Evans Analytics2e Ppt 13

7/24/2019 Evans Analytics2e Ppt 13

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Chapter 13

Linear Optimization


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Optimizationis the process of selecting values ofdecision variables that minimize ormaximizesome quantity of interest.

Optimization models have wide applicability inoperations and supply chains, finance, marketing,and other disciplines.

This chapter focuses only on linear optimization

models.

Linear Optimization


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1. Identify the decision variables the unknown valuesthat the model seeks to determine.

!. Identify the objective function the quantity we seekto minimize or ma"imize.

#. Identify all appropriate constraints limitations,requirements, or other restrictions that are imposed onany solution, either from practical or technologicalconsiderations or by management policy.

$. %rite the ob&ective function and constraints as mathematical e"pressions.

Building Linear Optimization Models


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''( sells two snow ski models ) *ordanelle + eercrest -anufacturing requires fabrication and finishing. The fabrication department has 1! skilled workers, each of

whom works hours per day. The finishing department has #

workers, who also work a )hour shift. /ach pair of *ordanelle skis requires #.0 labor)hours in the

fabricating department and 1 labor)hour in finishing. The eercrest model requires $ labor)hours in fabricating and

1.0 labor)hours in finishing. The company operates 0 days per week. ''( makes a net profit of 02 on the *ordanelle model and

30 on the eercrest model.

Example 131 !"len"a !"i Compan#$

%dentif#ing Model Components


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'tep 14 Identify the decision variables The company wants to determine how many of

each model should be produced on a daily basis

to ma"imize net profit. efine *ordanelle 5 number of pairs of *ordanelle skis

produced6day

eercrest 5 number of pairs of eercrest skisproduced6day

(learly specify the dimensions of the variables7

Example 131 Continued


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'tep !4 Identify the ob&ective function ''( wishes to ma"imize net profit, and we are given the

net profit figures for each type of ski. ''( makes a net profit of 02 on the *ordanelle model and 30

on the eercrest model.



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'tep #4 Identify the constraints

8ook for clues in the problem statement that describe limited resourcesthat are available, requirements that must be met, or other restrictions.

9oth the fabrication and finishing departments have limited numbersof workers, who work only hours each day: this limits the amountof production time available in each department4

;abrication4 Total labor hours used in fabrication cannot e"ceed the amount oflabor hours available.

;inishing4 Total labor hours used in finishing cannot e"ceed the amount of laborhours available.

The problem also notes that the company anticipates selling at leasttwice as many eercrest models as *ordanelle models4


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Aepresent decision variables by descriptivenames, abbreviations, or subscripted letters=X1,X2, etc.@

;or mathematical formulations involving manyvariables, subscripted letters are often moreconvenient.

In spreadsheet models, we recommend usingmore descriptive names to make the models and

solutions easier to understand.

&ranslating Model %nformation into

Mathematical Expressions


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Brofit per pair of skis sold4 02 for *ordanelle skis, 30 for eercrest skis

Ob&ective ;unction4

-a"imize total profit5 02 JordanelleC 30 Deercrest


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(onstraints are e"pressed as algebraic inequalities orequations, with all variables on the left side and constantterms on the right.

8ook for key words in word statements of constraints4 >(annot e"ceed? translates mathematically as >D?

>Et least,? would translate as >F?

>-ust contain e"actly,? would specify an >5 ? relationship.

Ell constraints in optimization models must be one of

these three forms.

&ranslating Constraints

Mathematicall#


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E constraint function is the left)hand side of aconstraint. /.g.4 Total labor)hours used in fabrication cannot

e"ceed the amount of labor hours available.

Constraint )unctions


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;abrication constraintEvailable fabrication labor hours4 =1! workers@= hours6day@ 5 G$

hours6day

Aequired fabrication labor hours per ski pair4 #.0 hours for*ordanelle, $ hours for eercrest

;abrication constraint4 #.0 JordanelleC $ DeercrestD G$

;inishing constraintEvailable finishing labor hours4 =# workers@= hours6day@ 5 !1

hours6day

Aequired finishing labor hours per ski pair4 1 hour for *ordanelle:1.0 hours for eercrest

;inishing constraint4 1 JordanelleC 1.0 DeercrestD !1

Example 133$ !!C ( Modeling the

Constraints


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-arket mi"ture constraint The number of pairs of eercrest skis must be at least

twice the number of *ordanelle skis.

