely rounding up or rounding do&n a result for a decision $ariable is not sufficient &hen they

must be expressed in &hole units, then a decision ma*er might instead use to analye

the situation.

Answer: integer programming


Difficulty: !oderateKeywords: integer, programming

"1. is an assumption that the decision $ariables must be either positi$e or ero.

Answer: 8onnegati$ity


Difficulty: Easy

Keywords: nonnegati$ity, assumption

"2. The assumption of allo&s a decision ma*er to combine the profit from one product

&ith the profit from another to realie the total profit from a feasible solution.

Answer: additi$ityReference: Basic oncepts

Difficulty: Easy

Keywords: additi$ity, assumption

"". The problem is a one4period type of aggregate planning problem, the solution of

&hich yields optimal output uantities of a group of products or ser$ices, sub%ect to resource capacity

and mar*et demand conditions.

Answer: product4mix



Keywords: product4mix, product, mix

"'. In linear programming, a is a point that lies at the intersection of t&o >or possibly

more? constraint lines on the boundary of the feasible region.

Answer: corner point



Keywords: corner, point, solution

"). A>n? forms the optimal corner and limits the ability to impro$e the ob%ecti$e function.

Answer: binding constraint



Keywords: binding, constraint, corner

"+. is the amount by &hich the left4hand side falls short of the right4hand side in a linear

programming model.

Answer: 5lac*



Keywords: slac*, left4hand, side, right4hand

"#"




". is the amount by &hich the left4hand side exceeds the right4hand side in a linear

programming model.

Answer: 5urplus



Keywords: surplus, left4hand, side, right4hand

"/. A modeler is limited to t&o or fe&er decision $ariables &hen using the .

Answer: graphical method


Difficulty: Easy

Keywords: decision, $ariables, graphical, method

". The is the upper and lo&er limit o$er &hich the optimal $alues of the decision$ariables remain unchanged.

Answer: range of optimalityReference: 5ensiti$ity AnalysisDifficulty: !oderateKeywords: range, optimality

'3. (or an = constraint, only points are feasible solutions.Answer: on the lineReference: 0raphic AnalysisDifficulty: EasyKeywords: eual, than, line, feasible, region

'1. A>n? is the marginal impro$ement in the ob%ecti$e function $alue caused by relaxing aconstraint by one unit.

Answer: shado& priceReference: 5ensiti$ity AnalysisDifficulty: !oderateKeywords: shado&, price, sensiti$ity, relax, constraint

'2. The inter$al o$er &hich the right4hand4side parameter can $ary &hile its shado& price remains $alidis the .

Answer: range of feasibilityReference: 5ensiti$ity AnalysisDifficulty: !oderateKeywords: range, feasibility

'". occurs in a linear programming problem &hen the number of nonero $ariables in the

optimal solution is fe&er than the number of constraints.Answer: #egeneracyReference: omputer 5olutionDifficulty: !oderateKeywords: degeneracy

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SHRT A$S(ER

''. 7hat are the assumptions of linear programming9 ro$ide examples of each.Answer: The assumptions are certainty, linearity, and nonnegati$ity. The assumption of certainty

is that a fact is *no&n &ithout doubt, such as an ob%ecti$e function coefficient, or the parametersin the right4 and left4hand sides of the constraints. The assumption of linearity implies proportionality and additi$ity, that is, that there are no cross products or suared or higher po&ersof the decision $ariables. The assumption of nonnegati$ity is that decision $ariables must either be positi$e or ero. Examples &ill $ary.Reference: 0raphic AnalysisDifficulty: !oderateKeywords: assumption, linearity, certainty, nonnegati$ity

'). 7hat is the meaning of a slac* or surplus $ariable9Answer: The amount by &hich the left4hand side falls short of the right4hand side is the slac*$ariable. The amount by &hich the left4hand side exceeds the right4hand side is the surplus$ariable.Reference: 0raphic AnalysisDifficulty: !oderateKeywords: slac*, surplus

'+. Briefly describe the meaning of a shado& price. ro$ide an example of ho& a manager could useinformation about shado& prices to impro$e operations9

Answer: The shado& price is the marginal impro$ement in Z caused by relaxing a constraint byone unit. Examples &ill $ary.Reference: 5ensiti$ity AnalysisDifficulty: !oderateKeywords: shado&, price

'. ro$ide three examples of operations management decision problems for &hich linear programmingcan be useful, and &hy.

