slides prepared by john s. loucks st. edward’s university

1 1© 2003 Thomson© 2003 Thomson/South-Western/South-Western Slide Slide

Slides Prepared bySlides Prepared by

JOHN S. LOUCKSJOHN S. LOUCKS

St. Edward’s UniversitySt. Edward’s University


Chapter 15Chapter 15Multicriteria Decision ProblemsMulticriteria Decision Problems

Goal ProgrammingGoal Programming Goal Programming: Formulation Goal Programming: Formulation

and Graphical Solutionand Graphical Solution Scoring ModelsScoring Models Analytic Hierarchy Process (AHP)Analytic Hierarchy Process (AHP) Establishing Priorities Using AHPEstablishing Priorities Using AHP Using AHP to Develop an Overall Priority Using AHP to Develop an Overall Priority

RankingRanking


Goal ProgrammingGoal Programming

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


Goal ProgrammingGoal Programming

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.


Goal Programming ApproachGoal Programming Approach

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 goal, for If a priority level has more than one goal, for each goal each goal ii decide the weight, decide the weight, wwi i , to , to

be placed be placed on the deviation(s), on the deviation(s), ddii++ and/or and/or ddii

--, , from the goal.from the goal.

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

reached.reached.


Goal Programming ApproachGoal Programming Approach

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

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


Example: Conceptual ProductsExample: Conceptual Products

Conceptual Products is a computer Conceptual Products is a computer company that produces the CP400 and the company that produces the CP400 and the CP500 computers. The computers use CP500 computers. The computers use different different mother boards produced in mother boards produced in abundant supply by the company, but use the abundant supply by the company, but use the same cases and disk drives. The CP400 same cases and disk drives. The CP400 models use two floppy disk drives and no zip models use two floppy disk drives and no zip disk drives whereas the CP500 models use one disk drives whereas the CP500 models use one floppy disk drive and one zip disk drive.floppy disk drive and one zip disk drive.



The disk drives and cases are bought from The disk drives and cases are bought from vendors. There are 1000 floppy disk drives, 500 vendors. There are 1000 floppy disk drives, 500 zip disk drives, and 600 cases available to zip disk drives, and 600 cases available to Conceptual Products on a weekly basis. It takes Conceptual Products on a weekly basis. It takes one hour to manufacture a CP400 and its profit is one hour to manufacture a CP400 and its profit is $200 and it takes one and one-half hours to $200 and it takes one and one-half hours to manufacture a CP500 and its profit is $500.manufacture a CP500 and its profit is $500.



The company has four goals which are given The company has four goals which are given below:below:

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. weekly. (Goal 2) (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 Priority 4: Use no more than 400 man-hours per per week. (Goal 4) week. (Goal 4)



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 is

deficient deficient

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

exceededexceeded

Functional ConstraintsFunctional Constraints

Availability of floppy disk drives: 2Availability of floppy disk drives: 2xx11 + + xx22 << 1000 1000

Availability of zip disk drives: Availability of zip disk drives: xx22 << 500 500

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



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



Objective FunctionsObjective Functions

Priority 1: Minimize the amount the state Priority 1: Minimize the amount the state contract is contract is not met: Min 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 Priority 3: Minimize the amount under $250,000 $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

++



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



Graphical Solution, Iteration 1Graphical Solution, Iteration 1

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

-- = 0).= 0).



Functional Constraints and Goal 1 GraphedFunctional Constraints and Goal 1 Graphed

10001000

800800

600600

400400

200200

200 400 600 800 1000 1200 200 400 600 800 1000 1200

22xx11 + + xx22 << 1000 1000

Goal 1: Goal 1: xx11 >> 200 200

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

Points SatisfyingPoints SatisfyingGoal 1Goal 1

xx11

xx22




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

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

-- = 0). = 0).



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

10001000

800800

600600

400400

200200

200 400 600 800 1000 1200 200 400 600 800 1000 1200

22xx11 + + xx22 << 1000 1000

Goal 1: Goal 1: xx11 >> 200 200

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

Points Satisfying BothPoints Satisfying BothGoals 1 and 2Goals 1 and 2

xx11

xx22

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




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

.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 points satisfy the previous functional constraints and goals and goals andand satisfy this constraint. satisfy this constraint.

