*Analytic hierarchy process

*What is AHP? A Process that Leads One to (Saaty, 1980) :Structure a problem as a hierarchy or as a system with dependence loops Elicit judgments that reflect ideas, feelings, and emotions Represent those judgments with meaningful numbers Synthesize results Analyze sensitivity to changes in judgments AHP also uses a weighted average approach idea, but it uses a method for assigning ratings and weights that is considered more reliable and consistent

*Purposes of AHP To structure complexity in gradual steps from the large to the small, or from the general to the particular, so we can relate them with greater accuracy according to our understanding To improve our awareness by richer synthesis of our knowledge and intuition; AHP is a learning tool rather than a means to discover the TRUTH .

*Phases in AHP Phase 1: Decompose the problem into a hierarchy 1 : Start with an identification of the criteria to be used in evaluating different alternatives, organized in a tree-like hierarchy Phase 2: Collect input data by pairwise comparisons of criteria at each level of the hierarchy and alternatives 2 : Phase 3: Estimate the relative importance (weights) of criteria and alternatives and check the consistency in the pair wise comparisons 3 : () Phase 4: Aggregate the relative weights of criteria and alternatives to obtain a global ranking of each alternative with regards to the goal 4 :

*Hierarchy GOALCRITERIAALTERNATIVES

*CRITERIAALTERNATIVESMaximize Overall Engineer-Job Satisfaction Research GrowthBenefitsLocationColleagues Job AReputationJob BJob CGOAL Hierarchy for a Engineer-Job Selection Decision

*Hierarchy (Cont.)How to Structure a Hierarchy Identify the overall objective or goalIdentify criteria to satisfy the goalIdentify, where appropriate, sub-criteria under each criterionIdentify alternatives to be evaluated in terms of the sub-criteria at the lowest levelIf the relative importance of the sub-criteria can be assessed and the alternatives can be evaluated in terms of the sub-criteria, the hierarchy is finishedOtherwise, continue inserting levels until it is possible to link levels and set priorities (relative weights) on the elements at each level in terms of the elements at the level above it ( )

*Structure of a Hierarchy

*Energy Decision in the Government

*Hierarchy (Cont.)How Large Should a Hierarchy Be?Large enough to capture decision makers major concernsSmall enough to remain sensitive to change in what is important

*Judgment and Preference In AHP, we use subjective judgment to express preference and its intensity e.g. Which of two apples is more red and how strongly more red we perceive it to be From this preference we derive a scale of relative strength of preference

*Define the relative importance of criteria at each level of the hierarchy and relative importance of alternatives by means of pairwise comparisons Pairwise Comparison Pairwise Comparison Matrix (Wi is the relative weight of criterion (alternative))

Criterion (Alternative) 1Criterion (Alternative) 2 Criterion (Alternative) nCriterion (Alternative) 1W1/W1W1/W2W1/WnCriterion (Alternative) 2W2/W1W2/W2W2/WnCriterion (Alternative) nWn/W1Wn/W2Wn/Wn

*Pairwise Comparison (Cont.) Scale for Pairwise Comparisons1. Equally preferred3. One is moderately preferred over the other5. One is strongly preferred over the other 7. One is very strongly preferred over the other 9. One is extremely preferred over the other 2,4,6,8 intermediate valuesReciprocals for inverse comparison 1. 3. 5. 7. 9. 2468

*Relative Weights Denote the pairwise comparison matrix and the weight matrix as Then Or is the eigenvalue of A and W is its corresponding right eigenvector A W There are n eigenvalues and n corresponding eigenvectors for Anxn

*Relative Weights (Cont.) ConsistencyTransitivitya1>a2 and a2>a3, then a1>a3Measurement consistencyaij ajk = aik (aij the cell at the ith row and jth column of the comparison matrix)Consistency Index (CI)A measure of deviation of consistency max is the maximum eigenvalue of the pairwise comparison matrix CI = 0 or max = n implies perfect consistency CI = 0.1 is the generally accepted threshold value