Answer: Ans&ers and %ustifications &ill $ary. ossible ans&ers include aggregate planning,distribution, in$entory, location, process management, and scheduling.Reference: ApplicationsDifficulty: !oderateKeywords: linear, programming, application

'/. 7hat are some potential abuses or misuses of linear programming >beyond $iolation of basicassumptions?9

Answer: Ans&ers &ill $ary, but may include a discussion of the inability of modeling techniuesto capture all of the rele$ant factors that may be as important as &hat can be uantified in an Lformulation. (actors such as aesthetics, ethics, ci$ility, character, etc., may be difficult to capturein an L. 5la$ish adhesion to the output from a linear programming formulation robs a managerof the freedom to in%ect reality or personality into a model. The rush to use a tool &ithoutunderstanding fully the &or*ings of it may render the output meaningless.Reference: ApplicationsDifficulty: Basic onceptsKeywords: linear, programming, application

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PR%LEMS

'. Cse the graphical techniue to find the optimal solution for this ob%ecti$e function and associated

constraints.!aximie6 J=/A < )B5ub%ect To6

onstraint 1 'A < )B @ /3onstraint 2 A < 'B @ 123A, B ; 3

a. 0raph the problem fully in the follo&ing space. Label the axes carefully, plot the constraints, shadethe feasibility region, identify all candidate corner points, and indicate &hich one yields theoptimal ans&er.

A

B

"#*




Answer:

>3,3? / 3 ) 3 3 Z = × + × =

>3,1+? / 3 ) 1+ 3 Z = × + × =

>3,1.1'? / 1.1' ) 3 1".1' Z = × + × =

>1'.", '.21? / 1'." ) '.21 1"/./ Z = × + × = optimal


Difficulty:!oderate

Keywords: graphic, analysis

"#6

Intersection of onstraint 1 K 2

> ' 123? )

>' ) /3? '

1 2/3

1'.", '.21

A B

A B

A

A B

+ = ×

− + = ×

=

= ∴ =




)3. A producer has three products, A, B, and , &hich are composed from many of the same ra&materials and subassemblies by the same s*illed &or*force. Each unit of product A uses 1) units ofra& material G, a single purge system subassembly, a case, a po&er cord, three labor hours in theassembly department, and one labor hour in the finishing department. Each unit of product B uses 13

units of ra& material G, fi$e units of ra& material D, t&o purge system subassemblies, a case, a po&er cord, fi$e labor hours in the assembly department, and 3 minutes in the finishing department. Eachunit of product uses fi$e units of ra& material G, 2) units of ra& material D, t&o purge systemsubassemblies, a case, a po&er cord, se$en labor hours in the assembly department, and three laborhours in the finishing department. Labor bet&een the assembly and finishing departments is nottransferable, but &or*ers &ithin each department &or* on any of the three products. There are threefull4time >'3 hours&ee*? &or*ers in the assembly department and one full4time and one half4time>23 hours&ee*? &or*er in the finishing department. At the start of this &ee*, the company has "33units of ra& material G, '33 units of ra& material D, +3 purge system subassemblies, '3 cases, and )3 po&er cords in in$entory. 8o additional deli$eries of ra& materials are expected this &ee*. There is aF3 profit on product A, a F123 profit on product B, and a F1)3 profit on product . The operationsmanager doesn:t ha$e any firm orders, but &ould li*e to ma*e at least fi$e of each product so he can

ha$e the products on the shelf in case a customer &anders in off the street.

(ormulate the ob%ecti$e function and all constraints, and clearly identify each constraint by the nameof the resource or condition it represents.

