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Chapter 3—Modeling and Solving LP Problems in a Spreadsheet

MULTIPLE CHOICE

1. An LP problem with a feasible region will havea. an optimal solution at some interior point. b. an optimal solution at some extreme point.

c. an optimal solution only at the origin.d. an optimal solution at two interior points.

ANS: P!S: 1

". #icrosoft® $xcel% &uattro Pro and Lotus 1'"'( contain built'in optimi)ers called

a. what'if engines. b. calculators.c. solvers.d. ris* analy)ers.

ANS: + P!S: 1

(. ,hich type of spreadsheet cell represents the ob-ective function in an LP modela. /b-ective cell b. +hanging variable cellc. +onstraint celld. +onstant cell

ANS: A P!S: 1

0. ,hich type of spreadsheet cell represents the decision variables in an LP modela. !arget or set cell b. ariable cellc. +onstraint celld. +onstant cell

ANS: P!S: 1

2. ,hich type of spreadsheet cell represents the left hand sides 3L4S5 formulas in an LP modela. !arget or set cell b. +hanging variable cellc. +onstraint celld. +onstant cell

ANS: + P!S: 1

6. !he constraints 71 ≥ 8 and 7" ≥ 8 are referred to as

a. positivity constraints. b. optimality conditions.c. left hand sides.d. nonnegativity conditions.

ANS: 9 P!S: 1

. ;n the <is* Solver Platform 3<SP5 dialog box simple upper and lower bounds for decision variablesare specified by

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a. referring directly to the decision variable cells in the +onstraints'ound area. b. re=uiring the addition of the bounds above and below the variable cells.c. resolving the problem with the bounds added.d. incorporating the bounds in the ob-ective function.

ANS: A P!S: 1

>. !he built'in Solver in $xcel is found under which tab on the ribbona. !ools

b. ;nsertc. 9atad. ,indow

ANS: + P!S: 1

?. ,hich tab in the <is* Solver Platform 3<SP5 tas* pane is used to define an optimi)ation problema. @uess b. #odelc. +hanged. 9elete

ANS: P!S: 1

18. Spreadsheet modeling is an ac=uired s*ill becausea. there is generally only one correct way to build a model. b. the spreadsheet is free'form providing many modeling options.c. using <is* Solver Platform 3<SP5 re=uires lots of experience.d. spreadsheets are not very easy to use.

ANS: P!S: 1

11. #odels which are setup in an intuitively appealing% logical layout tend to be the mosta. <eliable b. #odifiable

c. Auditabled. /rgani)ed

ANS: + P!S: 1

1". !he $nglish'reading eye scansa. <ight to left b. ottom to topc. Left to ottomd. Left to right

ANS: 9 P!S: 1

1(. Numeric constants should bea. embedded in formulas. b. placed in individual cellsc. placed in separate wor*boo*s.d. entered manually every time a model is solved.

ANS: P!S: 1

10. !he Analy)e ,ithout Solving tool in <is* Solver Platform 3<SP5 is useful for

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a. verifying the e=uations in a spreadsheet model. b. toggling between absolute and relative cell referencing.c. executing the $xcel spreadsheet layout ,i)ard.d. naming cells and cell ranges for easier modifiability.

ANS: A P!S: 1

12. !he /b-ective alue of option in the <is* Solver Platform 3<SP5 tas* pane may be used toa. find a solution at a maximum value.

b. find a solution at a minimum value.c. find a solution for a specific ob-ective function value.d. returns the best feasible solution.

ANS: + P!S: 1

16. !he /b-ective Sense option in the <is* Solver Platform 3<SP5 tas* pane may be used toa. return a heuristic solution to the problem. b. tell the Solver what value it should see* for your optimi)ation ob-ective.c. determine the value of the ob-ective based on specified decision variable cells.d. always wor*s correctly.

ANS: P!S: 1

1. ,hat action is re=uired to ma*e <is* Solver Platform 3<SP5 solve a specified problema. !ype go in cell A1. b. +lic* the /ptimi)e button on the <SP <ibbon% or the green arrow Solve in the !as*

Pane.c. +lic* the +lose button in the <SP Parameters dialog box.d. +lic* the @uess button in the <SP Parameters dialog box.

ANS: P!S: 1

1>. ,hat does the $xcel BSC#P</9C+!3A1:A2%+6D+185 command doa. Sums each range and multiplies the sums.

b. Sum each pair of cells and multiples each sum.c. #ultiplies the contents of cells containing the BSC#35 command.d. #ultiplies each pair of cells and sums each product.

ANS: 9 P!S: 1

1?. ,hat command is used to add the contents of cells A1% A" and A(a. BA1EA"EA( b. BA993A1:A(5c. B!/!AL3A1:A(5d. BP</9C+!3A1:A(5

ANS: A P!S: 1

"8. ,hich command is e=uivalent to BSC#P</9C+!3A1:A(%1:(5a. BSC#3P</9C+!33A1:A(%1:(55 b. BP</9C+!3SC#33A1:A(%1:(55c. BP</9C+!3A1E1%A"E"%A(E(55d. BA1F1EA"F"EA(F(

ANS: 9 P!S: 1

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"1. Problems which have only integer solutions are calleda. discrete programming problems b. integer programming problemsc. discrete programming problemsd. infeasible programming problems

ANS: P!S: 1

"". ,hat is the significance of an absolute cell reference in $xcel

a. !he cell reference will not change if the formula containing the reference is copied toanother location

b. !he cell will always contain the absolute value of any number entered into itc. !he cell reference changes if the formula containing the reference is copied to another

locationd. ;t is the only formula used to refer to a cell on another spreadsheet

ANS: A P!S: 1

"(. 4ow many decision variables are there in a transportation problem which has 2 supply points and 0demand pointsa. 0 b. 2c. ?d. "8

ANS: 9 P!S: 1

"0. 4ow many constraints are there in a transportation problem which has 2 supply points and 0 demand points 3ignore the non'negativity constraints5a. 0 b. 2c. ?d. "8

ANS: + P!S: 1

"2. A heuristic solution isa. used by <is* Solver Platform 3<SP5 when the @uess button is used. b. guaranteed to produce an optimal solution.c. used by <is* Solver Platform 3<SP5 if Standard @<@ Nonlinear method is selected.d. a rule'of'thumb for ma*ing decisions.

ANS: 9 P!S: 1

"6. Scaling problemsa. can cause <is* Solver Platform 3<SP5 to consider a linear problem as nonlinear. b. can cause problems in accuracy of solutions returned.

c. are caused by small numbers and large numbers used in the same problem.d. all of these.

ANS: 9 P!S: 1

". ,hich of the following describes 9ata $nvelopment Analysis 39$A5.a. 9$A finds the most effective company among some set of companies. b. 9$A determines if a company is converting inputs to outputs as effectively as possible.c. 9$A determines how effective a company converts inputs to outputs compared to other

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companies.d. 9$A compares how effective a company converts inputs to outputs compared to a

benchmar* composite of all companies.

ANS: + P!S: 1

">. 9ata $nvelopment Analysis 39$A5 is an LP'based methodology in which weighted sums of inputs andoutputs are calculated anda. the constraints capture the maximum effectiveness of each unit.

b. the ob-ective is to maximi)e every units output.c. the constraints ensure the sum of the weighted outputs is one.d. the ob-ective for each unit is to maximi)e the weighted sum of its outputs.

ANS: 9 P!S: 1

"?. Csing 9ata $nvelopment Analysis 39$A5 for an inefficient unit% a more efficient composite unit can be found bya. Solving its 9$A problem and retrieving the weights from the answer report. b. Solving its 9$A problem and examining those units whose final value is non')ero.c. Solving its 9$A problem and using the resulting shadow prices as composite weights.d. Solving its 9$A problem and using the positive resulting shadow prices as composite

weights.

ANS: + P!S: 1

Exhibit 3!

!he following =uestions are based on this problem and accompanying $xcel windows.

Gones Hurniture +ompany produces beds and des*s for college students. !he production processre=uires carpentry and varnishing. $ach bed re=uires 6 hours of carpentry and 0 hour of varnishing.$ach des* re=uires 0 hours of carpentry and > hours of varnishing. !here are (6 hours of carpentrytime and 08 hours of varnishing time available. eds generate I(8 of profit and des*s generate I08 of profit. 9emand for des*s is limited% so at most > will be produced.

Let 71 B Number of eds to produce7" B Number of 9es*s to produce

!he LP model for the problem is

#A7: (8 71 E 08 7"

Sub-ect to: 6 71 E 0 7" ≤ (6 3carpentry5

0 71 E > 7" ≤ 08 3varnishing5

7" ≤ > 3demand for des*s5

71% 7" ≥ 8

A + 9 $

1 Gones Hurniture

"

( eds 9es*s

0 Number to ma*e: !otal Profit:

2 Cnit profit: (8 08

6

+onstraints: Csed Available

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> +arpentry 6 0 (6

? arnishing 0 > 08

18 9es* demand 1 >

(8. <efer to $xhibit (.1. ,hat formula should be entered in cell $2 in the accompanying $xcelspreadsheet to compute total profita. B0F2E+0F+2 b. BSC#P</9C+!3>:+>%II0:I+I05c. BSC#32:+25d. BSC#3$>:$185

ANS: A P!S: 1

(1. <efer to $xhibit (.1. ,hat formula should be entered in cell 9> in the accompanying $xcelspreadsheet to compute the amount of carpentry useda. B0F2E+0F+2 b. BSC#P</9C+!3>:+>%II0:I+I05c. BSC#32:+25d. BSC#3$>:$185

ANS: P!S: 1

(". <efer to $xhibit (.1. ,hich cells should be changing cells in this problema. 0:+0 b. $2c. 9>:918d. $>:$18

ANS: A P!S: 1

((. <efer to $xhibit (.1. ,hich cells should be the constraint cells in this problema. 0:+0 b. $2c. 9>:918

d. $>:$18

ANS: + P!S: 1

(0. <efer to $xhibit (.1. ,hich of the following statements represent the carpentry% varnishing and limiteddemand for des*s constraintsa. 0:+0 ≤ 2:+2

b. $2 ≤ 8

c. 9>:918 ≤ $>:$18

d. $>:$18 ≤ 9>:918

ANS: + P!S: 1

Exhibit 3"


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!he yte computer company produces two models of computers% Plain and Hancy. ;t wants to planhow many computers to produce next month to maximi)e profits. Producing these computers re=uireswiring% assembly and inspection time. $ach computer produces a certain level of profits but faces onlya limited demand. !here are also a limited number of wiring% assembly and inspection hours availablein each month. !he data for this problem is summari)ed in the following table.

