1 1 Slide © 2005 Thomson/South-Western Lesson 10 Multicriteria Decisions within LP Framework n Goal...

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© 2005 Thomson/South-Western© 2005 Thomson/South-Western

Lesson 10Lesson 10Multicriteria Decisions within LP FrameworkMulticriteria Decisions within LP Framework

Goal ProgrammingGoal Programming Goal Programming: Formulation Goal Programming: Formulation

and Graphical Solutionand Graphical Solution Scoring Model for Job SelectionScoring Model for Job Selection

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Goal ProgrammingGoal Programming

Goal programmingGoal programming may be used to solve may be used to solve linear programs with multiple objectives, linear programs with multiple objectives, with each objective viewed as a "goal". with each objective viewed as a "goal".

In goal programming, In goal programming, ddii++ and and ddii

-- , , deviation deviation variablesvariables, are the amounts a targeted goal , are the amounts a targeted goal ii is overachieved or underachieved, is overachieved or underachieved, respectively.respectively.

The goals themselves are added to the The goals themselves are added to the constraint set with constraint set with ddii

++ and and ddii-- acting as the acting as the

surplus and slack variables.surplus and slack variables.

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Goal ProgrammingGoal Programming

One approach to goal programming is to satisfy One approach to goal programming is to satisfy goals in a goals in a priority sequencepriority sequence. Second-priority . Second-priority goals are pursued without reducing the first-goals are pursued without reducing the first-priority goals, etc.priority goals, etc.

For each priority level, the objective function is For each priority level, the objective function is to minimize the (weighted) sum of the goal to minimize the (weighted) sum of the goal deviations. deviations.

Previous "optimal" achievements of goals are Previous "optimal" achievements of goals are added to the constraint set so that they are not added to the constraint set so that they are not degraded while trying to achieve lesser priority degraded while trying to achieve lesser priority goals. goals.

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Goal Programming FormulationGoal Programming Formulation

Step 1: Decide the priority level of each goal.Step 1: Decide the priority level of each goal.

Step 2: Decide the weight on each goal.Step 2: Decide the weight on each goal.

If a priority level has more than one If a priority level has more than one goal, for goal, for each goal each goal ii decide the decide the weight, weight, wwi i , to be placed , to be placed on the on the deviation(s), deviation(s), ddii

++ and/or and/or ddii--, from the goal., from the goal.

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Step 3: Set up the initial linear program.Step 3: Set up the initial linear program.

Min Min ww11dd11++ + + ww22dd22

--

s.t. Functional Constraints, s.t. Functional Constraints, and Goal Constraints and Goal Constraints

Step 4: Solve the current linear program.Step 4: Solve the current linear program.

If there is a lower priority level, go to If there is a lower priority level, go to step 5. step 5. Otherwise, a final solution Otherwise, a final solution has been reached.has been reached.

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Step 5: Set up the new linear program.Step 5: Set up the new linear program.

Consider the next-lower priority level goals Consider the next-lower priority level goals and formulate a new objective function based on and formulate a new objective function based on these goals. Add a constraint requiring the these goals. Add a constraint requiring the achievement of the next-higher priority level achievement of the next-higher priority level goals to be maintained. goals to be maintained. The new linear The new linear program might be:program might be:

Min Min ww33dd33++ + + ww44dd44

--

s.t. Functional Constraints,s.t. Functional Constraints, Goal Constraints, andGoal Constraints, and

ww11dd11++ + + ww22dd22

-- = = kk

Go to step 4. (Repeat steps 4 and 5 until Go to step 4. (Repeat steps 4 and 5 until all priority levels have been examined.) all priority levels have been examined.)

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Example: Conceptual ProductsExample: Conceptual Products

Conceptual Products is a computer Conceptual Products is a computer company thatcompany that

produces the CP400 and CP500 computers. Theproduces the CP400 and CP500 computers. The

computers use differentcomputers use different

mother boards producedmother boards produced

in abundant supply by thein abundant supply by the

company, but use the samecompany, but use the same

cases and disk drives. Thecases and disk drives. The

CP400 models use two floppy disk drives and no CP400 models use two floppy disk drives and no zipzip

disk drives whereas the CP500 models use onedisk drives whereas the CP500 models use one

floppy disk drive and one zip disk drive.floppy disk drive and one zip disk drive.