Answer:

-b%ecti$e (unction6 F3 F123 F1)3 Max P A B C = + +

a& !aterial G6 1) 13 ) "33 A B C + + ≤

a& !aterial D6 3 ) 2) '33 A B C + + ≤

urge 5ystem 5ubassembly6 1 2 2 +3 A B C + + ≤

ase6 1 1 1 '3 A B C + + ≤

ord6 1 1 1 )3 A B C + + ≤

Assembly #epartment Labor6 " ) 123 A B C + + ≤

(inish #epartment Labor6 1 1.) " +3 A B C + + ≤

!inimum roduction for A6 1 3 3 ) A B C + + ≥

!inimum roduction for B6 3 1 3 ) A B C + + ≥

!inimum roduction for 6 3 3 1 ) A B C + + ≥

Reference: !ultiple sectionsDifficulty: EasyKeywords: linear, programming, ob%ecti$e, function, constraint

"#7




)1. A $ery confused manager is reading a t&o4page report gi$en to him by his student intern. M5he toldme that she had my problem sol$ed, ga$e me this, and then said she &as off to her productionmanagement course,N he &hined. MI ga$e her my best estimates of my on4hand in$entories andreuirements to produce, but &hat if my numbers are slightly off9 I recognie the names of our fourmodels 7, G, D, and J, but that:s about it. an you figure out &hat I:m supposed to do and &hy9N

Dou ta*e the report from his hands and note that it is the ans&er report and the sensiti$ity report fromExcel:s sol$er routine.

Explain each of the highlighted cells in layman:s terms and tell the manager &hat they mean inrelation to his problem.

Microsoft Excel 10.0 Answer Report

Worksheet: Supplement D

Report Create: 1!"#!"00$ 11:"#:%0 AM

&ar'et Cell (Max)

Cell Name Original Value Final Value

*A+*1" ,00 -----.----,

Austa/le Cells

Cell Name Original Value Final Value

**1" W 0 111.1111111

**1" 0 0

*2*1" 1.% 0

*AA*1" 2 0 0

Constraints

Cell Name Cell Value Formula Status Slack

*A+*1% 10000*A+*1%34*AC*1% +inin' 0

*A+*1$ 1111.111111*A+*1$34*AC*1$ 5ot +inin' 6---.-----,

*A+*1# %000*A+*1#34*AC*1# 5ot +inin' "%000

"#!




Microsoft Excel 10.0 Sensiti7it8 Report

Worksheet: Supplement D

Report Create: 1!"#!"00$ 11:"#:%0 AM

Austa/le Cells

Final Reduced Objective Alloable Alloable

Cell Name Value Cost Coe!!icient "ncrease #ecrease

**1" W 111.1111111 0 -00 1E960 -0

**1" 0 ,66.6666666 $00 ,66.6666666 1E960

*2*1" 0 ##.#######; #00 ##.#######; 1E960

*AA*1" 2 0 10%%.%%%%%# %00 10%%.%%%%%# 1E960

Constraints

Final S$ado Constraint Alloable Alloable

Cell Name Value Price R%&% Side "ncrease #ecrease

*A+*1% 10000 -.--------, 10000 6%000 10000

*A+*1$ 1111.111111 0 %000 1E960 6---.-----,

*A+*1# %000 0 60000 1E960 "%000

Answer:

Ans&er eportTarget ell !ax6 The target cell should be maximied, so the manager must ha$e pro$ided the

intern &ith profit information.(inal Oalue6 The final $alue is the greatest amount possible for the situation. If &e are

&or*ing &ith profit figures, this is the best return possible gi$en &hat &e estimate is on hand andho& it is to be produced. This may change if our in$entory or recipes are slightly off. The highest profit identified is F//,///./

Ad%ustable ells6 The ad%ustable cells sho& that &e considered any positi$e uantity of models

7HJ as possible outputs for the &ee*. 8ame6 The names are those of the models &e produce.(inal Oalue6 These are the exact amounts of each of our four models to produce to earn the

final $alue. In this case &e &ould ma*e 111.1 units of model 7 and none of the other fourmodels.

5tatus6 This sho&s &hat is limiting our ability to produce the models. A binding constraintdirectly limits our output although a nonbinding constraint means that factor does not limit us. Inthis case, the second and third constraints are nonbinding, so producing 111.1 units of model 7lea$es us &ith lefto$ers of &hate$er scarce resource they represent. The first constraint is binding,so &e are using up e$ery bit of that resource.