+omputer #odel

Profit per #odel 3I5

#aximumdemand for

product,iring 4ours

<e=uired

Assembly4ours

<e=uired

;nspection4ours

<e=uired

Plain (8 >8 .0 .2 ."Hancy 08 ?8 .2 .0 .(

4ours Available 28 28 ""

Let 71 B Number of Plain computers to produce7" B Number of Hancy computers to produce

#A7: (8 71 E 08 7"

Sub-ect to: .0 71 E .2 7" ≤ 28 3wiring hours5

.2 71 E .0 7" ≤ 28 3assembly hours5

." 71 E ." 7" ≤ "" 3inspection hours5

71 ≤ >8 3Plain computers demand5

7" ≤ ?8 3Hancy computers demand5

71% 7" ≥ 8

A + 9 $

1 yte +omputer +ompany

"

( Plain Hancy

0 Number to ma*e: !otal Profit:

2 Cnit profit: (8 08

6


> ,iring 8.0 8.2 28

? Assembly 8.2 8.0 28

18 ;nspection 8." 8.( ""

11 Plain 9emand 1 >8

1" Hancy 9emand 1 ?8

(2. <efer to $xhibit (.". ,hat formula should be entered in cell $2 in the accompanying $xcelspreadsheet to compute total profita. B0F+0E2F+2 b. BSC#P</9C+!30:+0%2:+25c. BSC#32:+25d. BSC#3$>:$185

ANS: P!S: 1

(6. <efer to $xhibit (.". ,hat formula should be entered in cell 9> in the accompanying $xcelspreadsheet to compute the amount of wiring useda. B0F2E+0F+2 b. BSC#P</9C+!3>:+>%II0:I+I05c. BSC#32:+25d. BSC#3$>:$185

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ANS: P!S: 1

(. <efer to $xhibit (.". ,hich cells should be changing cells in this problema. 0:+0 b. $2c. 9>:918d. $>:$18

ANS: A P!S: 1

(>. <efer to $xhibit (.". ,hich cells should be the constraint cells in this problema. 0:+0 b. $2c. 9>:91"d. $>:$1"

ANS: + P!S: 1

(?. <efer to $xhibit (.". ,hich of the following statements will represent the constraint for -ust assemblyhoursa. 0:+0 ≤ 2:+2

b. 9?≤

$?c. 9>:918 ≤ $>:$18

d. $>:$18 ≤ 9>:918

ANS: P!S: 1

Exhibit 33


Gac*Js distillery blends scotches for local bars and saloons. /ne of his customers has re=uested aspecial blend of scotch targeted as a bar scotch. !he customer wants the blend to involve two scotch

products% call them A and . Product A is a higher =uality scotch while product is a cheaper brand.!he customer wants to ma*e the claim the blend is closer to high =uality than the alternative. !hecustomer wants 28 1288 ml bottles of the blend. $ach bottle must contain at least 0>K of Product Aand at least 288 ml of . !he customer also specified that the blend have an alcohol content of at least>2K. Product A contains ?2K alcohol while product contains >K. !he blend is sold for I1".28 per bottle. Product A costs I per liter and product costs I( per liter. !he company wants to determinethe blend that will meet the customerJs re=uirements and maximi)e profit.

Let 71 B Number of liters of product A in total blend delivered7" B Number of liters of product in total blend delivered

#;N: 71 E ( 7"

Sub-ect to: 71 E 7" B 1.2 F 28 3!otal liters of mix5

71 ≥ 8.0> F 1.2 F 28 371 minimum5

7" ≥ 8.2 F 28 37" minimum5

.8.?2 71 E 8.> 7" ≥ 8.>2 F 1.2 F 28 3>2K alcohol minimum5

71% 7" ≥ 8

A + 9 $

1 Gac*sJ 9istillery

"

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( A

0 Liters to use !otal +ost:

2 Cnit cost: 18.2 0.2

6

+onstraints: Supplied <e=uirement

> !otal Liters 1 1 2

? A re=uired 1 (6

18 re=uired 1 "2

11 >2K alcohol 8.?2 8.> 6(.2

08. <efer to $xhibit (.(. ,hat formula should be entered in cell $2 in the accompanying $xcelspreadsheet to compute total costa. B0F+0E2F+2 b. BSC#P</9C+!30:+0%2:+25c. BSC#32:+25d. BSC#3$>:$185

ANS: P!S: 1

01. <efer to $xhibit (.(. ,hat formula should be entered in cell 911 in the accompanying $xcelspreadsheet to compute the total liters of alcohol supplied

a. B0F2E+0F+2 b. BSC#P</9C+!311:+11%II0:I+I05c. BSC#32:+25d. BSC#3$>:$185

ANS: P!S: 1

0". <efer to $xhibit (.(. ,hich cells should be changing cells in this problema. 0:+0 b. $2c. 9>:918d. $>:$18

ANS: A P!S: 1

0(. <efer to $xhibit (.(. ,hich cells should be the constraint cells in this problema. 0:+0 b. $2c. 9>:911d. $>:$18

ANS: + P!S: 1

00. <efer to $xhibit (.(. ,hich of the following statements could represent a constraint in this problema. 0:+0 ≤ 2:+2

b. $2≤

8c. 9> B $>d. $>:$11 ≤ 9>:911

ANS: + P!S: 1

Exhibit 3#


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A financial planner wants to design a portfolio of investments for a client. !he client has I(88%888 toinvest and the planner has identified four investment options for the money. !he followingre=uirements have been placed on the planner. No more than "2K of the money in any one investment%at least one third should be invested in long'term bonds which mature in seven or more years% and nomore than "2K of the total money should be invested in + or 9 since they are ris*ier investments. !he planner has developed the following LP model based on the data in this table and the re=uirements ofthe client. !he ob-ective is to maximi)e the total return of the portfolio.

;nvestment <eturn ears to #aturity <atingA 6.02K ? 1'$xcellent .18K > "'ery @ood+ >."8K 2 0'Hair 9 ?.88K > ('@ood

Let 71 B 9ollars invested in A7" B 9ollars invested in 7( B 9ollars invested in +70 B 9ollars invested in 9

#A7: .8602 71 E .81 7" E .8>" 7( E .8? 70

Sub-ect to: 71 E 7" E 7( E 70 ≤ (88888

71 ≤ 2888

7" ≤ 2888

7( ≤ 2888

70 ≤ 2888

71 E 7" E 70 ≥ 188888

7( E 70 ≤ 2888

71% 7"% 7(% 70 ≥ 8

A + 9 M

1 Amount #aximum M

" ond ;nvested "2.8K <eturn M( A I8 I2%888 6.02K M

0 I8 I2%888 .18K M

2 + I8 I2%888 >."8K M

6 9 I8 I2%888 ?.88K M

!otal ;nvested: I8 !otal: I8 M

> !otal Available: I(88%888 M

$ H @ 4

1 ears to E years @ood or worse

" #aturity 31'yes% 8'no5 <ating 31'yes% 8'no5

( ? 1 1'$xcellent 8

0 > 1 "'ery @ood 8 2 2 8 0'Hair 1

6 > 1 ('@ood 1

!otal: I8 !otal: I8

> <e=uired: I188%888 Allowed: I2%888

02. <efer to $xhibit (.0. ,hat formula should be entered in cell in the accompanying $xcelspreadsheet to compute total dollars investeda. BA993(:65

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b. BSC#3(:65c. B!/!AL3(:65d. B!ALL3(:65

ANS: P!S: 1

06. <efer to $xhibit (.0. ,hat formula should be entered in cell 9 in the accompanying $xcelspreadsheet to compute the total returna. BFSC#39(:965

b. BSC#P</9C+!3(:6%9(:965c. BSC#3(:65d. BSC#P</9C+!3(:$(%6:$65

ANS: P!S: 1

0. <efer to $xhibit (.0. ,hich cells are changing cells in the accompanying $xcel spreadsheeta. (:6 b. :;c. +d. $

ANS: A P!S: 1

Exhibit 3$


A company is planning production for the next 0 =uarters. !hey want to minimi)e the cost of production. !he production cost is stable but demand and production capacity vary from =uarter to=uarter. !he maximum amount of inventory which can be held is 1"%888 units and management wantsto *eep at least (%888 units on hand. &uarterly inventory holding cost is (K of the cost of production.!he company estimates the number of units carried in inventory each month by averaging the beginning and ending inventory for each month. !here are currently 2%888 units in inventory. !hecompany wants to produce at no less than one half of its maximum capacity in any =uarter.

&uarter

1 " ( 0

Cnit Production +ost I (88 I (88 I (88 I (88Cnits 9emanded "%888 ?%888 1"%888 11%888#aximum Production >%888 %888 >%888 ?%888

Let Pi B number of units produced in =uarter i% i B 1% ...% 0i B beginning inventory for =uarter i

#;N: (88 P1 E (88 P" E (88 P( E (88 P0 E?31 E "5O" E ?3" E (5O" E ?3( E 05O" E ?30 E 25O"

Sub-ect to: 0888 ≤ P1 ≤ >888(288 ≤ P" ≤ 888

0888 ≤ P( ≤ >888

0288 ≤ P0 ≤ ?888

(888 ≤ 1 E P1 − "888 ≤ 1"888

(888 ≤ " E P" − ?888 ≤ 1"888

(888 ≤ ( E P( − 1"888 ≤ 1"888

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(888 ≤ 0 E P0 − 11888 ≤ 1"888

" B 1 E P1 − "888

( B " E P" − ?888

0 B ( E P( − 1"888

2 B 0 E P0 − 11888

Pi% i ≥ 8

A + 9 $ H

1 &uarter " 1 " ( 0

( eginning ;nventory 2%888 11%888 ?%888 2%888

0 Cnits Produced >%888 %888 >%888 ?%888

2 Cnits 9emanded "%888 ?%888 1"%888 11%888

6 $nding ;nventory 11%888 ?%888 2%888 (%888

> #inimum Production 0%888 (%288 0%888 0%288

? #aximum Production >%888 %888 >%888 ?%888

18

11 #inimum ;nventory (%888 (%888 (%888 (%888

1" #aximum ;nventory 1"%888 1"%888 1"%888 1"%888

1(10 Cnit Production +ost I(88 I(88 I(88 I(88

12 Cnit +arrying +ost (.8K I?.88 I?.88 I?.88 I?.88

16

1 &uarterly Production +ost I"%088%888 I"%188%888 I"%088%888 I"%88%888

1> &uarterly +arrying +ost I"%888 I?8%888 I6(%888 I(6%888

1?