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

achieved when we Max .2achieved when we Max .2xx11 + .5 + .5xx22. Note that . Note that this occurs at 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.



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

10001000

800800

600600

400400

200200

200 400 600 800 1000 1200 200 400 600 800 1000 1200

22xx11 + + xx22 << 1000 1000

Goal 1: Goal 1: xx11 >> 200 200

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

Points Satisfying BothPoints Satisfying BothGoals 1 and 2Goals 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)


A Scoring Model for Job SelectionA Scoring Model for Job Selection

A graduating college student with a double major A graduating college student with a double major in Finance and Accounting has received the in Finance and Accounting has received the following three job offers:following three job offers:

•financial analyst for an investment firm in financial analyst for an investment firm in ChicagoChicago

•accountant for a manufacturing firm in Denveraccountant for a manufacturing firm in Denver

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



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

•““The financial analyst position provides the The financial analyst position provides the best opportunity for my long-run career best opportunity for my long-run career advancement.”advancement.”

•““I would prefer living in Denver rather than in I would prefer living in Denver rather than in Chicago or Houston.”Chicago or Houston.”

•““I like the management style and philosophy I like the management style and philosophy at the Houston CPA firm the best.”at the Houston CPA firm the best.”

Clearly, this is a multicriteria decision problem.Clearly, this is a multicriteria decision problem.



Considering only the Considering only the long-run career long-run career advancementadvancement criterion: criterion:

•The The financial analyst position in Chicagofinancial analyst position in Chicago is the is the best decision alternative.best decision alternative.

Considering only the Considering only the locationlocation criterion: criterion:

•The The accountant position in Denveraccountant position in Denver is the best is the best decision alternative.decision alternative.

Considering only the Considering only the stylestyle criterion: criterion:

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



Steps Required to Develop a Scoring ModelSteps Required to Develop a Scoring Model

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 Rate how well each decision alternative 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 highest score to lowest score. The score to lowest score. The alternative with alternative with the highest score is the the highest score is the recommended recommended alternative.alternative.



Mathematical ModelMathematical Model

SSjj = = wwii r rijij

ii

where:where:

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

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



Step 1: List the criteria (important factors).Step 1: List the criteria (important factors).

•Career advancement Career advancement

•LocationLocation

•ManagementManagement

•SalarySalary

•PrestigePrestige

• Job SecurityJob Security

•Enjoyable workEnjoyable work



Five-Point Scale Chosen for Step 2Five-Point Scale Chosen for Step 2

ImportanceImportance WeightWeight

Very unimportantVery unimportant 11

Somewhat unimportantSomewhat unimportant 22

Average importanceAverage importance 33

Somewhat importantSomewhat important 44

Very importantVery important 55



Step 2: Assign a weight to each criterion.Step 2: Assign a weight to each criterion.

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



Nine-Point Scale Chosen for Step 3Nine-Point Scale Chosen for Step 3

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



Step 3: Step 3: RateRate how well each decision alternative how well each decision alternative satisfies each criterion.satisfies each criterion.

Decision AlternativeDecision Alternative Analyst AccountantAnalyst Accountant

AuditorAuditor CriterionCriterion ChicagoChicago DenverDenver HoustonHouston

Career advancementCareer advancement 88 66 44LocationLocation 33 88 77ManagementManagement 55 66 99SalarySalary 66 77 55PrestigePrestige 77 55 44Job securityJob security 44 77 66Enjoyable workEnjoyable work 88 66 55



Step 4: Compute the score for each decision Step 4: Compute the score for each decision alternative.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



Step 4: Compute the score for each decision Step 4: Compute the score for each decision alternative.alternative.

SSjj = = wwii r rijij

ii

SS11 = = 5(8)+3(3)+4(5)+3(6)+2(7)+4(4)+5(8) = 1575(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) = 1675(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) = 1495(4)+3(7)+4(9)+3(5)+2(4)+4(6)+5(5) = 149



Step 4: Compute the Step 4: Compute the scorescore for each decision for each decision alternative.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



Step 5: Order the decision alternatives from Step 5: Order the decision alternatives from highest highest score to lowest score. The score to lowest score. The alternative with the alternative with the highest score is the highest score is the recommended alternative.recommended alternative.