*Find Eigenvalues and Eigenvectors, whereThe solution to the equation is given byYou can solve eigenvalue and eigenvector problems in Matlab using the command [V,E]=eig(A), where E is the eigenvalue and V is the corresponding eigenvector

*Suppose the comparison matrix of three criteria isThere is only one non-zero real eigenvalue for the pairwise comparison matrix in AHPIts corresponding eigenvector is (0.9161, 0.3715, 0.1506)T; the three numbers in the eigenvector are proportional to the relative weights of the three criteriamax =3.0385eigenvectorseigenvalue

*Relative Weights (Cont.)Because relative weights must sum up to 1, we have to normalize the eigenvector by dividing each number in it by the sum of all numberse.g. In the previous example, the eigenvector is (0.9161, 0.3715, 0.1506)TThe normalized eigenvector is =(0.64, 0.26, 0.10)T Relative weights for the three criteriaCI=(max-n)/(n-1) = (3.0385-3) /(3-1) = 0.0193

*Training Program Selection Example

*Pairwise Comparison Matrix of Six Criteria(L,F,SL,VT,CP, and MC denote Learning, Friends, School Life, Vocational Training, College Preparation, and Music Classes, respectively)

LFSLVTCPMCL143134F1/4111/312SL1/3111/51/22VT135113CP1/312112MC1/41/21/21/31/21

*Using Matlab, we can get max =6.2397, and its corresponding eigenvector is ( -0.6773, -0.2198, -0.1908, -0.5780, -0.3212, 0.1389)T The normalized eigenvector is (0.32, 0.10, 0.09, 0.27, 0.15, 0.07)TCI =( 6.2397 6) / (6 1) = 0.048 < 0.1

*

LFSLVTCPMCWeightL1431340.32F1/41731/510.10SL1/31/711/51/51/60.09VT11/35111/30.27CP1/3551130.15MC1/41631/310.07

*Pairwise Comparison Matrix of Three Alternative Programs With Respect to Learningmax=3.0536, , and the normalized eigenvector is (0.16, 0.59, 0.25)TCI =( 3.0536 3) / (3 1) = 0.027

ABCA11/31/2B313C21/31

*Pairwise Comparison Matrix of Three Alternative Programs With Respect to Friendsmax=3, and the normalized eigenvector is (1/3, 1/3, 1/3)TCI =( 3 3) / (3 1) = 0

ABCA111B111C111

*Pairwise Comparison Matrix of Three Alternative Programs With Respect to School Lifemax=3, and the normalized eigenvector is (0.46, 0.09, 0.46)TCI =( 3 3) / (3 1) = 0

ABCA151B1/511/5C151

*Pairwise Comparison Matrix of Three Alternative Programs With Respect to Vocational TrainingCI =( 3.2085 3) / (3 1) = 0.104max=3.2085, and the normalized eigenvector is (0.77, 0.05, 0.17)T

ABCA197B1/911/5C1/751

*Pairwise Comparison Matrix of Three Alternative Schools With Respect to College PreparationCI =( 3 3) / (3 1) = 0max=3, and the normalized eigenvector is (0.25, 0.50, 0.25)T

ABCA11/21B212C11/21

*Pairwise Comparison Matrix of Three Alternative Programs With Respect to Music ClassesCI =( 3.0536 3) / (3 1) = 0.027max=3.053, and the normalized eigenvector is (0.69, 0.09, 0.22)T

ABCA164B1/611/3C1/431

*Relative Weights (Cont.)Other than computing the eigenvector of a pairwise comparison matrix to find the weights of compared criteria or alternatives, we can also approximate the weight by:First, normalizing each column in the comparison matrixThen, calculating the average of each row in the normalized matrix as the estimate of the relative weight for its corresponding criterion or alternative

*T1T1T1Approximated WeightConsistency Measure

Criterion (Alternative) 1Criterion (Alternative) 2 Criterion (Alternative) nCriterion (Alternative) 1W1/W1W1/W2W1/WnCriterion (Alternative) 2W2/W1W2/W2W2/WnCriterion (Alternative) nWn/W1Wn/W2Wn/Wn

*Pairwise Comparison Matrix of Six CriterionSum3.1710.5012.503.877.0014.00Program Selection Example (Cont.)