"8 !otal +ost I?%>61%888

0>. <efer to $xhibit (.2. ,hat formula should be entered in cell +6 in the accompanying $xcelspreadsheet to compute ending inventorya. B+('+0E+2 b. B+(E+0'+2c. B+('3+0'+25d. B+2'+0'+(

ANS: P!S: 1

0?. <efer to $xhibit (.2. ,hat formula should be entered in cell +1> in the accompanying $xcelspreadsheet to compute the =uarterly carrying costsa. B+12F+(E+6 b. B+12F3+(E+65c. B+12F+(O"d. B+12F3+(E+65O"

ANS: 9 P!S: 1

28. <efer to $xhibit (.2. ,hich cells are changing cells in the accompanying $xcel spreadsheeta. +0:H0 b. +?:H?c. H"8d. +1":H1"

ANS: A P!S: 1

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21. <efer to $xhibit (.2. ,hat formula could be entered in cell H"8 in the accompanying $xcelspreadsheet to compute the !otal +ost for all four =uartersa. SC#P</9C+!3I+I0:IHI0%+1:H15 b. SC#3+1:H15 E SC#3+1>:H1>5c. P</9C+!3SC#3+10:H12%+1:H1>5d. SC#P</9C+!3+0:H0%+10:H105 E SC#P</9C+!3+6:H6%+12:H125

ANS: P!S: 1

P%O&LEM

2". ou have been given the following linear programming model and $xcel spreadsheet to solve this problem. ,hat numbers should be entered into cells 2:+2 and >:+18 to implement this model

#A7: " 71 E 7"

Sub-ect to: 2 71 E ? 7" ≤ ?8

? 71 E > 7" ≤ 100

7" ≤ >

71% 7" ≥ 8

A + 9 $

1

"

( 71 7"

0 Number to ma*e: /G. HN. ALC$

2 Cnit profit:

6


> 1 ?8

? " 100

18 ( >

ANS:

A + 9 $

1

"

( 71 7"


2 Cnit profit: "

6


> 1 2 ? ?8

? " ? > 100

18 ( 8 1 >

P!S: 1

2(. ou have been given the following linear programming model and $xcel spreadsheet to solve this problem. ,hat formulas should be entered into cells $2 and 9>:918 to implement this model

#A7: " 71 E 7"

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Sub-ect to: 2 71 E ? 7" ≤ ?8

? 71 E > 7" ≤ 100

7" ≤ >

71% 7" ≥ 8

A + 9 $

1

"

( 71 7"


2 Cnit profit: "

6


> 1 2 ? ?8

? " ? > 100

18 ( 8 1 >

ANS:+ell Hormula +opied to

$2 BSC#P</9C+!30:+0%2:+25

9> BSC#P</9C+!3II0:I+I0%>:+>5 9?:918

P!S: 1

20. ou have been given the following linear programming model and $xcel spreadsheet to solve this problem. ,hat cell references would you enter in the <is* Solver Platform 3<SP5 tas* pane for thefollowing

/b-ective +ell:

ariables +ells:

+onstraints +ells:

#A7: > 71 E 2 7"

Sub-ect to: ( 71 E 2 7" B 20

11 71 E 18 7" ≤ 100

71 ≥1"

71% 7" ≥ 8

A + 9 $

1

"

( 71 7"

0 Number to ma*e: /G. HN. ALC$2 Cnit profit: > 2

6


> 1 ( 2 20

? " 11 18 100

18 ( 1 8 1"

ANS:

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/b-ective +ell:I$I2

ariables +ells:II0:I+I0

+onstraints +ells:

II0:I+I0 ≥ 8

I9I> B I$I>

I9I? ≤ I$I?

I9I18≥

I$I18 3or II0≥

I$I185

P!S: 1

22. ou have been given the following linear programming model and $xcel spreadsheet to solve this problem. ,hat numbers should be entered into cells 2:+2 and >:+18 to implement this model

#A7: 0 71 E ( 7"

Sub-ect to: 6 71 E 7" ≤ >0

71 ≤ 18

7" ≤ >

71% 7" ≥ 8

A + 9 $

1

"

( 71 7"


2 Cnit profit:

6


> 1 >0

? " 18

18 ( >

ANS:

A + 9 $

1

"

( 71 7"


2 Cnit profit: 0 (

6


> 1 6 >0

? " 1 8 1818 ( 8 1 >

P!S: 1

26. ou have been given the following linear programming model and $xcel spreadsheet to solve this problem. ,hat formulas should be entered into cells $2 and 9>:918 to implement this model

#A7: 0 71 E ( 7"

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Sub-ect to: 6 71 E 7" ≤ >0

71 ≤ 18

7" ≤ >

71% 7" ≥ 8

A + 9 $

1

"

( 71 7"


2 Cnit profit: 0 (

6


> 1 6 >0

? " 1 8 18

18 ( 8 1 >


$2 BSC#P</9C+!30:+0%2:+25

9> BSC#P</9C+!3II0:I+I0%>:+>5 9?:918

P!S: 1


/b-ective +ell:

ariables +ells:

+onstraints +ells:

#A7: 1" 71 E ? 7"

Sub-ect to: ? 71 E 18.2 7" ≤ 1"6

71 ≥ 2

7" ≥ 6

71% 7" ≥ 8

A + 9 $

1

"

( 71 7"

0 Number to ma*e: /G. HN. ALC$2 Cnit profit: 1" ?

6


> 1 ? 18.2 1"6

? " 1 8 2

18 ( 8 1 6

ANS:

8/20/2019 11635aSamplech3-2


/b-ective +ell:I$I2

ariables +ells:II0:I+I0

+onstraints +ells:

II0:I+I0 ≥ 8

I9I> ≤ I$I>

I9I?:I9I18 ≥ I$I?:I$I18

P!S: 1

2>. ou have been given the following linear programming model and $xcel spreadsheet to solve this problem. ,hat numbers should be entered into cells 2:+2 and >:+18 to implement this model

#;N: > 71 E ( 7"

Sub-ect to: 7" ≥ >

> 71 E 2 7" ≥ >8

( 71 E 2 7" ≥ 68

71% 7" ≥ 8

A + 9 $1

"

( 71 7"


2 Cnit profit:

6


> 1 >

? " >8

18 ( 68

ANS:

A + 9 $

1

"

( 71 7"


2 Cnit profit: > (

6


> 1 8 1 >

? " > 2 >8

18 ( ( 2 68

P!S: 1

2?. ou have been given the following linear programming model and $xcel spreadsheet to solve this problem. ,hat formulas should be entered into cells $2 and 9>:918 to implement this model

#;N: > 71 E ( 7"


8/20/2019 11635aSamplech3-2


> 71 E 2 7" ≥ >8

( 71 E 2 7" ≥ 68

71% 7" ≥ 8

A + 9 $

1

"

( 71 7"

0 Number to ma*e: /G. HN. ALC$2 Cnit profit: > (

6


> 1 1 >

? " > 2 >8

18 ( ( 2 68


$2 BSC#P</9C+!30:+0%2:+259> BSC#P</9C+!3II0:I+I0%>:+>5 9?:918

P!S: 1


/b-ective +ell:

ariables +ells:

+onstraints +ells:

#;N: > 71 E ( 7"


> 71 E 2 7" ≥ >8

( 71 E 2 7" ≥ 68

71% 7" ≥ 8

A + 9 $

1

"

( 71 7"


2 Cnit profit:6


> 1 1 >

? " > 2 >8

18 ( ( 2 68

ANS:/b-ective +ell:

8/20/2019 11635aSamplech3-2


I$I2ariables +ells:

II0:I+I0+onstraints +ells:

II0:I+I0 ≥ 8

I9I>:I9I18 ≥ I$I>:I$I18

P!S: 1

61. A farmer is planning his spring planting. 4e has "8 acres on which he can plant a combination of +orn%Pump*ins and eans. 4e wants to maximi)e his profit but there is a limited demand for each crop.$ach crop also re=uires fertili)er and irrigation water which are in short supply. !here are only 28 acreft of irrigation available and only >%888 poundsOacre of fertili)er available. !he following tablesummari)es the data for the problem.

+ropProfit per Acre 3I5

ield per Acre 3lb5

#aximum9emand 3lb5

;rrigation3acre ft5

Hertili)er 3poundsOacre5

+orn "%188 "1%888 "88%888 " 288Pump*in ?88 18%888 1>8%888 ( 088eans 1%828 (%288 >8%888 1 (88

Hormulate the LP for this problem.

ANS:Let 71 B aces of corn

7" B acres of pump*in7( B acres of beans

#A7: "18871 E ?887" E 18287(

Sub-ect to: "171 ≤ "88

187" ≤ 1>8

(.27( ≤ >8

71 E 7" E 7( ≤ "8"71 E (7" E 17( ≤ 28

271 E 07" E (7( ≤ >8

71% 7"% 7( ≥ 8

P!S: 1

6". A farmer is planning his spring planting. 4e has "8 acres on which he can plant a combination of +orn%Pump*ins and eans. 4e wants to maximi)e his profit but there is a limited demand for each crop.$ach crop also re=uires fertili)er and irrigation water which are in short supply. !here are only 28 acreft of irrigation available and only >%888 poundsOacre of fertili)er available. !he following tablesummari)es the data for the problem.


ield per Acre 3lb5

#aximum9emand 3lb5

;rrigation3acre ft5


+orn "%188 "1%888 "88%888 " 288Pump*in ?88 18%888 1>8%888 ( 088eans 1%828 (%288 >8%888 1 (88

8/20/2019 11635aSamplech3-2


$nter the numbers in the appropriate cells of ranges 1":91" and $>:H1" in the $xcel spreadsheet tosolve this problem based on the following formulation.

Let 71 B aces of corn7" B acres of pump*in7( B acres of beans

#A7: "18871 E ?887" E 18287(

Sub-ect to: "171 ≤ "88

187" ≤ 1>8

(.27( ≤ >8

71 E 7" E 7( ≤ "8

"71 E (7" E 17( ≤ 28

271 E 07" E (7( ≤ >8

71% 7"% 7( ≥ 8

A + 9 $ H

1 Harm Planning Problem

"

( +orn Pump*in eans

0 Acres to plant !otal Profit:2 Profit per acre

6


> +orn demand

? Pump*in demand

18 ean demand

11 ,ater

1" Hertili)er

ANS:

A + 9 $ H1 Harm Planning Problem

"

( +orn Pump*in eans

0 Acres to plant !otal Profit:

2 Profit per acre "188 ?88 1828

6


> +orn demand "1888 "88888

? Pump*in demand 18888 1>8888

18 ean demand (288 >8888

11 ,ater " ( 1 28

1" Hertili)er 288 088 (88 >888

P!S: 1

6(. A farmer is planning his spring planting. 4e has "8 acres on which he can plant a combination of +orn%Pump*ins and eans. 4e wants to maximi)e his profit but there is a limited demand for each crop.$ach crop also re=uires fertili)er and irrigation water which are in short supply. !he following tablesummari)es the data for the problem.