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

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

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



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



Partial Spreadsheet Showing Formulas for Step 4Partial Spreadsheet Showing Formulas for Step 4

A B C D E12 SCORING CALCULATIONS1314 Analyst Accountant Auditor15 Criteria Chicago Denver Houston16 Career Advance. =B4*C4 =B4*D4 =B4*E417 Location =B5*C5 =B5*D5 =B5*E518 Management =B6*C6 =B6*D6 =B6*E619 Salary =B7*C7 =B7*D7 =B7*E720 Prestige =B8*C8 =B8*D8 =B8*E821 Job Security =B9*C9 =B9*D9 =B9*E922 Enjoyable Work =B10*C10 =B10*D10 =B10*E1023 Score =sum(C16:C22) =sum(D16:D22) =sum(E16:E22)



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

A B C D E12 SCORING CALCULATIONS1314 Analyst Accountant Auditor15 Criteria Chicago Denver Houston16 Career Advance. 40 30 2017 Location 9 24 2118 Management 20 24 3619 Salary 18 21 1520 Prestige 14 10 821 Job Security 16 28 2422 Enjoyable Work 40 30 2523 Score 157 167 149


Analytic Hierarchy ProcessAnalytic Hierarchy Process

The The Analytic Hierarchy Process (AHP)Analytic Hierarchy Process (AHP), is a , is a procedure designed to quantify managerial procedure designed to quantify managerial judgments of the relative importance of each of judgments of the relative importance of each of several conflicting criteria used in the decision several conflicting criteria used in the decision making process.making process.



Step 1: List the Overall Goal, Criteria, and Step 1: List the Overall Goal, Criteria, and Decision Decision Alternatives Alternatives

Step 2: Develop a Pairwise Comparison MatrixStep 2: Develop a Pairwise Comparison Matrix

Rate the relative importance between each Rate the relative importance between each pair of decision alternatives. The matrix lists the pair of decision alternatives. The matrix lists the alternatives horizontally and vertically and has the alternatives horizontally and vertically and has the numerical ratings comparing the horizontal (first) numerical ratings comparing the horizontal (first) alternative with the vertical (second) alternative.alternative with the vertical (second) alternative.

Ratings are given as follows:Ratings are given as follows:

. . . continued. . . continued

------- For each criterion, perform steps 2 through 5 -------------- For each criterion, perform steps 2 through 5 -------------- For each criterion, perform steps 2 through 5 -------------- For each criterion, perform steps 2 through 5 -------



Step 2: Pairwise Comparison Matrix (continued)Step 2: Pairwise Comparison Matrix (continued)

Compared to the secondCompared to the secondalternative, the first alternative isalternative, the first alternative is: : Numerical Numerical ratingrating

extremely preferred extremely preferred 99

very strongly preferred very strongly preferred 77

strongly preferred strongly preferred 55

moderately preferred moderately preferred 33

equally preferred equally preferred 11



Step 2: Pairwise Comparison Matrix (continued)Step 2: Pairwise Comparison Matrix (continued)

Intermediate numeric ratings of 8, 6, 4, 2 Intermediate numeric ratings of 8, 6, 4, 2 can be assigned. A reciprocal rating (i.e. 1/9, can be assigned. A reciprocal rating (i.e. 1/9, 1/8, etc.) is assigned when the second 1/8, etc.) is assigned when the second alternative is preferred to the first. The value of alternative is preferred to the first. The value of 1 is always assigned when comparing an 1 is always assigned when comparing an alternative with itself. alternative with itself.



Step 3: Develop a Normalized MatrixStep 3: Develop a Normalized Matrix

Divide each number in a column of the Divide each number in a column of the pairwise comparison matrix by its column sum.pairwise comparison matrix by its column sum.

Step 4: Develop the Priority VectorStep 4: Develop the Priority Vector

Average each row of the normalized Average each row of the normalized matrix. These row averages form the priority matrix. These row averages form the priority vector of alternative preferences with respect vector of alternative preferences with respect to the particular criterion. The values in this to the particular criterion. The values in this vector sum to 1.vector sum to 1.