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Conceptual Products is a computer company Conceptual Products is a computer company thatthatproduces the CP400 and CP500 computers. Many ofproduces the CP400 and CP500 computers. Many ofthe components used in the twothe components used in the two

computer models are produced incomputer models are produced inabundant supply by the abundant supply by the company.company.However, the memory modules,However, the memory modules,external hard drives, and cases areexternal hard drives, and cases arebought from suppliers.bought from suppliers.

The CP400 model uses two memory modules The CP400 model uses two memory modules andandno external hard drive, whereas the CP500 uses oneno external hard drive, whereas the CP500 uses onememory module and one external hard drive. Bothmemory module and one external hard drive. Bothmodels use one case.models use one case.

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Suppliers can provide Conceptual Products withSuppliers can provide Conceptual Products with

1000 memory modules, 500 external hard drives, and1000 memory modules, 500 external hard drives, and

600 cases on a weekly basis. It takes one hour to600 cases on a weekly basis. It takes one hour to

manufacture a CP400 and its profit is $200 and it takesmanufacture a CP400 and its profit is $200 and it takes

one and one-half hours to manufacture a CP500 andone and one-half hours to manufacture a CP500 and

its profit is $500.its profit is $500.

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The company has four goals:The company has four goals:

Priority 1: Meet a state contract of 200 CP400 Priority 1: Meet a state contract of 200 CP400 machines weekly. (Goal 1) machines weekly. (Goal 1)

Priority 2: Make at least 500 total computers Priority 2: Make at least 500 total computers weekly. (Goal 2) weekly. (Goal 2)

Priority 3: Make at least $250,000 weekly. Priority 3: Make at least $250,000 weekly. (Goal 3)(Goal 3)

Priority 4: Use no more than 400 man-hours per Priority 4: Use no more than 400 man-hours per week. (Goal 4) week. (Goal 4)

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VariablesVariables

xx11 = number of CP400 computers produced weekly= number of CP400 computers produced weekly

xx22 = number of CP500 computers produced weekly= number of CP500 computers produced weekly

ddii- - = amount the right hand side of goal = amount the right hand side of goal ii is deficient is deficient

ddii++ = amount the right hand side of goal = amount the right hand side of goal ii is exceeded is exceeded

Functional ConstraintsFunctional Constraints

Availability of memory modules: 2Availability of memory modules: 2xx11 + + xx22 << 1000 1000

Availability of external hard drives: Availability of external hard drives: xx22 << 500 500

Availability of cases:Availability of cases: xx11 + + xx22 << 600 600

Goal Programming: FormulationGoal Programming: Formulation

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GoalsGoals

(1) 200 CP400 computers weekly: (1) 200 CP400 computers weekly:

xx11 + + dd11-- - - dd11

++ = 200 = 200

(2) 500 total computers weekly: (2) 500 total computers weekly:

xx11 + + xx22 + + dd22-- - - dd22

++ = 500 = 500

(3) $250(in thousands) profit:(3) $250(in thousands) profit:

.2.2xx11 + .5 + .5xx22 + + dd33-- - - dd33

++ = 250 = 250

(4) 400 total man-hours weekly: (4) 400 total man-hours weekly:

xx11 + 1.5 + 1.5xx22 + + dd44-- - - dd44

++ = 400 = 400

Non-negativity: Non-negativity:

xx11, , xx22, , ddii--, , ddii

++ >> 0 for all 0 for all ii


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Objective FunctionsObjective Functions

Priority 1: Minimize the amount the state Priority 1: Minimize the amount the state contract contract is not met: Min is not met: Min dd11

--

Priority 2: Minimize the number under 500 Priority 2: Minimize the number under 500 computers produced weekly: computers produced weekly:

Min Min dd22--

Priority 3: Minimize the amount under $250,000 Priority 3: Minimize the amount under $250,000 earned weekly: Min earned weekly: Min dd33

--

Priority 4: Minimize the man-hours over 400 Priority 4: Minimize the man-hours over 400 used used weekly: Min weekly: Min dd44

++


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Formulation SummaryFormulation Summary