5lac*6 5lac* sho&s us ho& much of each resource &e ha$e left. -ur first constraint is binding, so &e ha$e none left o$er and therefore ha$e 3 slac*. -ur second and third constraintsare not binding, so &e ha$e plenty >",/// and 2),333 units respecti$ely? of these scarce resourcesleft o$er.

5ensiti$ity eportAd%ustable ells

educed ost6 This is the change in the optimum objective per unit change in the upper or

lower bounds of the variable. The objective function will increase by 0.-66, and so on, per unit

increase.

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Allo&able Increase6 These t&o >Allo&able Increase and Allo&able #ecrease? pro$ide arange for our current ans&er and the recipe &e used to arri$e at it. (or model 7, &e ha$eassumed that each unit gi$es us F/33 profit. If our estimate &as too high, and the return &as up toF/3 less per unit, &e &ould still arri$e at the same ans&er. If it &ere more than F/3 too high, ourans&er &ould change. The same holds true for the models &e are not ma*ing. If model D mademore than F+++.++ profit per unit, then our final product mix &ould change.

Allo&able #ecrease6 5ee analysis for Allo&able Increase.onstraints

5hado& rice6 This is the marginal return for ha$ing one more unit of each resource. ere&e ha$e a shado& price of F/.//, so if &e had one more unit of resource in the first constraint, &ecould ma*e an additional F/.//. This gi$es us an idea of the maximum &e &ould be &illing to pay for more of that resource.

Allo&able Increase6 These &or* the same as the allo&able increases and decreases for thead%ustable cells except they focus on the shado& prices. They indicate ho& far the 5 of theconstraint can change before the shado& price &ill change.

Allo&able #ecrease6 5ee discussion immediately preceding.Reference: 5ensiti$ity AnalysisDifficulty: !oderate

Keywords: sensiti$ity, analysis

)2. The J Pe&elry ompany produces t&o products6 >1? engagement rings and >2? %e&eled &atches. The production process for each is similar in that both reuire a certain number of hours of diamond &or*and a certain number of labor hours in the gold department. Each ring ta*es four hours of diamond&or* and t&o hours in the gold shop. Each &atch reuires three hours in diamonds and one hour inthe gold department. There are 2'3 hours of diamond labor a$ailable and 133 hours of golddepartment time a$ailable for the next month. Each engagement ring sold yields a profit of FQ each&atch produced may be sold for a F13 profit.a. 0i$e a complete formulation of this problem, including a careful definition of your decision

$ariables. Let the first decision $ariable, > X 1?, deal &ith rings, the second decision $ariable, > X 2?,&ith &atches, the first constraint &ith diamonds, and the second constraint &ith gold.

b. 0raph the problem fully in the follo&ing space. Label the axes carefully, plot the constraints, shadethe feasibility region, plot at least one isoprofit line that re$eals the optimal solution, circle thecorner points and highlight the optimal optimal corner point so found, and sol$e for italgebraically. >5ho& all your &or* to get credit.?

"1#

X 1

X 2




Answer:

a. !ax6 X 1 < 13 X 2s.t.' X 1 < " X 2 @ 2'3 hours of diamond &or*

2 X 1 < X 2 @ 133 hours of gold &or* X 1, X 2 ; 3

b.

Reference: !ultiple sectionsDifficulty: !oderateKeywords: ob%ecti$e, function, constraint, graphical

"11

X 1




)". 8D8EG must schedule round4the4cloc* co$erage for its telephone operators. To *eep the number ofdifferent shifts do&n to a manageable le$el, it has only four different shifts. -perators &or* eight4hour shifts and can begin &or* at either midnight, / a.m., noon, or ' p.m. -perators are neededaccording to the follo&ing demand pattern, gi$en in four4hour time bloc*s.

Time eriod -perators 8eededmidnight to ' a.m. '' a.m. to / a.m. +/ a.m. to noon 3 8oon to ' p.m. /)' p.m. to / p.m. ))/ p.m. to midnight 23

(ormulate this scheduling decision as a linear programming problem, defining fully your decision$ariables and then gi$ing the ob%ecti$e function and constraints.Answer:

Let X 1 = the number of telephone operators starting their shift at midnight.