LFSLVTCPMCL143134F1/4111/312SL1/3111/51/22VT135113CP1/312112MC1/41/21/21/31/21

*Normalized Pairwise Comparison Matrix of Six CriteriaApproximated Weight0.320.100.090.270.150.07Eigenvector Weight0.320.100.090.270.150.07

LFSLVTCPMCL1/3.17=0.324/10.5=0.383/12.5=0.241/3.87=0.263/7=0.434/14=0.29F(1/4)/3.17=0.081/10.5=0.101/12.5=0.08(1/3)/3.87=0.091/7=0.142/14=0.14SL(1/3)/3.17=0.111/10.5=0.101/12.5=0.08(1/5)/3.87=0.05(1/2)/7=0.072/14=0.14VT1/3.17=0.323/10.5=0.295/12.5=0.41/3.87=0.261/7=0.143/14=0.21CP(1/3)/3.17=0.111/10.5=0.102/12.5=0.161/3.87=0.261/7=0.142/14=0.14MC(1/4)/3.17=0.08(1/2)/10.5=0.05(1/2)/12.5=0.04(1/3)/3.87=0.09(1/2)/7=0.071/14=0.07

*Consistency MeasureCM2=Likewise, we can calculate CM3, CM4, CM5, and CM66.226.56.296.306.315.86CI CI =0.048 from eigenvalueApproximated Weight0.320.100.090.270.150.07

LFSLVTCPMCL143134F1/4111/312SL1/3111/51/22VT135113CP1/312112MC1/41/21/21/31/21

*Pairwise Comparison Matrix of Three Alternative Programs With Respect to LearningSum61.674.5Approximated Weight0.160.590.25Normalized Pairwise Comparison Matrix of Three Alternative Programs With Respect to Learning

ABCA11/31/2B313C21/31

ABCA1/6=0.17(1/3)/1.67=0.20(1/2)/4.5=0.11B3/6=0.501/1.67=0.603/4.5=0.67C2/6=0.33(1/3)/1.67=0.201/4.5=0.22

*Approximated Weight0.160.590.25CM1=CM2=CM3=Consistency Measure3.013.083.07CI

ABCA11/31/2B313C21/31

*Composition and Synthesis Combine the relative importance of criteria and alternatives to obtain a global ranking of each alternative with regards to the goal

Criteria Cj (j=1,2,,n) and their corresponding weightsWeights of alternatives Ai (i=1,2,,m) w.r.t. criteria Cj (j=1,2,,n)Composite impact OA1 = wC1wA1C1+wC2wA1C2++wCnwA1CnOAm = wC1wAmC1+wC2wAmC2++wCnwCnAmOA2 = wC1wA2C1+wC2wA2C2++wCnwA2Cn

wC1wC2wCnC1C2CnA1wA1C1wA1C2wA1CnA2wA2C1wA2C2wA2CnAmwAmC1wAmC2wAmCn

*In conclusion, Program A seems to be the best, and C seems to be the worstOA=0.32*0.16+0.10*0.33+0.09*0.45+0.27*0.77+0.15*0.25+0.07*0.69 = 0.42 OB=0.32*0.59+0.10*0.33+0.09*0.09+0.27*0.05+0.15*0.5+0.07*0.09 = 0.33 OC=0.32*0.25+0.10*0.33+0.09*0.46+0.27*0.17+0.15*0.25+0.07*0.22 = 0.25

0.320.100.090.270.150.07Composite Impact of ProgramsLFSLVTCPMCA0.160.330.450.770.250.690.42B0.590.330.090.050.50.090.33C0.250.330.460.170.250.220.25