Min Min PP11((dd11--) + ) + PP22((dd22

--) + ) + PP33((dd33--) + ) + PP44((dd44

++))

s.t. 2s.t. 2xx11 + +xx22 << 1000 1000

++xx22 << 500 500

xx11 + +xx22 << 600 600

xx11 + +dd11-- - -dd11

++ = 200 = 200

xx11 + +xx22 + +dd22-- - -dd22

++ = 500 = 500

.2.2xx11+ .5+ .5xx22 + +dd33-- - -dd33

+ + = 250 = 250

xx11+1.5+1.5xx22 + +dd44-- - -dd44

+ + = 400= 400

xx11, , xx22, , dd11--, , dd11

++, , dd22--, , dd22

++, , dd33--, , dd33

++, , dd44--, , dd44

++ >> 0 0


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Iteration 1Iteration 1

To solve graphically, first graph the functional To solve graphically, first graph the functional constraints. Then graph the first goal: constraints. Then graph the first goal: xx11 = 200. = 200. Note on the next slide that there is a set of points Note on the next slide that there is a set of points that exceed that exceed xx11 = 200 (where = 200 (where dd11

-- = 0). = 0).

Goal Programming:Goal Programming:Graphical SolutionGraphical Solution

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Functional Constraints and Goal 1 GraphedFunctional Constraints and Goal 1 Graphed

22xx11 + + xx22 << 1000 1000

Goal 1: Goal 1: xx11 >> 200 200

xx11 + + xx22 << 600 600xx2 2 << 500 500

PointsPointsSatisfyingSatisfyingGoal 1Goal 1

xx11

xx22


10001000

800800

600600

400400

200200

200 400 600 800 1000 1200200 400 600 800 1000 1200

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Now add Goal 1 as Now add Goal 1 as xx11 >> 200 and graph Goal 2: 200 and graph Goal 2:

xx11 + + xx22 = 500. Note on the next slide that there is = 500. Note on the next slide that there is still a set of points satisfying the first goal that also still a set of points satisfying the first goal that also satisfies this second goal (where satisfies this second goal (where dd22

-- = 0). = 0).


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Goal 1 (Constraint) and Goal 2 GraphedGoal 1 (Constraint) and Goal 2 Graphed

22xx11 + + xx22 << 1000 1000

Goal 1: Goal 1: xx11 >> 200 200

xx11 + + xx22 << 600 600xx2 2 << 500 500

Points SatisfyingPoints SatisfyingBoth Goals 1 and 2Both Goals 1 and 2

xx11

xx22

Goal 2: Goal 2: xx11 + + xx22 >> 500 500


200 400 600 800 1000 1200200 400 600 800 1000 1200

10001000

800800

600600

400400

200200

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Now add Goal 2 as Now add Goal 2 as xx11 + + xx22 >> 500 and Goal 3: 500 and Goal 3:

.2.2xx11 + .5 + .5xx22 = 250. Note on the next slide that no = 250. Note on the next slide that no points satisfy the previous functional constraints and points satisfy the previous functional constraints and goals goals andand satisfy this constraint. satisfy this constraint.

Thus, to Min Thus, to Min dd33--, this minimum value is achieved , this minimum value is achieved

when we Max .2when we Max .2xx11 + .5 + .5xx22. Note that this occurs at . Note that this occurs at xx11 = 200 and = 200 and xx22 = 400, so that .2 = 400, so that .2xx11 + .5 + .5xx22 = 240 or = 240 or dd33

-- = = 10.10.


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Goal 2 (Constraint) and Goal 3 GraphedGoal 2 (Constraint) and Goal 3 Graphed

22xx11 + + xx22 << 1000 1000

Goal 1: Goal 1: xx11 >> 200 200

xx11 + + xx22 << 600 600 xx2 2 << 500 500

Points SatisfyingPoints SatisfyingBoth Goals 1 and 2Both Goals 1 and 2

xx11

xx22

Goal 2: Goal 2: xx11 + + xx22 >> 500 500

Goal 3: .2Goal 3: .2xx11 + .5 + .5xx22 = 250 = 250

(200,400)(200,400)


200 400 600 800 1000 1200200 400 600 800 1000 1200

10001000

800800

600600

400400

200200

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Scoring Model for Job SelectionScoring Model for Job Selection

A graduating college student with a double A graduating college student with a double majormajor

in Finance and Accounting has receivedin Finance and Accounting has received

the following three job offers:the following three job offers:

• financial analyst for an investmentfinancial analyst for an investment

firm in Chicagofirm in Chicago

• accountant for a manufacturingaccountant for a manufacturing

firm in Denverfirm in Denver

• auditor for a CPA firm in Houstonauditor for a CPA firm in Houston

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The student made the following The student made the following comments:comments:

• ““The financial analyst positionThe financial analyst position

provides the best opportunity for myprovides the best opportunity for my

long-run career advancement.”long-run career advancement.”