DeercrestF ! Jordanelle,

or

! JordanelleC 1 DeercrestF 2


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-a"imize total profit 5 02 JordanelleC 30 Deercrest

#.0 JordanelleC $ DeercrestD G$

1 JordanelleC 1.0 DeercrestD !1

! JordanelleC 1 Deercrest

F 2

Jordanelle F 2

DeercrestF 2

!!C Optimization Model

The highlighted portions are the constraint functions


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'ome e"amples4 The amount of money spent on research and

development pro&ects cannot e"ceed the assignedbudget of #22,222. Emount spent on research and development D #22,222

(ontractual requirements specify that at least 022 unitsof product must be produced.


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E fertilizer mi"ture is made of two ingredients and must contain e"actly#2H nitrogen. IngredientXcontains !2H nitrogen. Ingredient Ycontains ##H nitrogen.

efinex5 the number of pounds ofXin the mi"ture and y5 thenumber of pounds of Yin the mi"ture

Emount of nitrogen in mi"ture 5 2.!2xC 2.##y

Total amount of mi"ture 5x C y

;raction of nitrogen in mi" 5 =2.!2xC 2.##y)6=" C y@

'ince the fraction of nitrogen must be 2.#2, the constraint

would be=2.!2xC 2.##y)6=" C y@ 5 2.#2, or simplified as )2.1x) 2.2#y5 2


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E linear optimization model =often called alinear program, or L,@ has two basic properties.

1. The ob&ective function and all constraints are

linear functions of the decision variables. This means that each function is simply a sum of terms,

each of which is some constant multiplied by a decisionvariable.

!.

Ell variables are continuous This means that they may assume any real value

=typically, nonnegative@.

Characteristics of Linear Optimization

Models


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But the ob&ective function coefficients, constraint coefficients, andright)hand values in a logical format in the spreadsheet.

;or e"ample, you might assign the decision variables to columns and theconstraints to rows

efine a set of cells =either rows or columns@ for the values of the

decision variables. The names of the decision variables should be listed directly above the

decision variable cells.

se shading or other formatting to distinguish these cells.

efine separate cells for the ob&ective function and each constraint

function =the left)hand side of a constraint@. se descriptive labels directly above these cells.

%mplementing Linear Optimization

Models on !preadsheets


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Example 13-$ * !preadsheet Model for

!"len"a !"is

ecision variables Ob&ective function (onstraint functions


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-a"imize JordanelleC 30 Deercrest#.0 JordanelleC $ DeercrestD G$

1 JordanelleC 1.0 DeercrestD !1

! JordanelleC 1 DeercrestF 2

Jordanelle F 2

DeercrestF 2

Correspondence Bet.een the Model and

the !preadsheet

-a"imize !! 5 9JK B14C (JK C1410 5 93K B14C (3K C14D 3

13 5 9K B14 C (K C14D

1J 5 C14) !K B14F 2

B14F 2

C14F 2


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In /"cel, the pairwise sum of products of termscan easily be computed using the '-BAO(Tfunction. 9JK 91$ C (JK(1$ 5 '-BAO(T=9J4(J,91$4(1$@

This often simplifies the model)building process,particularly when many variables are involved.

/sing the !/M,0O/C& )unction


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'everal common functions in /"cel can cause difficultieswhen attempting to solve linear programs using 'olverbecause they are discontinuous =or >nonsmooth?@ and donot satisfy the conditions of a linear model.

These include4 I;

-EL

I


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E feasible solution to an optimization problem isany solution that satisfies all of the constraints.

En optimal solution is the best of all the feasiblesolutions.

'oftware for determining optimal solutions Solver=>standard Solver?@ is a free add)in packaged with

/"cel for solving optimization problems. Premium Solverwhich is a part of!nalytic Solver

Platformhas better functionality, accuracy, reporting, andinterface.

!olving Linear Optimization Models


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DataM!nalysisM Solver in the /"cel ribbon se the Solver Parameters dialog to define the

ob&ective, decision variables, and constraints fromyour spreadsheet model.

/sing the !tandard !olver


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Example 132$ /sing !tandard !olver for

the !!C ,roblem

Solver Parameters dialogOb&ective function cell

ecision variables cells

(onstraintsto enter click!ddand fill in the!ddConstraint dialog4

(heck bo" for 'imple" 8B?


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Three reports4 Enswer,'ensitivity, and 8imits To add them to your /"cel

workbook, click on theones you want and then

click "#. o not check the bo"

"utline $e%orts: this isan /"cel feature thatproduces the reports inNoutlined format.N

!olver 0esults ialog


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Optimal !olution to !!C ,roblem


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Efter installing!nalytic Solver Platform, PremiumSolverwill be found under the!dd&'ns tab in the/"cel ribbon.

Premium Solver has a different user interface thanthe standard Solver.