8/20/2019 11635aSamplech3-2



ield per Acre 3lb5

#aximum9emand 3lb5

;rrigation3acre ft5


+orn "%188 "1%888 "88%888 " 288Pump*in ?88 18%888 1>8%888 ( 088eans 1%828 (%288 >8%888 1 (88

,hat are the *ey formulas for this $xcel spreadsheet implementation of the following formulation

Let 71 B aces of corn7" B acres of pump*in7( B acres of beans

#A7: "18871 E ?887" E 18287(

Sub-ect to: "171 ≤ "88

187" ≤ 1>8

(.27( ≤ >8

"71 E (7" E 17( ≤ 28

271 E 07" E (7( ≤ >8

71% 7"% 7( ≥ 8

A + 9 $ H1 Harm Planning Problem

"

( +orn Pump*in eans

0 Acres to plant !otal Profit:

2 Profit per acre "188 ?88 1828

6


> +orn demand "1888 "88888

? Pump*in demand 18888 1>8888

18 ean demand (288 >8888

11 ,ater " ( 1 28

1" Hertili)er 288 088 (88 >888


H2 BSC#P</9C+!30:90%2:925$> BSC#P</9C+!3II0:I9I0%>:9>5 $:$1"

P!S: 1

60. A hospital needs to determine how many nurses to hire to cover a "0 hour period. !he nurses mustwor* > consecutive hours but can start wor* at the start of 6 different shifts. !hey are paid differentwages depending on when they start their shifts. !he number of nurses re=uired per 0'hour time period

and their wages are shown in the following table.

!ime period <e=uired of Nurses ,age 3IOhr5

1" am − 0 am "8 12

0 am − > am (8 16

> am − 1" pm 08 1(

1" pm − 0 pm 28 1(

0 pm − > pm 08 10

8/20/2019 11635aSamplech3-2


> pm − 1" am (8 12


ANS:Let 7i B number of nurses wor*ing in time period iD i B 1%6

#;N: 171 E 17" E 17( E 170 E 172 E 176

Sub-ect to: 171 E 17" ≥ (8

17" E 17( ≥ 08

17( E 170 ≥ 28

170 E 172 ≥ 08

172 E 176 ≥ (8

171 E 176 ≥ "8

7i ≥ 8

P!S: 1

62. A hospital needs to determine how many nurses to hire to cover a "0 hour period. !he nurses mustwor* > consecutive hours but can start wor* at the start of 6 different shifts. !hey are paid differentwages depending on when they start their shifts. !he number of nurses re=uired per 0'hour time periodand their wages are shown in the following table.


1" am − 0 am "8 12

0 am − > am (8 16

> am − 1" pm 08 1(

1" pm − 0 pm 28 1(

0 pm − > pm 08 10

> pm − 1" am (8 12

$nter the numbers in the appropriate cells of ranges 6:@11 and 1(:@1( in the $xcel spreadsheet tosolve this problem based on the following formulation.

Let 7i B number of nurses wor*ing in time period iD i B 1%6

#;N: 171 E 17" E 17( E 170 E 172 E 176

Sub-ect to: 171 E 17" ≥ (8

17" E 17( ≥ 08

17( E 170 ≥ 28

170 E 172 ≥ 08

172 E 176 ≥ (8

171 E 176 ≥ "87i ≥ 8

A + 9 $ H @ 4 ;

1 Nurse 4iring

"

( 4ours for each shift

0 #id 0am >am Noon 0pm >pm Nurses ,ages per

2 Shift 0am >am Noon 0pm >pm #id Scheduled Nurse

8/20/2019 11635aSamplech3-2


6 1 I12

" I16

> ( I1(

? 0 I1(

18 2 I10

11 6 I12

1" Available: !otal ,ages:

1( <e=uired:

ANS:

A + 9 $ H @ 4 ;

1 Nurse 4iring

"




6 1 1 1 8 8 8 8 I12

" 8 1 1 8 8 8 I16

> ( 8 8 1 1 8 8 I1(

? 0 8 8 8 1 1 8 I1(

18 2 8 8 8 8 1 1 I1011 6 1 8 8 8 8 1 I12


1( <e=uired: "8 (8 08 28 08 (8

P!S: 1


!ime period <e=uired of Nurses ,age 3IOhr51" am − 0 am "8 12

0 am − > am (8 16

> am − 1" pm 08 1(

1" pm − 0 pm 28 1(

0 pm − > pm 08 10

> pm − 1" am (8 12



#;N: 171 E 17" E 17( E 170 E 172 E 176

Sub-ect to: 171 E 17" ≥ (8

17" E 17( ≥ 08

17( E 170 ≥ 28

170 E 172 ≥ 08

172 E 176 ≥ (8

171 E 176 ≥ "8

8/20/2019 11635aSamplech3-2


7i ≥ 8

A + 9 $ H @ 4 ;

1 Nurse 4iring

"




6 1 I12

" I16> ( I1(

? 0 I1(

18 2 I10

11 6 I12


1( <e=uired:


;1" BSC#P</9C+!346:411%;6:;1151" BSC#P</9C+!36:11%I4I6:I4I115 +1":@1"

P!S: 1



1" am − 0 am "8 12

0 am − > am (8 16

> am − 1" pm 08 1(

1" pm − 0 pm 28 1(

0 pm − > pm 08 10

> pm − 1" am (8 12

,hat values would you enter in the <is* Solver Platform 3<SP5 tas* pane for the following cells forthis $xcel spreadsheet implementation of the formulation for this problem

/b-ective +ell:

ariables +ells:

+onstraints +ells:


#;N: 171 E 17" E 17( E 170 E 172 E 176

Sub-ect to: 171 E 17" ≥ (8

17" E 17( ≥ 08

17( E 170 ≥ 28

8/20/2019 11635aSamplech3-2


170 E 172 ≥ 08

172 E 176 ≥ (8

171 E 176 ≥ "8

7i ≥ 8

A + 9 $ H @ 4 ;

1 Nurse 4iring

"

( 4ours for each shift0 #id 0am >am Noon 0pm >pm Nurses ,ages per


6 1 1 1 8 8 8 8 I12

" 8 1 1 8 8 8 I16

> ( 8 8 1 1 8 8 I1(

? 0 8 8 8 1 1 8 I1(

18 2 8 8 8 8 1 1 I10

11 6 1 8 8 8 8 1 I12


1( <e=uired: "8 (8 08 28 08 (8

ANS:/b-ective +ell:

I;I1"ariables +ells:

I4I6:I4I11+onstraints +ells:

I4I6:I4I11 ≥ 8

II1":I@I1" ≥ II1(:I@I1(

P!S: 1

6>. A company needs to purchase several new machines to meet its future production needs. ;t can

purchase three different types of machines A% % and +. $ach machine A costs I>8%888 and re=uires"%888 s=uare feet of floor space. $ach machine costs I28%888 and re=uires (%888 s=uare feet of floorspace. $ach machine + costs I08%888 and re=uires 2%888 s=uare feet of floor space. !he machines can produce "88% "28 and (28 units per day respectively. !he plant can only afford I288%888 for all themachines and has at most "8%888 s=uare feet of room for the machines. !he company wants to buy asmany machines as possible to maximi)e daily production.


ANS:Let 7i B number of machines of type i purchased

#A7: "8871 E "287" E (887(Sub-ect to: "71 E (7" E 27( ≤ "8

>871 E 287" E 087( ≤ 288

71% 7"% 7( ≥ 8

P!S: 1

8/20/2019 11635aSamplech3-2


6?. A company needs to purchase several new machines to meet its future production needs. ;t can purchase three different types of machines A% % and +. $ach machine A costs I>8%888 and re=uires"%888 s=uare feet of floor space. $ach machine costs I28%888 and re=uires (%888 s=uare feet of floorspace. $ach machine + costs I08%888 and re=uires 2%888 s=uare feet of floor space. !he machines can produce "88% "28 and (28 units per day respectively. !he plant can only afford I288%888 for all themachines and has at most "8%888 s=uare feet of room for the machines. !he company wants to buy asmany machines as possible to maximi)e daily production.

$nter the numbers in the appropriate cells of range 2:H18 in the $xcel spreadsheet to solve this

problem based on the following formulation.

Let 7i B number of machines of type i purchased

#A7: "8871 E "287" E (887(

Sub-ect to: "71 E (7" E 27( ≤ "8

>871 E 287" E 087( ≤ 288

71% 7"% 7( ≥ 8

A + 9 $ H

1 +apital $xpansion

"

( #achine !ypes

0 #achine 1 #achine " #achine (

2 Number to buy !otal /utput:

6 #achine output

> <e=uirements: Csed Available

? S=uare feet

18 +ost

ANS:

A + 9 $ H

1 +apital $xpansion

"

( #achine !ypes



6 #achine output "88 "28 (88


? S=uare feet "%888 (%888 2%888 "8%888

18 +ost >8%888 28%888 08%888 288%888

P!S: 1

8. A company needs to purchase several new machines to meet its future production needs. ;t can purchase three different types of machines A% % and +. $ach machine A costs I>8%888 and re=uires"%888 s=uare feet of floor space. $ach machine costs I28%888 and re=uires (%888 s=uare feet of floorspace. $ach machine + costs I08%888 and re=uires 2%888 s=uare feet of floor space. !he machines can produce "88% "28 and (28 units per day respectively. !he plant can only afford I288%888 for all themachines and has at most "8%888 s=uare feet of room for the machines. !he company wants to buy asmany machines as possible to maximi)e daily production.

8/20/2019 11635aSamplech3-2




#A7: "8871 E "287" E (887(

Sub-ect to: "71 E (7" E 27( ≤ "8

>871 E 287" E 087( ≤ 288

71% 7"% 7( ≥ 8

A + 9 $ H @

1 #achine A #achine #achine +

" Number to uy

(

0 Production Possible "88 "28 (28 !otal:

2

6 <esources 4ours <e=uired Csed Available:

Hloor Space <e=Jd " ( 0 "8

> Assemble >8 28 08 288

ANS:

+ell Hormula +opied to@0 BSC#P</9C+!3":9"%0:905$ BSC#P</9C+!3II":I9I"%:95 $>

P!S: 1

1. A company needs to purchase several new machines to meet its future production needs. ;t can purchase three different types of machines A% % and +. $ach machine A costs I>8%888 and re=uires"%888 s=uare feet of floor space. $ach machine costs I28%888 and re=uires (%888 s=uare feet of floorspace. $ach machine + costs I08%888 and re=uires 2%888 s=uare feet of floor space. !he machines can produce "88% "28 and (28 units per day respectively. !he plant can only afford I288%888 for all themachines and has at most "8%888 s=uare feet of room for the machines. !he company wants to buy as

many machines as possible to maximi)e daily production.

