•Decision making methodology and algorithm : AHP (Analytical Hierarchical Process) • • • • •Creation of Information Systems on the Basis of Team Work •Dalia Krikščiūnienė, VU, 2022 Analytical Hierarchical process AHP •Author: Saaty 1980. •Based on expert evaluation •Allows expert groups •Application of quantitative and qualitative criteria •Process of pairwise evaluation (instead of overall assignment of values) •Consistency check •Result – ranking by importance, weights 2 AHP algorithm •Two stage algorithm, where evaluation is done by pairwise comparison of all possible pairs of items: –Stage 1: Criteria assessment -weight assignment to the criteria, arranged as a hierarchical tree –Stage 2: Assessment of decision alternatives (a list of selected objects- weight assignment by each criterion and globally 3 Example – car selection •Goal- car •Criteria – style, reliability, fuel, price •Alternatives (list of items) –Civic Coupe, Saturn Coupe, Ford Escort, Mazda Miata 4 Stage 1- Criteria hierarchy 5 Civic Saturn Escort Miata STAGE 1 STAGE 2 Ranking criteria and alternatives •Expert ranking: scale 1-9 : symetric relation • • • • • • 6 Intensyvumas, rangas Apibrėžimas 1 Equal 3 Slightly more important 5 Definitely More important 7 Much more important 9 Absolutely more important 2,4,6,8 Intermediate values Evaluation of Criteria: experts assign importance value by analysis of pairs. Is the criterion i (row) more important to criterion j (column) ? 7 Style Reliability Fuel efficiency Style Reliability Fuel efficiency 1 1/2 3 2 1 4 1/3 1/4 1 Is Reliability (row) more important than Style (column) Automatic recalculation of symmetric relation Calculating weights •We analyse the matric [Ax = maxx] where: –A – matrix of citeria ranks n×n. (n- matrix rank equal to the number of items/criteria) –x eigenvalue vector n×1 – vector of output weights. –max eigenvalue, max ÎÂ > n. – •The output weights are estimated by calculating eigenvalue vector X: •1) Matrix A comlumns are normalized (the sum columns should become equal to 1) . •2) Calculate average values by rows. • 8 0.30 0.29 0.38 0.60 0.57 0.50 0.10 0.14 0.13 Sums of columns 3.33 1.75 8.00 1.00 1.00 1.00 A= 1 0.5 3 2 1 4 0.33 0.25 1.0 Normalization Row average 0.30 0.60 0.10 Weight vector X X= • STAGE 1 output: Criteria weights •Style 0.30 •Reliability 0.60 •Fuel efficiency 0.10 9 Style 0.30 Reliability 0.60 Fuel efficiency 0.10 Car selection 1.00 Consistency check, evaluation of CR (Consistency ratio) •Consistency Ratio (CR) shows if the expert evaluations were logical, without self-contradiction or random guess. •AHP relies on the common logics of expert evaluations – if A is more important than B, and criterion B is more important than C, then A should be more important than C. • •If CR is bigger that 0.1 (>10%) the evaluation matrix is not reliable and has high extent of random guess 10 CR calculation •In order to find eigenvalue max •we analyse the matric expression [Ax = max x] where x is the weight vector (eigenvector), and A is the primary (not normalized) expert evaluation matrix, n is matrix rank (number of rows, columns): • • 11 0.30 0.60 0.10 1 0.5 3 2 1 4 0.333 0.25 1.0 0.90 1.60 0.35 = = max λmax=average of divided values: {0.90/0.30, 1.60/0.6, 0.35/0.10}=3.06 0.30 0.60 0.10 A x Ax x nConsistency index is calculated : CI=(λmax-n)/(n-1)=(3.06-3)/(3-1)= 0.03 nConsistancy ratio CR=CI/RI=0.03/0.58=0.05. If CR<0.1 evaluations a logical and not random. If CR > 10% (0,1) , the experts should revise evaluation or be excluded Matrix rank RI-random index STAGE 2: Assessment of decision alternatives by EACH criterion 12 Style Civic Saturn Escort 1 1/4 4 1/6 4 1 4 1/4 1/4 1/4 1 1/5 Miata 6 4 5 1 Civic Saturn Escort Miata Miata Reliability Civic Saturn Escort 1 2 5 1 1/2 1 3 2 1/5 1/3 1 1/4 Miata 1 1/2 4 1 Civic Saturn Escort Miata 0.13 0.24 0.07 0.56 Priority vector 0.38 0.29 0.07 0.26 13 Fuel efficiency Civic Saturn Escort Miata Miata 34 27 24 28 113 KM/L Normalized .30 .24 .21 .25 1.0 Ranking values for including quantitative (not expert) evaluation nThe indicators of fuel consumption for 100km is recalculated to the indicator “How many km we can drive with 1 liter” (the bigger value is the better should be used for all evaluations – both quantitative and expert) nIf we’d like to reflect difference of fuel efficiency depending on seasons (winter/summer or surroundings (city/highway), the expert ranking can be used again 14 Civic 0.13 Saturn 0.24 Escort 0.07 Miata 0.56 Civic 0.38 Saturn 0.29 Escort 0.07 Miata 0.26 Civic 0.30 Saturn 0.24 Escort 0.21 Miata 0.25 Style 0.30 Reliability 0.60 Fuel efficiency 0.10 Car selection 1.00 The result of two-stage evaluation: criteria weights