X 2 = the number of telephone operators starting their shift at / a.m. X " = the number of telephone operators starting their shift at noon. X ' = the number of telephone operators starting their shift at ' p.m.

!in6 X 1 < X 2 < X " < X 'sub%ect to X 1 ; ' !idnight to ' a.m.

X 1 ; + ' a.m. to / a.m. X 2 ; 3 / a.m. to noon X 2 < X " ; /) noon to ' p.m.

X " < X ' ; )) ' p.m. to / p.m. X ' ; 23 / p.m. to midnight

X 1, X 2, X ", X ' ; 3Reference: Basic oncepts

Difficulty: !oderateKeywords: ob%ecti$e function, constraint

"1"




)'. The eally Big 5hoe ompany is a manufacturer of bas*etball shoes and football shoes. Ed 5ulli$an,the manager of mar*eting, must decide the best &ay to spend ad$ertising resources. Each footballteam sponsored reuires 123 pairs of shoes. Each bas*etball team reuires "2 pairs of shoes. (ootballcoaches recei$e F"33,333 for shoe sponsorship and bas*etball coaches recei$e F1,333,333. EdRs promotional budget is F"3,333,333. The eally Big 5hoe ompany has a $ery limited supply >'liters or ',333cc? of flubber, a rare and costly ra& material used only in promotional athletic shoes.

Each pair of bas*etball shoes reuires "cc of flubber, and each pair of football shoes reuires 1cc offlubber. Ed desires to sponsor as many bas*etball and football teams as resources allo&. o&e$er, hehas already committed to sponsoring 1 football teams and &ants to *eep his promises.a. 0i$e a linear programming formulation for Ed. !a*e the $ariable definitions and constraints line

up &ith the computer output appended to this exam. b. 5ol$e the problem graphically, sho&ing constraints, feasible region, and isoprofit lines. ircle the

optimal solution, ma*ing sure that the isoprofit lines dra&n ma*e clear &hy you chose this point.>5ho& all your calculations for plotting the constraints and isoprofit line on the left to get credit.?

c. 5ol$e algebraically for the corner point on the feasible region.d. art of EdRs computer output is sho&n follo&ing. 0i$e a full explanation of the meaning of the

three numbers listed at the end. Based on your graphical and algebraic analysis, explain &hythese numbers ma*e sense. >int6 e formulated the budget constraint in terms of F333.?

5ee the computer printout that follo&s.

"1'

X1

X2




Solver'Linear Programming

Solution

<aria/le <aria/le =ri'inal Coefficient>a/el <alue Coefficient Sensiti7it8

<ar1 1,.0000 1.0000 0<ar" 1;.,1#; 1.0000 0

Constraint =ri'inal Slack or Shaow>a/el R?< Surplus @rice

Const1 1, 0Const" 60000 #6-6 0Const6 $000 0 0.010$

=/ecti7e unction <alue: 6#.,1#####;

Sensitivit( Anal(sis and Ranges

=/ecti7e unction Coefficients

<aria/le >ower =ri'inal Bpper >a/el >imit Coefficient >imit

<ar1 5o >imit 1 1."%<ar" 0.- 1 5o >imit

Ri'ht?anSie <alues

Constraint >ower =ri'inal Bpper >a/el >imit <alue >imit

Const1 1"."-0; 1, 66.66666666Const" "6#1#.#; 60000 5o >imitConst6 ""-0 $000 $#1".-

(irst 8umber6 The shado& price of 3.313' for the Sonst"S constraint.5econd 8umber6 The slac* or surplus of +"/" for the Sonst1S constraint.Third 8umber6 The lo&er limit of 12.2/3for the Sonst1S constraint.

Answer:

a. Let X 1 = the number of football teams sponsored X 2 = the number of bas*etball teams sponsored

!ax X 1 < X 2s.t. X 1 ; 1 ommitments

"33 X 1 < 1333G2 @ "3333 Budget123 X 1 < +G2 @ '333 (lubber

"1)




G1, X 2 ; 3

"1*




b.