*Summary of AHPApplications of AHPe.g. resource allocation , conflict resolution, prediction, planning, etc.AdvantagesDecision hierarchy and pairwise comparisons make the AHP process easy to comprehendThe use of a subjective scale, such as strongly preferred, rather than a quantitative scale is particularly useful when it is difficult to formalize some criteria (attributes) quantitativelyIt is usually much easier to compare two items at a time than to compare many items all at onceDisadvantages The decision hierarchy in AHP assumes independence among criteria, which is not always appropriateThe subjective scale is subject to human errors and biasesThe number of pairwise comparisons becomes quite extensive when the number of attributes and alternatives is large

What is Pre-Qualification?Is the process of determining how much a prospective bidder will be eligible to bid before a tender is applied for.It is like an initial interview, for both sides of the potential contract.It helps answering almost half of the critical information questions for Contract B

* | Slide # *Qualification CriteriaSome of the critical information for Contract B are:QualificationExperience of staffAvailability of servicesDelivery schedule for goodsFinancial stabilityCompatibility with Existing Systems

* | Slide # *Need for Pre-Qualification.One of the prime difficulties with tendering for services is determining what the tendering organization is going to receive.To reduce the risks of intangibility, contract managers normally rely upon their own experience and the knowledge of othersDoes the provider have a good reputation with other companies, clients, or government ministries?Pre-Qualification of service providers becomes critically important. (CIVE 612 Lecture)

The ChallengesCompanies managers want: To consider only those who are qualified for the job.To make sure that the prequalification process is unbiasedWeighting bidders qualification based upon their performance functions, not on the personal relationship with the organization

The ApplicationThe companys contract manager wants: To select two qualified bidders out of four bidders who were interested in bidding for the undergoing organization project.To use the AHP technique based on the organization policy criteria in selecting the qualified bidders.

Why AHP?AHP is a technique that can be used to facilitate subjective decision makingIt answer the question Which one do we choose? or Which one is best ? by selecting the best alternative that matches all of the decision makers criteria.

Alternative and CriteriaThere are four alternatives bidders (B1, B2, B3, and B4)The organization policy criteria are categorized as following1- Performance1.1- QualificationAbility of professional personnelWorkload (have experience in similar project, size and budget?)Trades and Sub-trades Skills and reliability

Policy Criteria (Continued)

1.2- Past RecordMeeting Project ScheduleMeeting Project Budget1.3- Quality of worksOwners satisfactionOrganization management satisfaction2. Financial Status2.1- Work Loads2.2- Financial ability2.3- Financial performance history

Decision HierarchyThis involves breaking the decision problem down into a hierarchy of interrelated decision elements.At the top of the tree is a statement of the most general objective of the decision problem. Then the attributes of the decision are set out below. At the next level in the tree these attributes can be broken down into more detail, and so on.For example, we might consider that the main areas of concern in a choice of bidder are performance and financial. Within financial, we may wish to consider attributes such as the financial ability, the budget of undergoing workloads, the financial history.The lowest level of the hierarchy, the alternative courses of action are set out (Which to select?).

Decision Hierarchy (Fig.1)SELECTING THE 2 MOST QULIFIED BIDDERGOALPERFORMANCEFINANCIALQUALIFICATIONPAST RECORDQLTY. OF WORKABILITYWORKLOADSPRO. SKILLWORK LOADTRADSKILLSCHEDULEMEETING:BUDGETOWNERSATISFACTION OF:ORG.HISTORYB4B3B2B1B4B3B2B1B4B3B2B1B4B3B2B1B4B3B2B1B4B3B2B1B4B3B2B1B4B3B2B1B4B3B2B1B4B3B2B1

The techniqueMake Pairwise Comparisons of Attributes and Alternatives using AHP method.This pairwise comparison process is carried out at each level of the hierarchy The lowest level, the attraction of the alternative courses of action are compared in pairs with respect each of the attributes in the level above. For example, the attraction of B1 and B2 will be compared in terms of their financial history.

Pairwise comparison (Fig.2)In first level we have to criteria: Performance and Financial.

Usually there is no need for the pairwise comparison matrix if we are dealing with 2 criteria only.