,hat values would you enter in the <is* Solver Platform 3<SP5 tas* pane for the following cells forthis $xcel spreadsheet implementation of the formulation for this problem

/b-ective +ell:

ariables +ells:

+onstraints +ells:


#A7: "8871 E "287" E (887(

Sub-ect to: "71 E (7" E 27( ≤ "8

>871 E 287" E 087( ≤ 288

71% 7"% 7( ≥ 8

A + 9 $ H

1 +apital $xpansion

"

8/20/2019 11635aSamplech3-2


( #achine !ypes



6 #achine output "88 "28 (88


? S=uare feet "%888 (%888 2%888 "8%888

18 +ost >8%888 28%888 08%888 288%888

ANS:/b-ective +ell:

IHI6ariables +ells:

II2:I9I2+onstraints +ells:

II2:I9I2 ≥ 8

I$I?:I$I18 ≤ IHI?:IHI18

P!S: 1

". State Harm Supply has -ust received an order for 18%888 pounds of chic*en feed. !he farmer hasspecified certain that the feed meet minimum re=uirements for Protein% +arbohydrate% Hat anditamins. State Harm can blend four different feeds to produce the re=uired mix. !he farmer wouldli*e to pay the lowest possible price for the feed. !he data for the problem is summari)ed in thefollowing table.

State Harm SupplyPercent of Nutrient in: #inimum

Nutrient Heed 1 Heed " Heed ( Heed 0 <e=Jd Amt

Protein 12 "8 (8 12 1>+arbohydrate "8 18 18 12 1"Hat "8 (8 12 "8 "8itamin 1 1.28 8.2 8.28 1

+ostO1%888 lbs I288 I688 I228 I028


ANS:Let 7i B pounds of feed i used in mixture

#;N: .271 E .67" E .227( E .0270

Sub-ect to: .1271 E ."87" E .(7( E .1270 ≥ 1>88

."871 E .187" E .17( E .1270 ≥ 1"88

."871 E .(87" E .127( E ."870 ≥ "888

.8171 E .8127" E .8827( E .88270 ≥ 188171 E 17" E 17( E 170 B 18888

7i ≥ 8

P!S: 1

8/20/2019 11635aSamplech3-2


(. A paper mill has received an order for rolls of paper. !he customer wants 088 1" wide rolls% (88 1>rolls and "88 "0 rolls. !he company has 08 wide rolls of paper which it can slit to the appropriatewidth. !he company wants to minimi)e the number of rolls it must use to fill the order.


ANS:9efine the following cutting patterns.

Number of widths in roll

+utting pattern 1" 1> "0

1 ( 8 8" 1 1 8( 1 8 10 8 " 8

088 (88 "88

Let 7i B number of rolls cut in pattern i

#;N: 171 E 17" E 17( E 170

Sub-ect to: (71 E 17" E 17( ≥ 088

17" E "70 ≥ (88

17( ≥ "88

7i ≥ 8

P!S: 1

0. PeteJs Plastics manufactures plastic at plants in #iami% St. Louis and +leveland. Pete needs to ship plastic to customers in Pittsburgh% Atlanta and +hicago. 4e wants to minimi)e the cost of shipping the plastic from his plants to his customers. !he data for the problem is summari)ed in the following table.

9istance Hrom Plants to +ustomers

Plant Pittsburgh Atlanta +hicago Supply#iami 1"88 88 1(88 (8St. Louis 88 228 (88 08+leveland 1"2 62 (28 28

9emand 08 68 "8


ANS:Let 7i- B tons shipped from plant i to customer - 3i and - B 1% "% (5

#;N: 1"88711 E 8871" E 1(8871( E 887"1 E 2287"" E (887"(

E 1"27(1 E 627(" E (287((

Sub-ect to: 711 E 71" E 71( B (87"1 E 7"" E 7"( B 087(1 E 7(" E 7(( B 28

711 E 7"1 E 7(1 ≥ 08

71" E 7"" E 7(" ≥ 68

71( E 7"( E 7(( ≥ "8

7i- ≥ 8

8/20/2019 11635aSamplech3-2


P!S: 1

2. A financial planner wants to design a portfolio of investments for a client. !he client has I088%888 toinvest and the planner has identified four investment options for the money. !he followingre=uirements have been placed on the planner. No more than (8K of the money in any one investment%at least one half should be invested in long'term bonds which mature in six or more years% and no morethan 08K of the total money should be invested in or + since they are ris*ier investments. !he planner has developed the following LP model based on the data in this table and the re=uirements of

the client. !he ob-ective is to maximi)e the total return of the portfolio.

;nvestment <eturn ears to #aturity <ating

A 6.02K 6 1'$xcellent >.2K 2 ('@ood+ ?.88K > 0'Hair 9 .2K 0 "'ery @ood


ANS:Let 71 B 9ollars invested in A

7" B 9ollars invested in 7( B 9ollars invested in +70 B 9ollars invested in 9

#A7: .8602 71 E .8>2 7" E .8?8 7( E .82 70

Sub-ect to: 71 E 7" E 7( E 70 ≤ 088888

71 ≤ 1"8888

7" ≤ 1"8888

7( ≤ 1"8888

70 ≤ 1"8888

71 E 7( ≥ "88888

7" E 7( ≤ 16888871% 7"% 7(% 70 ≥ 8

P!S: 1

6. A financial planner wants to design a portfolio of investments for a client. !he client has I088%888 toinvest and the planner has identified four investment options for the money. !he followingre=uirements have been placed on the planner. No more than (8K of the money in any one investment%at least one half should be invested in long'term bonds which mature in six or more years% and no morethan 08K of the total money should be invested in or + since they are ris*ier investments. !he planner has developed the following LP model based on the data in this table and the re=uirements ofthe client. !he ob-ective is to maximi)e the total return of the portfolio.



Let 71 B 9ollars invested in A

8/20/2019 11635aSamplech3-2



#A7: .8602 71 E .8>2 7" E .8?8 7( E .82 70

Sub-ect to: 71 E 7" E 7( E 70 ≤ 088888

71 ≤ 1"8888

7" ≤ 1"8888

7( ≤ 1"888870 ≤ 1"8888

71 E 7( ≥ "88888

7" E 7( ≤ 168888

71% 7"% 7(% 70 ≥ 8

A + 9 M

1 Amount #aximum M

" ond ;nvested (8.8K <eturn M

( A I8 I1"8%888 6.02K M

0 I8 I1"8%888 >.2K M

2 + I8 I1"8%888 ?.88K M

6 9 I8 I1"8%888 .2K M !otal ;nvested: I8 !otal: I8 M

> !otal Available: I088%888 M

$ H @ 4

1 ears to 6E years @ood or worse


( 6 1 1'$xcellent 8

0 2 8 ('@ood 8

2 > 1 0'Hair 1

6 0 8 "'ery @ood 1

!otal: I8 !otal: I8

> <e=uired: I"88%888 Allowed: I168%888

,hat values would you enter in the <is* Solver Platform 3<SP5 tas* pane for the following cells forthis $xcel spreadsheet implementation of this problem

/b-ective +ell:

ariables +ells:

+onstraints +ells:

ANS:

/b-ective +ell:9ariables +ells:

(:6+onstraints +ells:

(:6 ≤ +(:+6

(:6 ≥ 8

≤ >

8/20/2019 11635aSamplech3-2


H ≥ H>

4 ≤ 4>

P!S: 1

. A financial planner wants to design a portfolio of investments for a client. !he client has I088%888 toinvest and the planner has identified four investment options for the money. !he followingre=uirements have been placed on the planner. No more than (8K of the money in any one investment%

at least one half should be invested in long'term bonds which mature in six or more years% and no morethan 08K of the total money should be invested in or + since they are ris*ier investments. !he planner has developed the following LP model based on the data in this table and the re=uirements ofthe client. !he ob-ective is to maximi)e the total return of the portfolio.



Let 71 B 9ollars invested in A


#A7: .8602 71 E .8>2 7" E .8?8 7( E .82 70

Sub-ect to: 71 E 7" E 7( E 70 ≤ 088888

71 ≤ 1"8888

7" ≤ 1"8888

7( ≤ 1"8888

70 ≤ 1"8888

71 E 7( ≥ "88888

7" E 7( ≤ 168888

71% 7"% 7(% 70 ≥ 8

A + 9 M

1 Amount #aximum M

" ond ;nvested (8.8K <eturn M

( A I8 I1"8%888 6.02K M

0 I8 I1"8%888 >.2K M

2 + I8 I1"8%888 ?.88K M

6 9 I8 I1"8%888 .2K M

!otal ;nvested: I8 !otal: I8 M

> !otal Available: I088%888 M

$ H @ 4

1 ears to 6E years @ood or worse


( 6 1 1'$xcellent 8

0 2 8 ('@ood 8

2 > 1 0'Hair 1

6 0 8 "'ery @ood 1

!otal: I8 !otal: I8

> <e=uired: I"88%888 Allowed: I168%888

8/20/2019 11635aSamplech3-2


,hat formulas are re=uired for the following cells in the $xcel spreadsheet implementation of theformulation9H4

ANS:

BSC#3(:659 BSC#P</9C+!3II(:II6%9(:965H BSC#P</9C+!3II(:II6%H(:H654 BSC#P</9C+!3II(:II6%4(:465

P!S: 1

>. A company is planning production for the next 0 =uarters. !hey want to minimi)e the cost of production. !he production cost% demand and production capacity vary from =uarter to =uarter. !hemaximum amount of inventory which can be held is 188 units and management wants to *eep at least28 units on hand. &uarterly inventory holding cost is 0K of the cost of production. !here are currently28 units in inventory. !he company wants to produce at no less than one half of its maximum capacity

in any =uarter.

&uarter

1 " ( 0

Cnit Production +ost I22 I28 I28 I02Cnits 9emanded 188 128 1>8 1"8#aximum Production 128 128 168 1(84olding cost I"." I" I" I1.>

Let Pi B number of units produced in =uarter i% i B 1% ...% 0i B beginning inventory for =uarter i

#;N: 22 P1 E 28 P " E 28 P( E 02 P0 E"." 31 E "5O" E " 3" E (5O" E " 3( E 05O" E 1.> 30 E 25O"

Sub-ect to: 2 ≤ P1 ≤ 128

2 ≤ P" ≤ 128

>8 ≤ P( ≤ 168

62 ≤ P0 ≤ 1(8

28 ≤ 1 E P1 − 188 ≤ 188

28 ≤ " E P" − 128 ≤ 188

28 ≤ ( E P( − 1>8 ≤ 188

28 ≤ 0 E P0 − 1"8 ≤ 188

" B 1 E P1 − 188

( B " E P" − 128

0 B ( E P( − 1>8

2 B 0 E P0 − 1"8

Pi% i ≥ 8

A + 9 $ H

1 &uarter

" 1 " ( 0

8/20/2019 11635aSamplech3-2


( eginning ;nventory 28 8 8 28

0 Cnits Produced 1"8 128 168 1"8

2 Cnits 9emanded 188 128 1>8 1"8

6 $nding ;nventory 8 8 28 28

> #inimum Production 2 2 >8 62

? #aximum Production 128 128 168 1(8

18

11 #inimum ;nventory 28 28 28 28

1" #aximum ;nventory 188 188 188 1881(

10 Cnit Production +ost I22 I28 I28 I02

12 Cnit +arrying +ost 0.8K I"."8 I".88 I".88 I1.>8

16

1 #onthly Production +ost I6%688 I%288 I>%888 I2%088

1> #onthly +arrying +ost I1(" I108 I1"8 I?8

1?