"16

1 16 1 1Commitments X X > → =

1 2

1 2

2 1

6 "33 1333 "3333

"33333, "3

1333

"33333Q 133

"33

get X X

X

X

+ <

= = =

= = =

1 2

1 2

2 1

(lubber 6123 + '333

'333 3Q '1.+

+

'333 3Q ""."

123

X X

if X X

if X X

+ <

= = =

= = =




c.

Therefore, X 1 = 1

d.

(irst 8umber6 The shado& price of 3.313' for the Sonst"S constraint.5econd 8umber6 The slac* or surplus of +"33 for the Sonst1S constraint.Third 8umber6 The lo&er limit of 12."+/' for the Sonst1S constraint.

The first number is the amount >.313'? by &hich the ob%ecti$e function &ill impro$e &ith a one4unitdecrease in the right4hand4side $alue. The second number means that +,"33,333 remains in the promisedcommitment. The third $alue is the amount by &hich the constraint can change and still *eep the current$alues of the shado& price.

Reference: !ultiple sectionsDifficulty: !oderateKeywords: constraint, ob%ecti$e, function

)). A portfolio manager is trying to balance in$estments bet&een bonds, stoc*s and cash. The return onstoc*s is 12 percent, percent on bonds, and " percent on cash. The total portfolio is F1 billion, andhe or she must *eep 13 percent in cash in accordance &ith company policy. The fundRs prospectus

promises that stoc*s cannot exceed ) percent of the portfolio, and the ratio of stoc*s to bonds musteual t&o. (ormulate this in$estment decision as a linear programming problem, defining fully yourdecision $ariables and then gi$ing the ob%ecti$e function and constraints.

Answer:

Let X 1 = the amount in$ested in bonds X 2 = the amount in$ested in stoc*s X " = the amount in$ested in cash

!ax6 z = .3 X 1 < .12 X 2 <.3" X "s.t. X 1 < X 2 < G" ≤ 1,333,333,333 ortfolio $alue

G1 = 133,333,333 13 minimum stoc*

X 2 ≤ )3,333,333 ) maximum cash

2 X 1 H X 2 = 3 261 ratio stoc*s to bonds X 1, X 2, X " ; 3

Reference: Basic onceptsDifficulty: !oderateKeywords: ob%ecti$e, function, constraint

"17

1 2

1

2

2

corner point

123 + '333

> 1? 123

+ 123

1.1+

X X

X

X

X

+ =

= × −

=

=




)+. A small oil company has a refining budget of F233,333 and &ould li*e to determine the optimal production plan for profitability. The follo&ing table lists the costs associated &ith its three products.

!ar*eting has a budget of F)3,333, and the company has )3,333 gallons of crude oil a$ailable.Each gallon of gasoline contributes 1' cents of profits, heating oil pro$ides 13 cents, and plasticresin "3 cents per unit. The refining process results in a ratio of t&o units of heating oil for eachunit of gasoline produced. This problem has been modeled as a linear programming problem andsol$ed on the computer. The output follo&s6

Solution

<aria/le <aria/le =ri'inal Coefficient

>a/el <alue Coefficient Sensiti7it8

<ar1 0.0000 0.1$00 0<ar" 1%0000.0000 0.1000 0<ar6 0.0000 0.6000 0

Constraint =ri'inal Slack or Shaow>a/el R?< Surplus @rice

Const1 "00000 1-%000 0Const" %0000 $"%00 0Const6 ;%0000 0 0.0"00

=/ecti7e unction <alue: 1%000

Sensitivit( Anal(sis and Ranges

=/ecti7e unction Coefficients

<aria/le >ower =ri'inal Bpper >a/el >imit Coefficient >imit

<ar1 5o >imit 0.1$ 0."<ar" 0.0;% 0.1 5o >imit<ar6 5o >imit 0.6 0.$

Ri'ht?anSie <alues

Constraint >ower =ri'inal Bpper >a/el >imit <alue >imit

Const1 1%000 "00000 5o >imitConst" ;%00 %0000 5o >imitConst6 0 ;%0000 %000000

"1!




a. 0i$e a linear programming formulation for this problem. !a*e the $ariable definitions andconstraints line up &ith the computer output.

b. 7hat product mix maximies the profit for the company using its limited resources9c. o& much gasoline is produced if profits are maximied9d. 0i$e a full explanation of the meaning of the three numbers listed follo&ing.