/sing Premium Solver


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Solver Parameters

dialog ;irst, click on "(ective

and then click the!dd

button. The!dd"(ectivedialogappears, prompting youfor the cell reference for

the ob&ective functionand the type of ob&ective=min or ma"@.

Example 13$ /sing Premium Solver for

the !!C Model


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(heck this bo"

'elect >'tandard

8B6uadratic? for thesolving method



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(ompletedPremium Solver

dialog



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The SolverEnswer Aeport provides basic information about thesolution, including the values of the original and optimal ob&ectivefunction =in the "(ective Cell section@ and decision variables =in theDecision +aria(le Cells section@.

In the Constraints section, Cell +alue refers to the value of the

constraint function using the optimal values of the decisionvariables.

E binding constraint is one for which the (ell Palue is equal to theright)hand side of the value of the constraint.

The Statuscolumn tells whether each constraint is binding or not

binding. !lac"refers to the difference between the left) and right)hand sides

of the constraints for the optimal solution.

Solver*ns.er 0eport


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Example 134$ %nterpreting the !!C *ns.er

0eport


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nderstanding slack values


-a"imize profit 5 02 JordanelleC 30 Deercrest #.0 JordanelleC $ DeercrestD G$ =fabrication@ 1 JordanelleC 1.0 DeercrestD !1 =finishing@ ! JordanelleC 1 DeercrestF 2 =marketmi"@ Jordanelle F 2

DeercrestF 2

Optimal solution4 Jordanelle5 0.!0: Deercrest5 12.0;abrication constraint4 #.0=0.!0@ C $=12.0@ 5 32.#0 D G$ !#.3!0 e"cess fabrication hours

;inishing constraint4 1=0.!0@ C 1.0=12.0@ 5 !1 D !1


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The set of feasible solutions is called the feasible region.

;or a problem with only two decision variables,x1 andx2, we can draw the

feasible region on a two)dimensional coordinate system by plotting theequations corresponding to each constraint.


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;abrication constraint4 #.0 *ordanelle C $ eercrest D G$ Blot the equation4 #.0 *ordanelle C $ eercrest 5 G$

'et *ordanelle 5 2: eercrest 5 !1

'et eercrest 5 2: *ordanelle 5 !$

Example 136$ 5raphing the Constraints in the !!C

,roblem


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;inishing constraint4 1 *ordanelle C 1.0 eercrest D !1 Blot the equation4 1 *ordanelle C 1.0 eercrest 5 !1

'et *ordanelle 5 2: eercrest 5 1$

'et eercrest 5 2: *ordanelle 5 !1



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-arket mi" constraint4 )! *ordanelle C 1 eercrest F 2 Blot the equation4 )! *ordanelle C 1 eercrest 5 2

'et *ordanelle 5 0: eercrest 5 12

'et eercrest 5 2: *ordanelle 5 2



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;easible region

Example 1317 %dentif#ing the )easible 0egion and

Optimal !olution


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The points at which the constraint lines intersect alongthe feasible region are called corner points.

If an optimal solution e"ists, then it will occur at a cornerpoint.

Corner ,oints


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9ecause our ob&ective is

to ma"imize profit, weseek a corner point thathas the largest value ofthe ob&ective function TotalBrofit 5 02 *ordanelle C 30eercrest.

Qraph the profit line andmove in an improvingdirection until it passesthrough the last cornerpoint of the feasibleregion.

'olve the two intersectingequations simultaneouslyto find the optimalsolution.



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'olver uses a mathematical algorithm called the sim%lexmet,od, which was developed in 1J$ by the late r.Qeorge antzig. The simple" method characterizes feasible solutions algebraically

by solving systems of linear equations.

It moves systematically from one corner point to another toimprove the ob&ective function until an optimal solution is found=or until the problem is deemed infeasible or unbounded@.

It is quick and efficient.

8o. !olver 9or"s


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(rebo -anufacturing produces $ types of structuralsupport fittings. -achining centers have a capacity of !G2,222 minutes

per year. Qross margin6unit and machining requirements4

Row many units of each product type should beproduced to ma"imize gross profit marginS

Example 1311$ Crebo Manufacturing


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efineX1,X!,X#, andX$as the number of plugs, rails,

rivets, and clips to produce. Ob&ective4 -a"imize gross profit margin 5 2.#X1 C 1.#X! C 2.0X# C 1.!X$

(onstraints4

1X1 C !.0X! C 1.0X# C !X$D !G2,222X1,X!,X#,X$F 2



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The simple" method evaluates the impact of constraintsin terms of their contribution to the ob&ective function foreach variable.

;or the simple case of only one constraint, the optimal=ma"imum@ solution is found by simply choosing the

variable with the highest ratio of the ob&ective coefficientto the constraint coefficient.