• ““I would prefer living in DenverI would prefer living in Denver

rather than in Chicago or Houston.”rather than in Chicago or Houston.”

• ““I like the management style andI like the management style and

philosophy at the Houston CPA firmphilosophy at the Houston CPA firm

the best.”the best.” Clearly, this is a multicriteria decision.Clearly, this is a multicriteria decision.

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Considering only the Considering only the long-run careerlong-run career

advancementadvancement criterion: criterion:

• The The financial analyst position infinancial analyst position in

ChicagoChicago is the best decision alternative. is the best decision alternative. Considering only the Considering only the locationlocation criterion: criterion:

• The The accountant position in Denveraccountant position in Denver

is the best decision alternative.is the best decision alternative. Considering only the Considering only the stylestyle criterion: criterion:

• The The auditor position in Houstonauditor position in Houston is is the the best alternative.best alternative.

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Steps Required to Develop a Scoring Steps Required to Develop a Scoring ModelModel

Step 1:Step 1: List the decision-making criteria. List the decision-making criteria. Step 2:Step 2: Assign a weight to each criterion. Assign a weight to each criterion. Step 3:Step 3: Rate how well each decision alternative Rate how well each decision alternative

satisfies each criterion.satisfies each criterion. Step 4:Step 4: Compute the score for each decision Compute the score for each decision

alternative.alternative. Step 5:Step 5: Order the decision alternatives from Order the decision alternatives from

highest score to lowest score. highest score to lowest score. The The alternative with the highest alternative with the highest score is the score is the recommended recommended alternative.alternative.

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Mathematical ModelMathematical Model

SSjj = = wwii r rijij

ii

where:where:

rrijij = rating for criterion = rating for criterion ii and decision alternative and decision alternative jj

SSjj = = score for decision alternativescore for decision alternative j j


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Scoring Model: Step 1Scoring Model: Step 1

List of CriteriaList of Criteria

• Career advancement Career advancement

• LocationLocation

• ManagementManagement

• SalarySalary

• PrestigePrestige

• Job SecurityJob Security

• Enjoyable workEnjoyable work

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Five-Point Scale ChosenFive-Point Scale Chosen

ImportanceImportance WeightWeight

Very unimportantVery unimportant 11

Somewhat unimportantSomewhat unimportant 22

Average importanceAverage importance 33

Somewhat importantSomewhat important 44

Very importantVery important 55

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Assigning a Weight to Each CriterionAssigning a Weight to Each Criterion

CriterionCriterion ImportanceImportance WeightWeight Career advancementCareer advancement Very importantVery important 55 LocationLocation Average importanceAverage importance 33 ManagementManagement Somewhat importantSomewhat important 44 SalarySalary Average importanceAverage importance 33 PrestigePrestige Somewhat unimportantSomewhat unimportant 22 Job securityJob security Somewhat importantSomewhat important 44 Enjoyable workEnjoyable work Very importantVery important 55

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Nine-Point Scale ChosenNine-Point Scale Chosen

Level of SatisfactionLevel of Satisfaction RatingRating

Extremely lowExtremely low 11 Very lowVery low 22 LowLow 33 Slightly lowSlightly low 44 AverageAverage 55

Slightly highSlightly high 66 HighHigh 77 Very highVery high 88 Extremely highExtremely high 99


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RateRate how well each decision alternative satisfies how well each decision alternative satisfies each criterion. each criterion.

Decision AlternativeDecision Alternative Analyst AccountantAnalyst Accountant Auditor Auditor CriterionCriterion ChicagoChicago DenverDenver HoustonHouston

Career advancementCareer advancement 88 6 6 4 4LocationLocation 33 8 8 7 7ManagementManagement 55 6 6 9 9SalarySalary 66 7 7 5 5PrestigePrestige 77 5 5 4 4Job securityJob security 44 7 7 6 6Enjoyable workEnjoyable work 88 6 6 5 5


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Compute the score for each decision alternative.Compute the score for each decision alternative.