(irst 8umber6 5lac* or surplus of '2)33 for constraint 2.

5econd 8umber6 5hado& price of 3 for constraint 1.Third 8umber6 An upper limit of Sno limitS for the right4hand4side $alue constraint 1.

Answer:

a. Let X 1 = gallons of gasoline refined X 2 = gallons of heating oil refined X " = gallons of plastic resin refined

!ax6 .1' X 1 < .13 X 2 < ."3 X "s.t. .'3 X 1 < .13 X 2 < .+3 X " @ 233,333 efining budget

.13 X 1 < .3) X 2 < .3 X " @ )3,333 !ar*eting budget

13 X 1 < ) X 2 < 23 X " @ )3,333 rude oil a$ailable

X 2 H 2 X 1 = 3 atio

X 1, X 2, X " ; 3

b. X 1 = 3 gallons, X 2 = 1)3,333 gallons, and X " = 3 gallonsc. 8o gasoline is produced if profits are maximied.

d. F'2,)33 remains in the mar*eting budget. A ero implies that increasing the refining budget &ill not

impro$e the $alue of the ob%ecti$e function. A no4limit implies that the right4hand side can be increased

by any amount and the shado& price &ill remain the same.

Reference: !ultiple sectionsDifficulty: !oderateKeywords: ob%ecti$e, function, constraint

). A snac* food producer runs four different plants that supply product to four different regionaldistribution centers. The di$ision operations manager is focused on one product, so he creates a tablesho&ing each plant:s monthly capacity and each distribution center:s monthly demand >both amounts

in cases? for the product. The di$ision manager supplements this table &ith the cost data to ship onecase from each plant to each distribution center. (ormulate an ob%ecti$e function and constraints that&ill sol$e this problem using linear programming.

enter 1 enter 2 enter " enter ' !onthlyapacity

lant A F2 F F) F' /333

lant B F F' F F+ 12333

lant F F+ F' F" )33

lant # F' F/ F" F) )333

!onthly#emand

333 /)33 /333 333

Answer:

This is a cost minimiation problem &ith 1+ decision $ariables, one for each combination of plantand centerQ there are / constraints, one for each plant:s capacity and one for each center:sdemand.

1 2 " ' 1 2 " ' 1

2 " ' 1 2 " '

F2 F F) F' F F' F F+ F

F+ F' F" F' F/ F" F)

A A A A B B B B C

C C C D D D D

O!ecti"e Min Z x x x x x x x x x

x x x x x x x

= + + + + + + + +

+ + + + + + +

"19




1 2 " '

1 2 " '

1 2 " '

1 2 " '

1 1 1 1

2 2 2 2

" "

6 /333

6 12333

6 )33

6 )333 16 333

2 6 /)33

" 6

A A A A

B B B B

C C C C

D D D D

A B C D

A B C D

A B C

#u!ect to

P$ant A x x x x

P$ant B x x x x

P$ant C x x x x

P$ant D x x x xCente% x x x x

Cente% x x x x

Cente% x x x

+ + + =

+ + + =

+ + + =

+ + + =

+ + + =

+ + + =

+ +" "

' ' ' '

/333

' 6 333

D

A B C D

x

Cente% x x x x

+ =

+ + + =

(or those of you *eeping score at home, the optimal solution is6

Center 1 Center 2 Center 3 Center 4 Monthly

Capacity

Plant A 8,000 8,000

Plant B 8,500 3,500 12,000

Plant C 500 7,000 7,500

Plant D 1,000 4,000 5,000

Monthly

Demand

9,000 8,500 8,000 7,000 $113,500

Reference: Basic onceptsDifficulty: !oderateKeywords: ob%ecti$e, function, constraint

""#

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Krajewski TIF Supplement E