Pairwise comparison (Fig.3)SELECTING THE 2 MOST QULIFIED BIDDERGOALPERFORMANCEFINANCIAL0.6670.333QUALIFICATIONPAST RECORDQLTY. OF WORKABILITYWORKLOADSHISTORYAfter getting the weight of each criteria of the top level we move on to the next one perform the pairwise comparison matrix and assign each of this level attributes with its weight

Pairwise comparison (Fig.4)QUALIFICATIONPAST RECORDPAST RECORDQUALIFICATION11/3311.6674.5WEIGHTNormalized ComparisonsQLTY. OF WORKQLTY. OF WORK3261/31/21QUALIFICATIONPAST RECORD0.6000.2000.6670.222QLTY. OF WORK0.5000.3330.2000.1110.1670.5890.2520.159Consistency Ratio: 0.060 < 0.10

Pairwise comparison (Fig.5)SELECTING THE 2 MOST QULIFIED BIDDERGOALPERFORMANCEFINANCIAL0.6670.333QUALIFICATIONPAST RECORDQLTY. OF WORKABILITYWORKLOADSHISTORY0.5890.2520.1590.3250.2350.440

Pairwise comparisonFigure 6 shows the decision hierarchy again with the weights of the various attributes and alternatives. The relative attraction of a course of action is found by identifying all the paths through the tree which end in that course of action. For each path all the weights are multiplied together and the resulting products are summed for all the paths involving the course of action.

Pairwise comparison (Fig.6)SELECTING THE 2 MOST QULIFIED BIDDERGOALPERFORMANCE (1)FINANCIAL0.6670.333QUALIFICATION (1.1)PAST RECORD (1.2)QLTY. OF WORK(1.3)ABILITYWORKLOADSHISTORY0.5890.2520.1590.3250.2350.440PRO. SKILLWORK LOADTRADSKILLSCHEDULEMEETING:BUDGETOWNERSATISFACTION OF:ORG.B4B3B2B1B4B3B2B1B4B3B2B10.060.290.540.110.060.150.310.480.230.29O.250.230.530.170.170.130.070.130.460.340.080.360.160.400.300.200.180.320.6330.1060.2600.050.240.490.22Bidders are compared in pairs with respect of their financial ability 0.170.190.470.170.480.310.150.060.560.440.600.40

WHAT IS THE DECISION?BIDDER (B1)BIDDER (B2)0.3070.275BIDDER (B3)0.274BIDDER (B4)BIDDERS IDWEIGHT0.144Based on each of the four alternatives bidders weight, our model suggested that the contract manager should select the first and second highest bidders. This is the model first iteration result, we noticed that the difference between B2 and B3 is so small. In order to make better decision, we suggest reviewing some of the criteria evaluation process and try to come with better consistent evaluation. This revision could be done at the criteria that have highest weight or the ones that having lower degree of consistency ratio ( CR very close to 0.1) B1 weight = 0.667 x 0.589 x 0.633 x 0.048 + 0.667 x 0.589 x 0.106 x 0.130 + 0.667 x 0.589 x 0.260 x 0.340 + 0.667 x 0.252 x 0.560 x 0.400 + 0.667 x 0.252 x 0.440 x 0.320 + 0.667 x 0.159 x 0.600 x 0.230 + 0.667 x 0.159 x 0.400 x 0.110 + 0.333 x 0.325 x 0.005 + 0.333 x 0.235 x 0.480 + 0.333 x 0.440 x 0.017=0.307

AHP Advantages.The use of pairwise verbal comparisons makes the elicitation of judgements easy.The AHP process requires more comparisons to be made than are needed to establish weights. This 'over specification' allows consistency checks to be made on the decision maker's judgements. Clarity is provided by the formal structuring of the decision problem - though this should be a feature of all decision analysis methods. Tools are available and are easy to use, all you need is good judgement skills and an Excel software

AHP Disadvantages.The eigenvalue method for obtaining weights will not be transparent to most decision makers.The failure to distinguish between options and attributes reduces the clarity with which a problem is perceived.The number of comparisons that need to be made can make the method extremely time consuming if a large number of attributes or options need to be compared (e.g. 5 options compared with respect to 5 attributes above them in the hierarchy would need 60 pairwise comparisons to be made).