Decision Alternative 1 - Analyst in ChicagoDecision Alternative 1 - Analyst in Chicago

CriterionCriterion Weight ( Weight (wwi i ) Rating () Rating (rrii11) ) wwiirrii11

Career advancementCareer advancement 5 5 x x 8 8 = = 4040LocationLocation 3 3 3 3 9 9ManagementManagement 4 4 5 5 2020SalarySalary 3 3 6 6 1818PrestigePrestige 2 2 7 7 1414Job securityJob security 4 4 4 4 1616Enjoyable workEnjoyable work 5 5 8 8 4040

ScoreScore 157 157


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Compute the score for each decision alternative.Compute the score for each decision alternative.

SS11 = 5(8)+3(3)+4(5)+3(6)+2(7)+4(4)+5(8) = 157 = 5(8)+3(3)+4(5)+3(6)+2(7)+4(4)+5(8) = 157

SS22 = 5(6)+3(8)+4(6)+3(7)+2(5)+4(7)+5(6) = 167 = 5(6)+3(8)+4(6)+3(7)+2(5)+4(7)+5(6) = 167

SS33 = 5(4)+3(7)+4(9)+3(5)+2(4)+4(6)+5(5) = 149 = 5(4)+3(7)+4(9)+3(5)+2(4)+4(6)+5(5) = 149

j i iji

s wrj i iji

s wr


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Compute the Compute the scorescore for each decision alternative. for each decision alternative.

Decision AlternativeDecision Alternative Analyst AccountantAnalyst Accountant

AuditorAuditor CriterionCriterion ChicagoChicago DenverDenver HoustonHouston

Career advancementCareer advancement 4040 3030 2020LocationLocation 9 9 2424 2121ManagementManagement 2020 2424 3636SalarySalary 1818 2121 1515PrestigePrestige 1414 1010 8 8Job securityJob security 1616 2828 2424Enjoyable workEnjoyable work 4040 3030 2525

ScoreScore 157 157 167 167 149 149


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Order the decision alternatives from highestOrder the decision alternatives from highestscore to lowest score. The alternative with the highestscore to lowest score. The alternative with the highestscore is the recommended alternative.score is the recommended alternative.

• The The accountant position in Denveraccountant position in Denver has the highest has the highest score and is the score and is the recommended decision alternativerecommended decision alternative..

• Note that the analyst position in Chicago ranks first Note that the analyst position in Chicago ranks first in 4 of 7 criteria compared to only 2 of 7 for the in 4 of 7 criteria compared to only 2 of 7 for the accountant position in Denver.accountant position in Denver.

• But when the weights of the criteria are considered, But when the weights of the criteria are considered, the Denver position is superior to the Chicago job.the Denver position is superior to the Chicago job.


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Partial Spreadsheet Showing Steps 1 - 3Partial Spreadsheet Showing Steps 1 - 3

A B C D E1 RATINGS2 Analyst Accountant Auditor3 Criteria Weight Chicago Denver Houston4 Career Advance. 5 8 6 45 Location 3 3 8 76 Management 4 5 6 97 Salary 3 6 7 58 Prestige 2 7 5 49 Job Security 4 4 7 610 Enjoyable Work 5 8 6 5

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Partial Spreadsheet Showing Formulas of Step 4Partial Spreadsheet Showing Formulas of Step 4

A B C D E12 SCORING CALCULATIONS13 Analyst Accountant Auditor14 Criteria Chicago Denver Houston15 Career Advance. =B4*C4 =B4*D4 =B4*E416 Location =B5*C5 =B5*D5 =B5*E517 Management =B6*C6 =B6*D6 =B6*E618 Salary =B7*C7 =B7*D7 =B7*E719 Prestige =B8*C8 =B8*D8 =B8*E820 Job Security =B9*C9 =B9*D9 =B9*E921 Enjoyable Work =B10*C10 =B10*D10 =B10*E1022 Score =sum(C16:C22) =sum(D16:D22) =sum(E16:E22)

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Partial Spreadsheet Showing Results of Step 4Partial Spreadsheet Showing Results of Step 4

A B C D E12 SCORING CALCULATIONS13 Analyst Accountant Auditor14 Criteria Chicago Denver Houston15 Career Advance. 40 30 2016 Location 9 24 2117 Management 20 24 3618 Salary 18 21 1519 Prestige 14 10 820 Job Security 16 28 2421 Enjoyable Work 40 30 2522 Score 157 167 149

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End of Lesson 10End of Lesson 10

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