"8 !otal +ost I"%?>"

,hat formulas are re=uired for cells 9(% 96% 9>% 912% 91 and 91> in the $xcel spreadsheetimplementation of the formulation

ANS:9( B+696 B9(E90'929> B9?O"912 BII12F9191 B910F90591> B912F39(E965O"5

P!S: 1

?. A grain store has six types of grain% each varying in cost% =uality% and nutritional content. Periodically%

excess inventory of these grains are consolidated into two local products% Heed'#'All and Supreme'Heed. Heed'#'All sells for I6.28 for a 18'pound bag while Supreme'Heed sells for I>.28 for a 18' pound bag. !hese feeds are advertised as having the following nutritional content:

@rain #inimum Protein #inimum Hat #aximum +arbohydrates

Heed'#'All 16K 1>K 18KSupreme'Heed 1>K 1>K ?K

!he component grains have the following content characteristics:

@rain +ostO18 lbs &uality Protein Hat +arbohydrates Pounds Avail.

A I0.2 0 12K 18K 18K ?8

I0.88 " "8K "8K >K 1"8+ I(.2 1 18K "2K 2K 1289 I0."2 ( 12K "8K 18K 1"2$ I0.28 ( "8K "8K 18K >2H I2.88 0 "2K 12K 1"K 162

8/20/2019 11635aSamplech3-2


!argets for Heed'#'All are a cost of I 0.(2 per 18'pound bag% a =uality rating of "."2% along with theminimum percentages of protein and fat% and the maximum percentage of carbohydrates. Similartargets are set for Supreme'Heed with cost set at I 0.68 and =uality at ".02. !here must be at least a8K'(8K mix among these two local feeds.

Hormulate an LP model for this product mix problem.

ANS:Let 7i- B amount of grain i in feed - where

i B A% % +% 9% $% H and - B 13Heed'#'All5% "3Supreme'Heed5 - B total amount of feed - produced

#A7: I6.281 E I>.28"

Sub-ect to:1 B 711 E 7"1 E 7(1 E 701 E 721 E 761 9efine - values" B 71" E 7"" E 7(" E 70" E 72" E 76"

711 E 71" ≤ ?8 @rain availability

7"1 E 7"" ≤ 1"8

7(1 E 7(" ≤ 128

701 E 70" ≤

1"2721 E 72" ≤ >2

761 E 76" ≤ 162

""8.2 ≤ 1 ≤ 210.2 #ix re=uirements

""8.2 ≤ " ≤ 210.2

0711 E "7"1 E 7(1 E (701 E (721 E 0761 ≥ "."21 &uality targets

071" E "7"" E 7(" E (70" E (72" E 076" ≥ ".02"

0.2711 E 07"1 E (.27(1 E 0."2701 E 0.2721 E 2761 ≤ 0.(21 +ost targets

0.271" E 07"" E (.27(" E 0."270" E 0.272" E 276" ≤ 0.68"

18711 E "87"1 E 187(1 E 12701 E "8721 E "2761 ≥ 161 Protein targets

1871" E "87"" E 187(" E 1270" E "872" E "276" ≥ 1>"

18711 E "87"1 E "27(1 E "8701 E "8721 E 12761 ≥ 1>1 Hat targets1871" E "87"" E "27(" E "870" E "872" E 1276" ≥ 1>"

18711 E >7"1 E 27(1 E 18701 E 18721 E 1"761 ≤ 181 +arbohydrate targets

1871" E >7"" E 27(" E 1870" E 1872" E 1"76" ≤ ?"

7i- ≥ 8 for all i and -% - ≥ 8 for all -.

P!S: 1

>8. +arlton construction is supplying building materials for a new mall construction pro-ect in Qansas.!heir contract calls for a total of "28%888 tons of material to be delivered over a three'wee* period.+arltonJs supply depot has access to three modes of transportation: a truc*ing fleet% railway delivery%

and air cargo transport. !heir contract calls for 1"8%888 tons delivered by the end of wee* one% >8K ofthe total delivered by the end of wee* two% and the entire amount delivered by the end of wee* three.+ontracts in place with the transportation companies call for at least 02K of the total delivered bedelivered by truc*ing% at least 08K of the total delivered be delivered by railway% and up to 12K of thetotal delivered be delivered by air cargo. Cnfortunately% competing demands limit the availability ofeach mode of transportation each of the three wee*s to the following levels 3all in thousands of tons5:

,ee* !ruc*ing Limits <ailway Limits Air +argo Limits

1 02 68 12

8/20/2019 11635aSamplech3-2


" 28 22 18( 22 02 2

+osts 3I per 1888 tons5 I"88 I108 I088

Hormulate an LP model for this logistics problem.

ANS:Let 7i- B amount shipped by mode i in wee* -

where i B 13!ruc*5% "3<ail5% (3Air5 and - B 1% "% (,Li- B wee*ly limit of mode i in wee* - 3as provided in above table5

#;N: "883711 E 71" E 71(5 E 10837"1 E 7"" E 7"(5 E 28837(1 E 7(" E 7((5Sub-ect to:

7i- ≤ ,Li- for all i and - ,ee*ly limits by mode

711 E 71" E 71( E 7"1 E 7"" E 7"( E 7(1 E 7(" E 7(( ≥ "28 !otal at end of three wee*s

711 E 7"1 E 7(1 E 71" E 7"" E 7(" ≥ "88 !otal at end of two wee*s

711 E 7"1 E 7(1 ≥ 1"8 !otal at end of first wee*

711 E 71" E 71( ≥ 8.02F"28 !ruc* mix re=uirement

7"1 E 7"" E 7"( ≥ 8.08F"28 <ail mix re=uirement

7(1 E 7(" E 7(( ≤ 8.12F"28 Air mix limit7i- ≥ 8 for all i and -

P!S: 1

>1. +arlton construction is supplying building materials for a new mall construction pro-ect in Qansas.!heir contract calls for a total of "28%888 tons of material to be delivered over a three'wee* period.+arltonJs supply depot has access to three modes of transportation: a truc*ing fleet% railway delivery%and air cargo transport. !heir contract calls for 1"8%888 tons delivered by the end of wee* one% >8K ofthe total delivered by the end of wee* two% and the entire amount delivered by the end of wee* three.+ontracts in place with the transportation companies call for at least 02K of the total delivered bedelivered by truc*ing% at least 08K of the total delivered be delivered by railway% and up to 12K of thetotal delivered be delivered by air cargo. Cnfortunately% competing demands limit the availability ofeach mode of transportation each of the three wee*s to the following levels 3all in thousands of tons5:

,ee* !ruc*ing Limits <ailway Limits Air +argo Limits

1 02 68 12" 28 22 18( 22 02 2


!he following is the LP model for this logistics problem.

Let 7i- B amount shipped by mode i in wee* - where i B 13!ruc*5% "3<ail5% (3Air5 and - B 1% "% (,Li- B wee*ly limit of mode i in wee* - 3as provided in above table5




8/20/2019 11635aSamplech3-2


711 E 7"1 E 7(1 E 71" E 7"" E 7(" ≥ "88 !otal at end of two wee*s

711 E 7"1 E 7(1 ≥ 1"8 !otal at end of first wee*



7(1 E 7(" E 7(( ≤ 8.12F"28 Air mix limit

7i- ≥ 8 for all i and -

A + 9 $ H

1 +osts I"88.88 I108.88 I288.88" by !ruc* by <ail by Air !otals <e=uired

( ,ee* 1 #$ '( !$ !"( 1"80 ,ee* " $( $$ ( ""$ "882 ,ee* ( !3 !" ( "$( "28

6 Shipped by !() !"* !$

Percentage 02K 08K 12K> !otal Limit 18> 188 (.2?18 !otal +ost +#',))(((

11 ,ee*ly Limits1" !ruc* <ail Air

1( ,ee* 1 02 68 12

10 ,ee* " 28 22 1812 ,ee* ( 22 02 2

,hat formula goes in cells H18% $(% $0% $2% and 6 of this $xcel spreadsheet

ANS:H18 BSC#P</9C+!3II1:I9I1%II6:I9I65$( BSC#3II(:I9I(5$0 BSC#3II0:I9I05$2 BSC#3II2:I9I256 BSC#3II(:II25

P!S: 1

>". +arlton construction is supplying building materials for a new mall construction pro-ect in Qansas.!heir contract calls for a total of "28%888 tons of material to be delivered over a three'wee* period.+arltonJs supply depot has access to three modes of transportation: a truc*ing fleet% railway delivery%and air cargo transport. !heir contract calls for 1"8%888 tons delivered by the end of wee* one% >8K ofthe total delivered by the end of wee* two% and the entire amount delivered by the end of wee* three.+ontracts in place with the transportation companies call for at least 02K of the total delivered bedelivered by truc*ing% at least 08K of the total delivered be delivered by railway% and up to 12K of thetotal delivered be delivered by air cargo. Cnfortunately% competing demands limit the availability ofeach mode of transportation each of the three wee*s to the following levels 3all in thousands of tons5:

,ee* !ruc*ing Limits <ailway Limits Air +argo Limits1 02 68 12" 28 22 18( 22 02 2


!he following is the LP model for this logistics problem.