8o. the !implex Method 9or"s


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(lips have the highest marginal profit per unit of resource consumed.

-a"imum possible production of clips 5 !G2,222 minutes minutes6unit 5 !G2,222 ! 5 1$2,222

Brofit for ma"imum production of clips 5 gross margin6unit K ma" possible production 5 1.!2 K 1$2,222 5 13G,222

Example 131'$ !olving the Crebo

Manufacturing Model


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Solver assigns names to4 Target cells

(hanging cells

(onstraint function cells


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!!C Example

Brofit (ontribution C Total Brofit?

uantity Broduced C *ordanelle?

(ell 9104 uantity Broduced C eercrest?

;abrication C Rours sed?

(ell 134 >;inishing C Rours sed?

(ell 1J4 >-arket mi"ture C /"cesseercrest?


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/ni;ue optimal solution there is e"actly one solution that will result in the ma"imum =or

minimum@ ob&ective.

*lternative


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Example 313$ * Model .ith *lternative

Optimal !olutions

1#)


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Aemove the finishing and fabrication constraints from the 'klenka'ki problem.

Solvermessage4

Example 131+$ * Model .ith an /nbounded

!olution


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'uppose, by mistake, the modeler in the 'klenka 'ki problem useda F sign in the fabrication constraint =instead of D@4

Example 131-$ *n %nfeasible Model


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-odels should be used to provide insight formaking better decisions. %hat might happen should the model assumptions

change or when the data used in the model are

uncertainS %ith Solver, answers to such questions can easily be

found by simply changing the data and re)solving themodel.

/sing Optimization Models for ,rediction

and %nsight


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;our questions are posed by the managers of'klenka 'ki company4

1. If the JordanelleskiUs profit increased 126pair,how would the optimal solution changeS

!. If the JordanelleskiUs profit decreased 126pair,how would the optimal solution changeS

#. If 12 additional finishing hours were available,how would manufacturing plans be affectedS

$. If ! fewer finishing hours were available, howwould manufacturing plans be affectedS

Example 1312$ /sing Solverfor 9hat>%f

*nal#sis


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'ummary of %hat)If scenarios4



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The 'ensitivity Aeport allows us to understand how the optimal ob&ective value and optimal decision variables

are affected by changes in the ob&ective functioncoefficients,

the impact of forced changes in certain decision variables,or

the impact of changes in the constraint resource limitationsor requirements.

The 'ensitivity Aeport information applies to changes in

only one of the model parameters at a time: all othersare assumed to remain at their original values.

Solver!ensitivit# 0eport

! iti it 0 t f th !!C


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!ensitivit# 0eport for the !!C

,roblem

E l 13 1 % t ti ! iti it


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0educed Cost4 Row much the ob&ective function coefficient needs to bereduced for a nonnegative variable that is zero in the optimal solution tobecome positive. If a variable is positive in the optimal solution, its reducedcost is zero.

If the ob&ective coefficient of any one variable that has positive value in thecurrent solution changes but stays within the range specified by the EllowableIncrease and Ellowable ecrease, the optimal decision variables will stay thesame: however, t,e o(ective function value -ill c,an.e.

Example 131$ %nterpreting !ensitivit#

%nformation for ecision ?ariables

E l 13 14 / d t di :


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'ensitivity report for the 'klenka 'ki model afterchanging the profit on Jordanelleskis from 02 to $2

Example 1314$ /nderstanding :onzero

0educed Costs

E l 13 16 % t ti ! iti it


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!hado. ,rice ) how much the ob&ective function will change as theright hand side of a constraint is increased by 1.

%henever a constraint has positive slack, the shadow price is zero.%hen a constraint involves a limited resource, the shadow pricerepresents the economic value of having an additional unit of thatresource.

Example 1316$ %nterpreting !ensitivit#

%nformation for Constraints


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'hows the upper and lower limits that eachdecision variable can assume while satisfying allconstraints and holding the other variablesconstant.

Limits 0eport


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If a change in an ob&ective function coefficient remainswithin the Ellowable Increase and Ellowable ecreaseranges in the Decision +aria(le Cells section of thereport, then the optimal values of the decision variables

will not change. Rowever, you must recalculate the valueof the ob&ective function using the new value of thecoefficient.

If a change in an ob&ective function coefficient e"ceedsthe Ellowable Increase or Ellowable ecrease limits inthe Decision +aria(le Cells section of the report, thenyou must re)solve the model to find the new optimalvalues.