Step 5: Calculate a Consistency RatioStep 5: Calculate a Consistency Ratio

The consistency of the subjective input in The consistency of the subjective input in the pairwise comparison matrix can be the pairwise comparison matrix can be measured by calculating a consistency ratio. A measured by calculating a consistency ratio. A consistency ratio of less than .1 is good. For consistency ratio of less than .1 is good. For ratios which are greater than .1, the subjective ratios which are greater than .1, the subjective input should be re-evaluated.input should be re-evaluated.

Step 6: Develop a Priority MatrixStep 6: Develop a Priority Matrix

After steps 2 through 5 has been After steps 2 through 5 has been performed for all criteria, the results of step 4 performed for all criteria, the results of step 4 are summarized in a priority matrix by listing the are summarized in a priority matrix by listing the decision alternatives horizontally and the criteria decision alternatives horizontally and the criteria vertically. The column entries are the priority vertically. The column entries are the priority vectors for each criterion. vectors for each criterion.



Step 7: Develop a Criteria Pairwise Development Step 7: Develop a Criteria Pairwise Development Matrix Matrix

This is done in the same manner as that This is done in the same manner as that used to construct alternative pairwise used to construct alternative pairwise comparison matrices by using subjective ratings comparison matrices by using subjective ratings (step 2). Similarly, normalize the matrix (step 3) (step 2). Similarly, normalize the matrix (step 3) and develop a criteria priority vector (step 4). and develop a criteria priority vector (step 4).

Step 8: Develop an Overall Priority VectorStep 8: Develop an Overall Priority Vector

Multiply the criteria priority vector (from Multiply the criteria priority vector (from step 7) by the priority matrix (from step 6).step 7) by the priority matrix (from step 6).


Determining the Consistency RatioDetermining the Consistency Ratio

Step 1:Step 1: For each row of the pairwise comparison For each row of the pairwise comparison

matrix, determine a weighted sum by summing matrix, determine a weighted sum by summing the multiples of the entries by the priority of its the multiples of the entries by the priority of its corresponding (column) alternative.corresponding (column) alternative.

Step 2:Step 2: For each row, divide its weighted sum by For each row, divide its weighted sum by

the priority of its corresponding (row) alternative.the priority of its corresponding (row) alternative. Step 3:Step 3:

Determine the average, Determine the average, maxmax, of the results , of the results of step 2.of step 2.


Determining the Consistency RatioDetermining the Consistency Ratio

Step 4:Step 4:

Compute the consistency index, CI, of the Compute the consistency index, CI, of the nn alternatives by: CI = (alternatives by: CI = (maxmax - - nn)/()/(nn - 1). - 1).

Step 5:Step 5:

Determine the random index, RI, as follows:Determine the random index, RI, as follows:

Number of RandomNumber of Random Number of RandomNumber of Random Alternative (Alternative (nn)) Index (RI)Index (RI) Alternative (Alternative (nn)) Index (RI)Index (RI)

3 0.583 0.58 6 6 1.24 1.24 4 0.904 0.90 7 7 1.32 1.32 5 1.125 1.12 8 8 1.41 1.41

Step 6:Step 6:

Compute the consistency ratio: CR = CR/RI.Compute the consistency ratio: CR = CR/RI.


Example: Gill GlassExample: Gill Glass

Designer Gill Glass must decide which of Designer Gill Glass must decide which of three manufacturers will develop his three manufacturers will develop his "signature" toothbrushes. Three factors seem "signature" toothbrushes. Three factors seem important to Gill: (1) his costs; (2) reliability of important to Gill: (1) his costs; (2) reliability of the product; and, (3) delivery time of the the product; and, (3) delivery time of the orders.orders.