Let 7i- B amount shipped by mode i in wee* -

8/20/2019 11635aSamplech3-2


where i B 13!ruc*5% "3<ail5% (3Air5 and - B 1% "% (,Li- B wee*ly limit of mode i in wee* - 3as provided in above table5




711 E 7"1 E 7(1 E 71" E 7"" E 7(" ≥

"88 !otal at end of two wee*s711 E 7"1 E 7(1 ≥ 1"8 !otal at end of first wee*



7(1 E 7(" E 7(( ≤ 8.12F"28 Air mix limit

7i- ≥ 8 for all i and -

A + 9 $ H1 +osts I"88.88 I108.88 I288.88" by !ruc* by <ail by Air !otals <e=uired

( ,ee* 1 #$ '( !$ !"( 1"80 ,ee* " $( $$ ( ""$ "88

2 ,ee* ( !3 !" ( "$( "286 Shipped by !() !"* !$

Percentage 02K 08K 12K> !otal Limit 18> 188 (.2?18 !otal +ost +#',))(((

11 ,ee*ly Limits1" !ruc* <ail Air

1( ,ee* 1 02 68 1210 ,ee* " 28 22 1812 ,ee* ( 22 02 2

,hat values would you enter in the <is* Solver Platform 3<SP5 tas* pane for the cells in this $xcel

spreadsheet implementation of this problem

/b-ective +ell:

ariables +ells:

+onstraints +ells:

ANS:/b-ective +ell:

H18ariables +ells:

(:92+onstraints +ells:

(:9( ≤ 1(:91(

$(:$2 ≥ H(:H2

6:+6 ≥ >:+>

96 ≤ 9>

P!S: 1

8/20/2019 11635aSamplech3-2


>(. <obert 4ope received a welcome surprise in this management science classD the instructor has decidedto let each person define the percentage contribution to their grade for each of the graded instrumentsused in the class. !hese instruments were: homewor*% an individual pro-ect% a mid'term exam% and afinal exam. <obertJs grades on these instruments were 2% ?0% >2% and ?"% respectively. 4owever% theinstructor complicated <obertJs tas* somewhat by adding the following stipulations:

• homewor* can account for up to "2K of the grade% but must be at least 2K of the gradeD

• the pro-ect can account for up to "2K of the grade% but must be at least 2K of the gradeD

• the mid'term and final must each account for between 18K and 08K of the grade butcannot account for more than 8K of the grade when the percentages are combinedD and

• the pro-ect and final exam grades may not collectively constitute more than 28K of thegrade.

Hormulate an LP model for <obert to maximi)e his numerical grade.

ANS:Let ,1 B weight assigned to homewor*

," B weight assigned to the pro-ect,( B weight assigned to the mid'term,0 B weight assigned to the final

#A7: 2,1 E ?0," E >2,( E ?",0

Sub-ect to: ,1 E ," E ,( E ,0 B 1

,( E ,0 ≤ 8.8

,( E ,0 ≥ 8.28

8.82 ≤ ,1 ≤ 8."2

8.82 ≤ ," ≤ 8."2

8.18 ≤ ,( ≤ 8.08

8.18 ≤ ,0 ≤ 8.08

P!S: 1

>0. <obert 4ope received a welcome surprise in this management science classD the instructor has decidedto let each person define the percentage contribution to their grade for each of the graded instrumentsused in the class. !hese instruments were: homewor*% an individual pro-ect% a mid'term exam% and afinal exam. <obertJs grades on these instruments were 2% ?0% >2% and ?"% respectively. 4owever% theinstructor complicated <obertJs tas* somewhat by adding the following stipulations:

• homewor* can account for up to "2K of the grade% but must be at least 2K of the gradeD

• the pro-ect can account for up to "2K of the grade% but must be at least 2K of the gradeD

• the mid'term and final must each account for between 18K and 08K of the grade butcannot account for more than 8K of the grade when the percentages are combinedD and

• the pro-ect and final exam grades may not collectively constitute more than 28K of thegrade.

!he following LP model allows <obert to maximi)e his numerical grade.

Let ,1 B weight assigned to homewor* ," B weight assigned to the pro-ect,( B weight assigned to the mid'term,0 B weight assigned to the final

8/20/2019 11635aSamplech3-2


#A7: 2,1 E ?0," E >2,( E ?",0

Sub-ect to: ,1 E ," E ,( E ,0 B 1

,( E ,0 ≤ 8.8

,( E ,0 ≥ 8.28

8.82 ≤ ,1 ≤ 8."2

8.82 ≤ ," ≤ 8."2

8.18 ≤ ,( ≤ 8.08

8.18 ≤ ,0 ≤ 8.08

A + 9 $1 Percentage Limits" @rade to grade Lower Cpper

( #id !erm >2 8.08 8.18 8.080 Hinal ?" 8."2 8.18 8.082 Pro-ect ?0 8."2 8.82 8."26 4omewor* 2 8.18 8.82 8."2

1.88> 188K 1.88?18 @rade >>.88

11 !otal Limit1" oth $xams 8.62 8.81( Hinal R Pro-ect 8.2 8.28

,hat values would you enter in the <is* Solver Platform 3<SP5 tas* pane for the cells in this $xcelspreadsheet implementation of this problem

/b-ective +ell:

ariables +ells:

+onstraints +ells:

ANS:/b-ective +ell:

+18ariables +ells:

+(:+6+onstraints +ells:

+(:+6 ≤ 9(:96

+(:+6 ≥ $(:$6

+ B +>

1" ≤ +1"

1( ≥ +1(

P!S: 1

8/20/2019 11635aSamplech3-2


>2. !he hospital administrators at New 4ope% +ounty @eneral% and +ity $ast recently received notice of animpending state inspection of their facilities. Cnder new guidelines established to improve the overallhealth care system% state inspectors will be assessing the efficiency of each hospital. !he staff at New4ope has suggested a mutual assistance program in preparation for the inspections and have proposedusing 9$A as a means to assess the efficiency of each facility. !he data collected thus far issummari)ed in the following table. All data reflects averages compiled over the past six months.

4ospital

New 4ope +ounty @eneral +ity $ast

;nput #easuresed days unused 31888s5 >(.8 182.8 180.1Supply expense 3I1888s5 1"(.> 16".( 120.8Hull'time staff ""2.8 "88.8 "(1.8

/utput #easuresPatient'days 31888s5 182.8 1.8 >". Nurses =ualified "2(.8 ?".8 12.8Assistants on staff 1"2.8 02.8 62.8+ustomer satisfaction ?>.8 >>.8 >(.8

a. Hormulate a 9$A LP model to evaluate the efficiency of +ity $ast.

b. ;mplement a spreadsheet model for this problem and compute the 9$A efficiency for eachfacility. ,hich facilities are efficient

ANS:a. Let wi B weight assigned to output -% - B 1% ...%0

vi B weight assigned to input i% i B 1%...%(

#A7: >". w1 E 12.8 w" E 62.8 w( E >(.8 w0

Sub-ect to:

182.1 w1 E "2(.8 w" E 1"2.8 w( E ?>.8 w0 − >(.8 v1 − 1"(.> v" − ""2.8 v( ≤ 8

1.8 w1 E ?".8 w" E 02.8 w( E >>.8 w0 − 182.8 v1 − 16".( v" − "88 v( ≤ 8

>". w1 E 12.8 w" E 62.8 w( E >(.8 w0 −

180.1 v1 −

120.8 v" −

"(1.8 v( ≤

8180.1 v1 E 120.8 v" E "(1.8 v( B 1

w1% w"% w(% w0% v1% v"% v( ≥ 8

b. A + 9 $ M

1 Patient -sst M

" .a/s 01rses on C1st M

( Hospital 2!(((s 41al Sta55 Sat M

0 New 4ope 182.18 "2(.88 1"2.8 ?> M

2 +nty. @eneral 1.88 ?".88 02.8 >> M

6 +ity $ast >".8 12.88 62.8 >( M

M

> 6eights 8.88"88? 8 8 8.88> M? M

18 U0IT ( M

11 O1tp1t 8.>1""2? M

1" Inp1t 1 M

H @ 4 ; G Q

1 &ed7.a/s S1ppl/ 81ll

" Un1sed Expense Time 6gt 6gt

8/20/2019 11635aSamplech3-2


( 2!(((s 2+!(((s Sta55 O1tp1t Inp1t .i55

0 >(.88 1"(.>8 ""2.88 ?K ?K 8.8888

2 182.88 16".(8 "88.88 >(K >K −8.8(>1

6 180.18 120.88 "(1.88 >1K 188K −8.1>

> 8 8 8.880("?

?

18

11

1"

<esults:9$A

Cnit $fficiency

New 4ope 1.8888+ounty @eneral 8.?"?+ity $ast 8.>1"(

New 4ope is an efficient facility.

P!S: 1

>6. !he hospital administrators at New 4ope% +ounty @eneral% and +ity $ast recently received notice of animpending state inspection of their facilities. Cnder new guidelines established to improve the overallhealth care system% state inspectors will be assessing the efficiency of each hospital. !he staff at New4ope has suggested a mutual assistance program in preparation for the inspections and have proposedusing 9$A as a means to assess the efficiency of each facility. !he data collected thus far issummari)ed in the following table. All data reflects averages compiled over the past six months.

4ospital




ased on the following formulation% is +ity $ast efficient ;f not% what input and output values shouldthey aspire to in order to become efficient

Let wi B weight assigned to output -% - B 1% ...% 0vi B weight assigned to input i% i B 1%...%(

#A7: >". w1 E 12.8 w" E 62.8 w( E >(.8 w0

Sub-ect to:

182.1 w1 E "2(.8 w" E 1"2.8 w( E ?>.8 w0 − >(.8 v1 − 1"(.> v" − ""2.8 v( ≤ 8

1.8 w1 E ?".8 w" E 02.8 w( E >>.8 w0 − 182.8 v1 − 16".( v" − "88 v( ≤ 8

>". w1 E 12.8 w" E 62.8 w( E >(.8 w0 − 180.1 v1 − 120.8 v" − "(1.8 v( ≤ 8

180.1 v1 E 120.8 v" E "(1.8 v( B 1

8/20/2019 11635aSamplech3-2


w1% w"% w(% w0% v1% v"% v( ≥ 8

ANS: No% +ity $ast is not efficient. !he following shows that >.6?K of New 4ope input and outputs produces a composite unit with outputs greater than or e=ual to those of +ity $ast re=uiring less inputthan +ity $ast.

''' /utputs ''' M

Patient -sst M

.a/s 01rses on C1st M

Unit 2!(((s 41al Sta55 Sat M

New 4ope 182.18 "2(.88 1"2.8 ?> M

+ounty @eneral 1.88 ?".88 02.8 >> M

+ity $ast >".8 12.88 62.8 >( M

M

Comp 9als >". 1??.1 ?>.0 .1 M

''' ;nputs '''

&ed7.a/s S1ppl/ 81ll

Un1sed Expense Time Composite

Unit 2!(((s 2+!(((s Sta55 6eight

New 4ope >(.88 1"(.>8 ""2.88 8.>6?

+ounty @eneral 182.88 16".(8 "88.88 8

+ity $ast 180.18 120.88 "(1.88 8

Comp 9als 62.( ?.0 1.8

Note% however% the drop in customer satisfaction. +ity $ast will not want to aspire to that particularlevel. !hese composite values will ma*e +ity $ast efficient.