/sing the !ensitivit# 0eport



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If a change in the right)hand side of a constraint remainswithin the Ellowable Increase and Ellowable ecrease rangesin the Constraintssection of the report, then the shadow priceallows you to predict how the ob&ective function value willchange. -ultiply the change in the right)hand side =positive if

an increase, negative if a decrease@ by the value of theshadow price. Rowever, you must re)solve the model to findthe new values of the decision variables.

If a change in the right)hand side of a constraint e"ceeds theEllowable Increase or Ellowable ecrease limits in theConstraintssection of the report, then you cannot predict howthe ob&ective function value will change using the shadowprice. Vou must re)solve the problem to find the new solution.



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'uppose that the unit profit on *ordanelle skis is increased by 12.Row will the optimal solution changeS %hat is the best product mi"S

Is the increase in the ob&ective function coefficient is within the range ofthe Ellowable Increase and Ellowable ecrease in the ecision Pariable(ells portion of the reportS

9ecause 12 is less than the Ellowable Increase of infinity, we can safely

conclude that the optimal quantities of the decision variables will notchange.

Rowever, because the ob&ective function changed, we need to computethe new value of the total profit4 0.!0=32@ C 12.0=30@ 5 JJ.02.

Example 13'7$ /sing !ensitivit#

%nformation to Evaluate !cenarios


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'uppose that the unit profit on *ordanelle skis isdecreased by 12 because of higher material costs. Rowwill the optimal solution changeS %hat is the bestproduct mi"S

The change in the unit profit e"ceeds the Ellowable ecrease=3.3@. %e can conclude that the optimal values of the decisionvariables will change, although we must re)solve the problem todetermine what the new values would be.

Example 13'7 Continued


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'uppose that 12 additional finishing hours becomeavailable through overtime. Row will manufacturingplans be affectedS

(heck if the change in the right)hand)side value is within therange of the Ellowable Increase and Ellowable ecrease in theConstraintssection of the report.

Ten additional finishing hours e"ceeds the Ellowable Increase.Therefore, we must re)solve the problem to determine the newsolution.



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%hat if the number of finishing hours available isdecreased by ! hours because of planned equipmentmaintenanceS Row will manufacturing plans beaffectedS

E decrease of ! hours in finishing capacity is within the Ellowableecrease. Total profit will decrease by the value of the shadow

price for each hour that finishing capacity is decreased.Therefore, we can predict that the total profit will decrease by !W$0 5 J2 to G00. Rowever, we must re)solve the model inorder to determine the new values of the decision variables.


,arameter *nal#sis in Analytic Solver


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Solvercan be used to perform sensitivity analysisby either4 /"amining the sensitivity reports or

(hanging data in the model and re)solving it

!nalytic Solver Platform offers an alternativeapproach to sensitivity analysis called parameteranal#sis, which allows you to run multiple

optimizations while varying model parameterswithin predefined ranges.

,arameter *nal#sis inAnalytic Solver

Platform

Example 13 '1$ !ingle ,arameter


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Investigate the impact ofchanging finishing hourcapacity over a range from12 to 32. (hoose an empty cell in the

spreadsheet =e.g., ;#@

;rom!nalytic Solver Platformribbon, click Parametersbuttonand choose "%timization.

efine range in ;unctionErguments dialog.

Aeplace the value in cell by5;#

Example 13'1$ !ingle ,arameter

*nal#sis for the !!C ,roblem


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;rom the $e%ortsbutton in the

!nalysisgroup in the!nalyticSolver Platform ribbon, select"%timization $e%orts and thenParameter !nalysis

-ove decision variable and

ob&ective function cells to the right$esult Cells pane, and move theparameter cell ;# to the rightParameterspane.

In the drop)down bo", select +ary

!ll Selected ParametersSimultaneously/

'et 0aor !xis Points to numberof parameter values to test.



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Barameter analysis results

Aeformat the results to make them easier to understand.;or e"ample, name the columns with descriptive labelsinstead of cell references: use charts to visualize the

results.


Example 13 ''$ Multiple ,arameter


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'uppose that we wish to e"amine the effect on theoptimal profit of changing both the ;abrication and;inishing hour limitations, similar to a two)way

data table. ;ollow the procedure in /"ample 1#.!1 to define

the parameter for the ;inishing limitation. In the unction !r.uments dialog, set the range for the

;abrication limitation between 02 and 122.

Example 13''$ Multiple ,arameter

*nal#sis for the !!C ,roblem


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In the 0ulti%le "%timizations$e%ort dialog, choose bothparameter cells ;! and ;#:however, we can only chooseone result cell. In this case,

choose !!, which representsthe ob&ective function value. In the drop)down bo", select +ary

-o Selected Parameters

'nde%endently.

Example 13'' Continued


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Aesults

Example 13'' Continued

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