The three manufacturers are Cornell The three manufacturers are Cornell Industries, Brush Pik, and Picobuy. Cornell Industries, Brush Pik, and Picobuy. Cornell Industries will sell toothbrushes to Gill Glass for Industries will sell toothbrushes to Gill Glass for $100 per gross, Brush Pik for $80 per gross, and $100 per gross, Brush Pik for $80 per gross, and Picobuy for $144 per gross. Gill has decided Picobuy for $144 per gross. Gill has decided that in terms of price, Brush Pik is moderately that in terms of price, Brush Pik is moderately preferred to Cornell and very strongly preferred preferred to Cornell and very strongly preferred to Picobuy. In turn Cornell is strongly to very to Picobuy. In turn Cornell is strongly to very strongly preferred to Picobuy.strongly preferred to Picobuy.



Hierarchy for the Manufacturer Selection Hierarchy for the Manufacturer Selection ProblemProblem

Select the Best Toothbrush ManufacturerSelect the Best Toothbrush ManufacturerSelect the Best Toothbrush ManufacturerSelect the Best Toothbrush Manufacturer

CostCost CostCost ReliabilityReliabilityReliabilityReliability Delivery TimeDelivery TimeDelivery TimeDelivery Time

CornellCornellBrush PikBrush PikPicobuyPicobuy






Overall GoalOverall Goal

CriteriaCriteria

DecisionDecisionAlternativesAlternatives



Forming the Pairwise Comparison Matrix For CostForming the Pairwise Comparison Matrix For Cost

•Since Brush Pik is moderately preferred to Since Brush Pik is moderately preferred to Cornell, Cornell's entry in the Brush Pik row is Cornell, Cornell's entry in the Brush Pik row is 3 and Brush Pik's entry in the Cornell row is 3 and Brush Pik's entry in the Cornell row is 1/3.1/3.

•Since Brush Pik is very strongly preferred to Since Brush Pik is very strongly preferred to Picobuy, Picobuy's entry in the Brush Pik row Picobuy, Picobuy's entry in the Brush Pik row is 7 and Brush Pik's entry in the Picobuy row is is 7 and Brush Pik's entry in the Picobuy row is 1/7.1/7.

•Since Cornell is strongly to very strongly Since Cornell is strongly to very strongly preferred to Picobuy, Picobuy's entry in the preferred to Picobuy, Picobuy's entry in the Cornell row is 6 and Cornell's entry in the Cornell row is 6 and Cornell's entry in the Picobuy row is 1/6.Picobuy row is 1/6.



Pairwise Comparison Matrix for CostPairwise Comparison Matrix for Cost

Cornell Brush Pik PicobuyCornell Brush Pik Picobuy

CornellCornell 1 1/3 1 1/3 6 6

Brush PikBrush Pik 3 3 1 1 7 7

PicobuyPicobuy 1/6 1/6 1/7 1/7 1 1



Normalized Matrix for CostNormalized Matrix for Cost

Divide each entry in the pairwise comparison Divide each entry in the pairwise comparison matrix by its corresponding column sum. For matrix by its corresponding column sum. For example, for Cornell the column sum = 1 + 3 + 1/6 example, for Cornell the column sum = 1 + 3 + 1/6 = 25/6. This gives:= 25/6. This gives:


CornellCornell 6/25 7/31 6/25 7/31 6/14 6/14

Brush PikBrush Pik 18/25 21/31 18/25 21/31 7/14 7/14

PicobuyPicobuy 1/25 3/31 1/25 3/31 1/14 1/14


Priority Vector For CostPriority Vector For Cost

The priority vector is determined by The priority vector is determined by averaging the row entries in the normalized averaging the row entries in the normalized matrix. Converting to decimals we get:matrix. Converting to decimals we get:

Cornell: ( 6/25 + 7/31 + 6/14)/3 Cornell: ( 6/25 + 7/31 + 6/14)/3 = .298 = .298

Brush Pik: (18/25 + 21/31 + 7/14)/3 Brush Pik: (18/25 + 21/31 + 7/14)/3 = .632 = .632

Picobuy: ( 1/25 + 3/31 + 1/14)/3 Picobuy: ( 1/25 + 3/31 + 1/14)/3 = .069 = .069



Checking ConsistencyChecking Consistency

•Multiply each column of the pairwise Multiply each column of the pairwise comparison matrix by its priority:comparison matrix by its priority:

1 1/3 1 1/3 6 .923 6 .923

.298 3 + .632 1 + .069 7 = .298 3 + .632 1 + .069 7 = 2.009 2.009

1/6 1/7 1/6 1/7 1 .209 1 .209

•Divide these number by their priorities to get:Divide these number by their priorities to get:

.923/.298 = 3.097.923/.298 = 3.097

2.009/.632 = 3.1792.009/.632 = 3.179

.209/.069 = 3.029.209/.069 = 3.029





•Average the above results to get Average the above results to get maxmax..

maxmax = (3.097 + 3.179 + 3.029)/3 = 3.102 = (3.097 + 3.179 + 3.029)/3 = 3.102

•Compute the consistence index, CI, for two terms.Compute the consistence index, CI, for two terms.

CI = (CI = (maxmax - - nn)/()/(nn - 1) = (3.102 - 3)/2 = .051 - 1) = (3.102 - 3)/2 = .051

•Compute the consistency ratio, CR, by CI/RI, Compute the consistency ratio, CR, by CI/RI, where RI = .58 for 3 factors:where RI = .58 for 3 factors:

CR = CI/RI = .051/.58 = .088CR = CI/RI = .051/.58 = .088

Since the consistency ratio, CR, is less than .10, this Since the consistency ratio, CR, is less than .10, this is well within the acceptable range for consistency. is well within the acceptable range for consistency.



Gill Glass has determined that for Gill Glass has determined that for reliabilityreliability, Cornell is very strongly preferable to , Cornell is very strongly preferable to Brush Pik and equally preferable to Picobuy. Brush Pik and equally preferable to Picobuy. Also, Picobuy is strongly preferable to Brush Pik.Also, Picobuy is strongly preferable to Brush Pik.



Pairwise Comparison Matrix for ReliabilityPairwise Comparison Matrix for Reliability


CornellCornell 1 7 1 7 2 2

Brush PikBrush Pik 1/7 1/7 1 1 5 5

PicobuyPicobuy 1/2 1/2 1/5 1/5 1 1



Normalized Matrix for ReliabilityNormalized Matrix for Reliability

Divide each entry in the pairwise Divide each entry in the pairwise comparison matrix by its corresponding column comparison matrix by its corresponding column sum. For example, for Cornell the column sum = sum. For example, for Cornell the column sum = 1 + 1/7 + 1/2 = 23/14. This gives:1 + 1/7 + 1/2 = 23/14. This gives:


CornellCornell 14/23 35/41 14/23 35/41 2/8 2/8


PicobuyPicobuy 7/23 1/41 7/23 1/41 1/8 1/8



Priority Vector For ReliabilityPriority Vector For Reliability

The priority vector is determined by The priority vector is determined by averaging the row entries in the normalized matrix. averaging the row entries in the normalized matrix. Converting to decimals we get: Converting to decimals we get:

Cornell: (14/23 + 35/41 + 2/8)/3 = .571 Cornell: (14/23 + 35/41 + 2/8)/3 = .571

Brush Pik: ( 2/23 + 5/41 + 5/8)/3 = .278 Brush Pik: ( 2/23 + 5/41 + 5/8)/3 = .278

Picobuy: ( 7/23 + 1/41 + 1/8)/3 = .151 Picobuy: ( 7/23 + 1/41 + 1/8)/3 = .151


Gill Glass’ responses to reliability could be Gill Glass’ responses to reliability could be checked for consistency in the same manner as checked for consistency in the same manner as was cost.was cost.



Gill Glass has determined that for Gill Glass has determined that for delivery delivery timetime, Cornell is equally preferable to Picobuy. , Cornell is equally preferable to Picobuy. Both Cornell and Picobuy are very strongly to Both Cornell and Picobuy are very strongly to extremely preferable to Brush Pik.extremely preferable to Brush Pik.



Pairwise Comparison Matrix for Delivery TimePairwise Comparison Matrix for Delivery Time


CornellCornell 1 8 1 8 1 1

Brush PikBrush Pik 1/8 1/8 1 1 1/8 1/8

PicobuyPicobuy 1 1 8 8 1 1



Normalized Matrix for Delivery TimeNormalized Matrix for Delivery Time

Divide each entry in the pairwise Divide each entry in the pairwise comparison matrix by its corresponding column comparison matrix by its corresponding column sum. sum.