A + 9 $ M

1 Patient -sst M


( Hospital 2!(((s 41al Sta55 Sat M0 New 4ope 182.18 "2(.88 1"2.8 ?> M

2 +nty. @eneral 1.88 ?".88 02.8 >> M

6 +ity $ast >".8 1??.8 ?>.0 >( M

M

> 6eights 8.880"? 8 8 8.88>0 M

? M

18 U0IT ( M

11 O1tp1t 1 M

1" Inp1t 1 M

H @ 4 ; G Q

1 &ed7.a/s S1ppl/ 81ll " Un1sed Expense Time 6gt 6gt

( 2!(((s 2+!(((s Sta55 O1tp1t Inp1t .i55

0 >(.88 1"(.>8 ""2.88 1"1K 1"K −8.82>6

2 182.88 16".(8 "88.88 ??K 11(K −8.1011

6 62.( ?.0 1 188K 188K 8.8888

> 8 8 8.88260?

?

8/20/2019 11635aSamplech3-2


18

11

1"

P!S: 1

>. !he hospital administrators at New 4ope% +ounty @eneral% and +ity $ast recently received notice of animpending state inspection of their facilities. Cnder new guidelines established to improve the overallhealth care system% state inspectors will be assessing the efficiency of each hospital. !he staff at New4ope has suggested a mutual assistance program in preparation for the inspections and have proposedusing 9$A as a means to assess the efficiency of each facility. !he data collected thus far issummari)ed in the following table. All data reflects averages compiled over the past six months.

4ospital




$nter the numbers in the appropriate cells of ranges 0:46 in the $xcel spreadsheet to solve this problem based on the following formulation.


#A7: >". w1 E 12.8 w" E 62.8 w( E >(.8 w0

Sub-ect to:

182.1 w1 E "2(.8 w" E 1"2.8 w( E ?>.8 w0 − >(.8 v1 − 1"(.> v" − ""2.8 v( ≤ 8

1.8 w1 E ?".8 w" E 02.8 w( E >>.8 w0 − 182.8 v1 − 16".( v" − "88 v( ≤ 8

>". w1 E 12.8 w" E 62.8 w( E >(.8 w0 − 180.1 v1 − 120.8 v" − "(1.8 v( ≤ 8

180.1 v1 E 120.8 v" E "(1.8 v( B 1

w1% w"% w(% w0% v1% v"% v( ≥ 8

A + 9 $ M

1 Patient -sst M



0 New 4ope M

2 +nty. @eneral M6 +ity $ast M

M

> 6eights 8 8 8 8 M

? M

18 U0IT ( M

11 O1tp1t 8.>1 M

1" Inp1t 1.8 M

8/20/2019 11635aSamplech3-2


H @ 4 ; G Q



( 2!(((s 2+!(((s Sta55 O1tp1t Inp1t .i55

0 ?K ?K 8.8888

2 >(K >K −8.8(>1

6 >1K 188K −8.1>

> 8 8 8

?

18

11

1"

ANS:

A + 9 $ M

1 Patient -sst M



0 New 4ope 182.18 "2(.88 1"2.8 ?> M

2 +nty. @eneral 1.88 ?".88 02.8 >> M6 +ity $ast >".8 12.88 62.8 >( M

M

> 6eights 8.88"88? 8 8 8.88> M

? M

18 U0IT ( M

11 O1tp1t 8.>1""2? M

1" Inp1t 1 M

H @ 4 ; G Q



( 2!(((s 2+!(((s Sta55 O1tp1t Inp1t .i55 0 >(.88 1"(.>8 ""2.88 ?K ?K 8.8888

2 182.88 16".(8 "88.88 >(K >K −8.8(>1

6 180.18 120.88 "(1.88 >1K 188K −8.1>

> 8 8 8.880("?

?

18

11

1"

P!S: 1

>>. !he hospital administrators at New 4ope% +ounty @eneral% and +ity $ast recently received notice of animpending state inspection of their facilities. Cnder new guidelines established to improve the overallhealth care system% state inspectors will be assessing the efficiency of each hospital. !he staff at New4ope has suggested a mutual assistance program in preparation for the inspections and have proposedusing 9$A as a means to assess the efficiency of each facility. !he data collected thus far issummari)ed in the following table. All data reflects averages compiled over the past six months.

4ospital

8/20/2019 11635aSamplech3-2




/utput #easuresPatient'days 31888s5 182.8 1.8 >". Nurses =ualified "2(.8 ?".8 12.8Assistants on staff 1"2.8 02.8 62.8satisfaction ?>.8 >>.8 >(.8



#A7: >". w1 E 12.8 w" E 62.8 w( E >(.8 w0

Sub-ect to:

182.1 w1 E "2(.8 w" E 1"2.8 w( E ?>.8 w0 − >(.8 v1 − 1"(.> v" − ""2.8 v( ≤ 8

1.8 w1 E ?".8 w" E 02.8 w( E >>.8 w0 − 182.8 v1 − 16".( v" − "88 v( ≤ 8

>". w1 E 12.8 w" E 62.8 w( E >(.8 w0 −

180.1 v1 −

120.8 v" −

"(1.8 v( ≤

8180.1 v1 E 120.8 v" E "(1.8 v( B 1

w1% w"% w(% w0% v1% v"% v( ≥ 8

A + 9 $ M

1 Patient -sst M



0 New 4ope 182.18 "2(.88 1"2.8 ?> M

2 +nty. @eneral 1.88 ?".88 02.8 >> M

6 +ity $ast >".8 12.88 62.8 >( M

M

> 6eights 8.88"88? 8 8 8.88> M? M

18 U0IT ( M

11 O1tp1t 8.>1""2? M

1" Inp1t 1 M

H @ 4 ; G Q



( 2!(((s 2+!(((s Sta55 O1tp1t Inp1t .i55

0 >(.88 1"(.>8 ""2.88 ?K ?K 8.8888

2 182.88 16".(8 "88.88 >(K >K −8.8(>1

6 180.18 120.88 "(1.88 >1K 188K −8.1>

> 8 8 8.880("?

?

18

11

1"

ANS:

8/20/2019 11635aSamplech3-2


+ell Hormula +opied to

;0 BSC#P</9C+!3II>:I$I>%0:$05 ;2:;6G0 BSC#P</9C+!3IHI>:I4I>%H0:405 G2:G6Q0 B;0'G0 Q2:Q611 B;N9$73;0:;6%18%151" B;N9$73G0:G6% 18%15

P!S: 1

P%O:ECT

>?. Pro;e<t 3! The .iet Problem= Ordering Meals 5rom M<.onald>s

ased on: <obert A. osch% ig #ac Attac*: !he 9iet Problem revisited% $ating at #c9onaldJs%OR/MS Today, August 1??(% pp (8'(1.

!ina Simpson is a new fourth'grade teacher at Horest <idge $lementary. !he first teacher wor*shop for the upcoming school year is next #onday and by ma-ority vote% #c9onaldJs was selected as the foodof choice. As the new person% !ina is tas*ed with developing the meal for the wor*shop. #c9onaldJshas graciously offered to deliver whatever food !ina decides to order% along with a variety of

condiments applicable to whatever is ordered. <ather than offer a menu choice% !ina has decided tosimply order the same meal for each person in the wor*shop.

!o get started% !ina too* a trip to #c9onaldJs and obtained their published information on thenutritional content of their food. !hat data is summari)ed in the table below.

Price Protein Hat Sodium M#enu ;tem 3I5 +alories 3grams5 3mg5 M

4amburger 8.2? "22 1" ? 0?8 M#cLean 9lx 1.? ("8 "" 18 68 Mig #ac 1.62 288 "2 "6 >?8 MSmall Hries 8.6> ""8 ( 1" 118 M#cNuggets 1.26 "8 "8 12 2>8 M4oney 8.88 02 8 8 8 M+hef Salad ".6? 18 1 ? 088 M@arden Salad 1.?6 28 0 " 8 M$gg #c#uffin 1.(6 ">8 1> 11 18 M,heaties 1.8? ?8 " 1 ""8 Mogurt +one 8.6( 182 0 1 >8 M#il* 8.26 118 ? " 1(8 M/range Guice 8.>> >8 1 8 8 M@rapefruit -uice 8.6> >8 1 8 8 MApple Guice 8.6> ?8 8 8 2 M

Prices recorded August% 1??1 in /berlin /hio M

it A it + it 1 it " Niacin +alcium ;ron #enu ;tem K C.S. <ecommended 9aily Allowance 3<9A5

4amburger 0 0 "8 18 "8 18 12 #cLean 9lx 18 18 "2 "8 (2 12 "8 ig #ac 6 " (8 "2 (2 "2 "8 Small Hries F 12 18 F 18 F " #cNuggets F F > > 08 F 6 4oney F F F F F F F +hef Salad 188 (2 "8 12 "8 12 > @arden Salad ?8 (2 6 6 " 0 >

8/20/2019 11635aSamplech3-2


$gg #c#uffin 18 F (8 "8 "8 "2 12 ,heaties "8 "8 "8 "8 "8 " "8 ogurt +one " F " 18 " 18 F #il* 18 0 > (8 F (8 F /range Guice F 1"8 18 F F F F @rapefruit -uice F 188 0 " " F F Apple Guice F " " F F F 0

Prices recorded August% 1??1 in /berlin /hio F +ontains less than "K of the C.S. <9A of these nutrients

!ina wants the meal to be nutritionally complete. !he National <esearch +ouncil publishes their<ecommended 9aily Allowances. ;n this publication% they contend that a diet 3in this case the meal5should provide at least 188 percent of the C.S. <9A of numerous nutrients. !he specific amount of the<9A depends on such factors as age% weight and gender. ;n addition% the council recommends dailysodium and cholesterol inta*es be *ept to at most ".0 grams of sodium and (88 milligrams ofcholesterol. Hurther% at most (8 percent of the calories consumed should come from fat% and at most 18 percent from saturated fat. $ach gram of fat contains ? calories.

ased on the above information% !ina wants to design a least'cost meal that provides at least 188K ofthe C.S. <9A of vitamins A% +% 1% "% niacin% calcium% and ironD supplies at least 22 grams of proteinDcontains at most ( grams of sodiumD and contains at most (8 percent of its calories from fat. /nlythose foods list in the table above are available for the meal.

Hormulate the LP model for !inaJs problem. 9evelop a spreadsheet model of the problem and use$xcel Solver to determine the least'cost meal that meets all the stated re=uirements.

,hat is the recommended meal ;s this meal reasonable ;f not% modify the model to obtain what you believe to be a reasonable meal that meets the stated re=uirements.

ANS:Answer not provided.

P!S: 1

11635aSamplech3-2

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