and evaluation of alternatives by all criteria Global evaluation- weights are calculated by matrix multiplication 15 Civic Escort Miata Miata Saturn .13 .38 .30 .24 .29 .24 .07 .07 .21 .56 .26 .25 x .30 .60 .10 = .30 .27 .08 .35 Criteria weights Stage 1 Weight matrix Stage 2 Global evaluations Cost criterion for decision making •CIVIC 12K 0.22 0.30 0.73 •SATURN 15K 0.28 0.27 1.03 •ESCORT 9K 0.17 0.08 2.13 •MIATA 18K 0.33 0.35 0.92 16 Price Normalized price Cost/Benefit ratio Cost can be included to the lost of all criteria for expert pairwise evaluation/ However, it tends to dominate and can hide importance of some criteria. Instead, it be can use it for COST/BENEFIT analysis . The normalized price is used to estimate COST (price) and is divided by BENEFIT (global evaluation). The smaller value, the more attractive is alternative, as the benefit exceeds cost (CIVIC is most attractive for smallest ratio 0.73, ESCORT is overpriced comparing to its overall benefit – ratio 2.13)) Benefit (global weights) Regression analysis: Graphical visualization of result •Regression analysis is applied • • • • • • • • •Benefit exceeds the cost (beneficial for us) for all cases above the trendline • • 17 Civic Escort Saturn Miata Choosing place to study. Criteria (Location, budget, recommendations). Alternatives (universities) 18 Complex decisions 19 •Many levels of criteria and sub-criteria exists for complex problems. Group Decision Making 20 The AHP allows group decision making, where group members can use their experience, values and knowledge to break down a problem into a hierarchy and solve. Doing so provides: §Understand the conflicting ideas in the organization and try to reach a consensus. §Minimize dominance by a strong member of the group. §Members of the group may vote for the criteria to form the AHP tree. (Overall priorities are determined by the weighted averages of the priorities obtained from members of the group.) However; The GDSS does not replace all the requirements for group decision making. Open meetings with the involvement of all members are still an asset. More about AHP: Pros and Cons 21 •There are hidden assumptions like consistency. Repeating evaluations is cumbersome. •Difficult to use when the number of criteria or alternatives is high, i.e., more than 7. • •Difficult to add a new criterion or alternative • •Difficult to take out an existing criterion or alternative, since the best alternative might differ if the worst one is excluded. Users should be trained to use AHP methodology. Use GDSS Use constraints to eliminate some alternatives Use cost/benefit ratio if applicable •It allows multi criteria decision making. • •It is applicable when it is difficult to formulate criteria evaluations, i.e., it allows qualitative evaluation as well as quantitative evaluation. •It is applicable for group decision making environments Example 1: : Evaluation of Job Offers 22 Ex: Peter is offered 4 jobs from Acme Manufacturing (A), Bankers Bank (B), Creative Consulting (C), and Dynamic Decision Making (D). He bases his evaluation on the criteria such as location, salary, job content, and long-term prospects. Step 1: Decide upon the relative importance of the selection criteria: Location Content Long-term Salary 1 1/5 1/3 1/2 5 1 2 4 3 1/2 1 3 2 1/2 1/3 1 Location Salary Content Long-term Example 1: Priority Vectors: 23 1) Normalize the column entries by dividing each entry by the sum of the column. 2) Take the overall row averages Location Content Long-term Salary 0.091 0.102 0.091 0.059 0.455 0.513 0.545 0.471 0.273 0.256 0.273 0.353 0.182 0.128 0.091 0.118 Location Salary Content Long-term Average 0.086 0.496 0.289 0.130 + + 1 1 1 1 1 Example 1: Evaluation of Job Offers 24 Step 2: Evaluate alternatives w.r.t. each criteria A B C D 1 1/2 1/3 5 2 1 1/2 7 3 2 1 9 1/5 1/7 1/9 1 A B C D Relative Location Scores Location Scores A B C D 0.161 0.137 0.171 0.227 0.322 0.275 0.257 0.312 0.484 0.549 0.514 0.409 0.032 0.040 0.057 0.045 A B C D Avg. 0.174 0.293 0.489 0.044 Example 1: Calculation of Relative Scores 25 Relative Scores for Each Criteria A B C D 0.174 0.050 0.210 0.510 0.293 0.444 0.038 0.012 0.489 0.312 0.354 0.290 0.044 0.194 0.398 0.188 Location Salary Content Long-Term 0.086 0.496 0.289 0.130 Relative weights for each criteria x = Relative scores for each alternative 0.164 0.256 0.335 0.238 Example 2: AHP in project management 26 Prequalification of contractors aims at the elimination of incompetent contractors from the bidding process. It is the choice of the decision maker to eliminate contractor E from the AHP evalution since it is not “feasible” at all !! Contractor A Contractor B Contractor C Contractor D Contractor