CornellCornell 8/17 8/17 8/17 8/17 8/17 8/17


PicobuyPicobuy 8/17 8/17 8/17 8/17 8/17 8/17



Priority Vector For Delivery TimePriority Vector For Delivery Time


Cornell: (8/17 + 8/17 + 8/17)/3 = .471 Cornell: (8/17 + 8/17 + 8/17)/3 = .471

Brush Pik: (1/17 + 1/17 + 1/17)/3 = .059 Brush Pik: (1/17 + 1/17 + 1/17)/3 = .059

Picobuy: (8/17 + 8/17 + 8/17)/3 = .471 Picobuy: (8/17 + 8/17 + 8/17)/3 = .471


Gill Glass’ responses to delivery time could Gill Glass’ responses to delivery time could be checked for consistency in the same manner be checked for consistency in the same manner as was cost.as was cost.



The accounting department has The accounting department has determined that in terms of determined that in terms of criteriacriteria, cost is , cost is extremely preferable to delivery time and very extremely preferable to delivery time and very strongly preferable to reliability, and that strongly preferable to reliability, and that reliability is very strongly preferable to delivery reliability is very strongly preferable to delivery time.time.



Pairwise Comparison Matrix for CriteriaPairwise Comparison Matrix for Criteria

Cost Reliability DeliveryCost Reliability Delivery

CostCost 1 7 1 7 9 9

ReliabilityReliability 1/7 1/7 1 1 7 7

DeliveryDelivery 1/9 1/9 1/7 1/7 1 1



Normalized Matrix for CriteriaNormalized Matrix for Criteria

Divide each entry in the pairwise Divide each entry in the pairwise comparison matrix by its corresponding column comparison matrix by its corresponding column sum.sum.

Cost Reliability Cost Reliability DeliveryDelivery

CostCost 63/79 49/57 63/79 49/57 9/179/17

ReliabilityReliability 9/79 7/57 9/79 7/57 7/177/17

DeliveryDelivery 7/79 1/57 7/79 1/57 1/17 1/17



Priority Vector For CriteriaPriority Vector For Criteria


Cost: Cost: (63/79 + 49/57 + 9/17)/3 (63/79 + 49/57 + 9/17)/3 = .729 = .729

Reliability: Reliability: ( 9/79 + 7/57 + 7/17)/3 ( 9/79 + 7/57 + 7/17)/3 = .216 = .216

Delivery: Delivery: ( 7/79 + 1/57 + 1/17)/3 ( 7/79 + 1/57 + 1/17)/3 = .055= .055


Overall Priority VectorOverall Priority Vector

The overall priorities are determined by The overall priorities are determined by multiplying the priority vector of the criteria by multiplying the priority vector of the criteria by the priorities for each decision alternative for the priorities for each decision alternative for each objective.each objective.

Priority VectorPriority Vector

for Criteriafor Criteria [ .729 .216 [ .729 .216 .055 ] .055 ]

Cost Reliability DeliveryCost Reliability Delivery

Cornell Cornell .298 .571 .298 .571 .471 .471

Brush PikBrush Pik .632 .278 .632 .278 .059 .059

PicobuyPicobuy .069 .151 .069 .151 .471 .471




Overall Priority Vector (continued)Overall Priority Vector (continued)

Thus, the overall priority vector is:Thus, the overall priority vector is:

Cornell:Cornell: (.729)(.298) + (.216)(.571) + (.055)(.729)(.298) + (.216)(.571) + (.055)(.471) = .366(.471) = .366

Brush Pik: (.729)(.632) + (.216)(.278) + (.055)Brush Pik: (.729)(.632) + (.216)(.278) + (.055)(.059) = .524(.059) = .524

Picobuy: (.729)(.069) + (.216)(.151) + (.055)Picobuy: (.729)(.069) + (.216)(.151) + (.055)(.471) = .109(.471) = .109

Brush Pik appears to be the overall Brush Pik appears to be the overall recommendation.recommendation.


End of Chapter 15End of Chapter 15

slides prepared by john s. loucks st. edward’s university

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