E Experience 5 years experience 7 years experience 8 years experience 10 years experience 15 years experience Two similar projects One similar project No similar project Two similar projects No similar project Special procurement experience 1 international project Financial stability $7 M assets $10 M assets $14 M assets $11 M assets $6 M assets High growth rate $5.5 M liabilities $6 M liabilities $4 M liabilities $1.5 M liabilities No liability Part of a group of companies Good relation with banks Quality performance Good organization Average organization Good organization Good organization Bad organization C.M. personnel C.M. personnel C.M. team Good reputation Unethical techniques Good reputation Two delayed projects Government award Many certi®cates One project terminated Many certi®cates Safety program Good reputation Cost raised in some projects Average quality Safety program QA/QC program Manpower resources 150 labourers 100 labourers 120 labourers 90 labourers 40 labourers 10 special skilled labourers 200 by subcontract Good skilled labors 130 by subcontract 260 by subcontract Availability in peaks 25 special skilled labourers Example 2 (cont.’d) 27 Contractor A Contractor B Contractor C Contractor D Contractor E Equipment resources 4 mixer machines 6 mixer machines 1 batching plant 4 mixer machines 2 mixer machines 1 excavator 1 excavator 2 concrete transferring trucks 1 excavator 10 others 15 others 1 bulldozer 2 mixer machines 9 others 2000 sf steel formwork 20 others 1 excavator 6000 sf wooden formwork 15,000 sf steel formwork 1 bulldozer 16 others 17,000 sf steel formwork Current works load 1 big project ending 2 projects ending (1 big+ 1 medium) 1 medium project started 2 big projects ending 2 small projects started 2 projects in mid (1 medium +1 small) 2 projects ending (1 big + 1 medium) 1 medium project in mid 3 projects ending (2 small + 1 medium) Example 2: Hierarchy Tree 28 Selecting the most suitable contractor Financial Stability Experience Quality Performence Manpower Resources Equipment Resources Current workload Contractor A Contractor B Contractor C Contractor D Contractor E Example 2: AHP in project management 29 Step 1: Evaluation of the weights of the criteria Step 2: a) Pairwise comparison matrix for experience Example 2: AHP in project management 30 Calculation of priority vector: x = Note that a DSS supports the decision maker, it can not replace him/her. Thus, an AHP Based DSS should allow the decision maker to make sensitivity analysis of his judgements on the overall priorities ! Probably Contractor-E should have been eliminated. It appears to be the worst. Multi Criteria Decision Making Models: PROMETHEE 31 nOne of the most efficient and easiest MCDM methodologies. nDeveloped by Jean-Pierre Brans and Bertrand Mareschal at the ULB and VUB universities since 1982 nConsiders a set of criteria and alternatives. Criteria weights are determined that indicate the relative importance nUtilizes a function reflecting the degree of advantage of one alternative over the other, along with the degree of disadvantage that the same alternative has with respect to the other alternative. nIn scaling, there are six options allowing the user to express meaningful differences by minimum gaps between observations. When type I is used, only relative advantage matters; type 6 is based on standardization with respect to normal distribution. Example 3: Media Selection for a Bicycle Co. 32 A bicycle manufacturing company is intending to advertise its products. Six marketing actions are considered: Advertising in the international newspaper, News; in the newspaper Herald; by mean of advertising boards in large cities; of a personal mailing; by TV spots on channels CMM or NCB. Units: Cost ($ 1,000), Target (10,000 people), Duration (days), Efficiency (0-100) Manpower (# people involved in the company) Partial anf full rankings with Promethee I and II 33 34 Ranking of the alternatives can be obtained for the selected weights Including optimization: Additional constraints 35 nIt is often necessary that several alternatives have to be selected subject to a set of goals. nIn this case an LP can be constructed with binary decision variables, which gives the selection of r actions, among n alternatives. Let xi=1 if media i is selected and 0 otherwise, i=1,2,...,6. φ(Ai) are the relative weight of media i, i=1,2,...,6. Max φ(A1) x1 + … + φ(A6) x6 Subject to x1 + x2 + x3 + x4 + x5 + x6 ≥ 2 (at least 2 media should be selected) x1 + x2 + x3 + x4 + x5 + x6 ≤ 4 (at most 4 media should be selected.) x1 + x2 = 1 (choose exactly one newspaper) x5 + x6 = 1 ((choose exactly 1 TV channel) 625 x1 + 550 x2 + 250 x3 + 150 x4 + 175 x5 + 750 x6 ≥ 1200 (min. expected return) - 60 x1 - 30 x2 + 40 x3 + 92 x4 + 52 x5 + 80 x6 ≥ 0 (cost of advertising in newspapers should be less than 50% of total costs) AHP web tool •https://bpmsg.com/ahp/ •Top link- Register (for the leader of AHP evaluation) •After registration an Sign in- use - My AHP Projects •For New project press “New hierarchy”. It will consist of two stages: 1 stage: criteria hierarchy is created and evaluated and saved (project type in My projects is “H”. •2 stage project will be generated automatically by listing alternatives after stage 1 evaluation phase. This will be shown in the My project list as project type A, same title as H. •For creating project for stage 1, press “New hierarchy” and edit it (keeping the syntax signs- column, semicolumn. E.g. •ATOSTOGOS:Saule,Gamta,Pramogos,Lankytini objektai; •Hierarchy is extended by explaining criteria – e.g. „Pramogos“ has to be explained, so the new line is written in the same format: •Pramogos: Jaunimui, Vaikams, Sportas; •After creating (save, submit) hierarchy, copy the link for expert evaluation to all members (they must not be registered or signed to the system) 36 AHP web tool •After receiving the link (or the session code), the experts can do evaluation. If they have a session code, they use AHP Group Session meniu. Each place where they see AHP, they have to do evaluation. They evaluate pairs of criteria : chose a dot for the more important criterion, assign value, how much more it is important. •After submitting evaluation, the „consistency ratio“ is computed by system, and , if CR value > 10%, the suggestions for changing evaluation are highlighted. The experts are allowed to revise to reach CR<10 % (if this is not achieved, the evaluation will be excluded forom the decision). •For initiating STAGE 2, project leader reviews the Stage 1 group results, selects "View results" press "Define alternatives" and writes the number of alternatives for discussion in the stage 2 etapui. Then the window opens for writing down the list of alternatives. After Save, the new code for voting is assigned. The new link with the new code is sent out to all experts. •The same procedure for evaluation (following each AHP in red square) and adjusting CR is done. •The team report will include the visualizations and tables about voting : criteria weights, alternative weights, table of voting by group members and their results . If some experts have CR>10, their results have to be excluded. • 37 Decision making method: mindmapping •Mind maps •https://www.mindmeister.com/ •It is an idea organization and visualization tool for grouping the ideas, summarizing them for further applying other (quantitative) decision making methods. 38 Decision alternatives – evaluation by “6 hats” method •Present evaluation of alternatives by taking a role, expressed by the hat color: •White hat – relies on information and facts; •Yellow – positive support, discuss pros, potential; •Red – based on emotional argumentation, convincing, taking perspective of market response; •Black – reserved, critical, cautious thinking and argumentation; •Green – creativity aspect, comparative evaluation of innovation and creativity influence for development; •Blue- management aspects, control and summarizing all pros and cons 39 Delphi method •Delphi is based on the principle that forecasts (or decisions) from a structured group of individuals are more accurate than those from unstructured groups. • The experts answer questionnaires in two or more rounds. After each round, a facilitator or change agent provides an anonymised summary of the experts' forecasts from the previous round as well as the reasons they provided for their judgments. •Thus, experts are encouraged to revise their earlier answers in light of the replies of other members of their panel. •Finally, the process is stopped after a predefined stop criterion (e.g., number of rounds, achievement of consensus, stability of results), and the mean or median scores of the final rounds determine the results 40 References 41 Al Harbi K.M.A.S. (1999), Application of AHP in Project Management, International Journal of Project Management, 19, 19-27. Haas R., Meixner, O., (2009) An Illustrated Guide to the Analytic Hierarchy Process, Lecture Notes, Institute of Marketing & Innovation, University of Natural Resources, retrieved from http://www.boku.ac.at/mi/ on October 2009. Saaty, T.L., Vargas, L.G., (2001), Models, Methods, Concepts & Applications of the Analytic Hierarchy Process, Kluwer’s Academic Publishers, Boston, USA. Brans, J.P., Mareschal, B., (2010) “How to Decide with Promethee, retrieved from http://www.visualdecision.com on October 2010.