Multi‑Criteria Decision Making

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Multi‑Criteria Decision Making
Dr. Shahram Yazdani

The Nature of Decision-Making in Healthcare System
Process is interactive - involves group of persons Multiple criteria (objectives/attributes) Interaction among criteria/objectives Need for a standard process - need for consistency and continuity Time dependent: both short- and long-term

Multi‑Criteria Decision Making
Ben Franklin over 200 years ago recognized the presence of multiple attributes in everyday decisions and suggested a workable solution Major development in theory and practice since 1970

Decision-Making Strategies
Optimizing Satisficing Elimination-by-aspects Incrementalism Mixed scanning Analytic Hierarchy Process

Decision-Making Strategies : Optimization
Select the alternative that gives the best overall value Identify criteria to judge alternatives Can be expressed in mathematical terms and implemented using computer programs Difficult to solve when model involves qualitative criteria Optimization of “Utility Super Function”

Decision-Making Strategies : Satisficing
Recognizes that “utility super functions” are difficult to formulate and In most cases one doesn’t want optimality Understand decision process (flow) (Sequence in which alternatives are identified and considered; usually governed by heuristics) Decision maker is therefore part of the multi-objective problem Select the first alternative that is good enough with respect to some minimal criteria Cutoff level of constraints Simon: satisficing or finding solutions that are good enough e.g. Goal Programming Keen: viewed requirement of assigning weights or priorities to be problematic. Noninferior sets.

Decision-Making Strategies : Elimination-by-Aspects
Elimination of all alternatives that fail with respect to one aspect, then consider another aspect … An aspect is like a constraint involving one or more criteria Order of aspects can strongly influence results An alternative that superior in many aspects may be eliminated

Decision Making Strategies: Incrementalism
Compare alternative courses of action to the current course of action Look for alternatives that can overcome shortcomings of the current course of action A decision that results in incremental improvement

Decision Making Strategies: Mixed Scanning
Scanning: Collection, processing, evaluating and weighing of information Importance of decision determines the degree of scanning and choice Each alternative is briefly considered Reject alternatives for which strong objections are detected

Decision Making Strategies: Analytic Hierarchy Process
Decomposes the overall decision objective into a hierarchic structure of criteria, subcriteria, and alternatives Pairwise comparison matrix for criteria, subcriteria and alternatives Matrices are mathematically processed to calculate relative weights of criteria and sub criteria Relative weights are used to arrive at a score for each alternative

Decision Making : Other Strategies
Dominance rule Select the alternative that is better than other alternative(s) on at least one attribute and not worse on other attributes Lexicographic rule Starts with the most important attribute and selects the attribute that ranks highest on that attribute If two or more are tied, proceed to the next important attribute Maximizing number of attributes with greater attractiveness rule Classify each alternative as better, equal or worse on each attribute Select the alternative with the greater number of favorable attributes

Decision Making Strategies: Other Strategies
Conjunctive decision making Compare all attributes of one alternative against all criteria Reject the alternatives that do not meet the criteria Additive linear rule Start with a set of predetermined weights of each alternative on each attribute (A) Allocate weights against the attributes (B) Multiply (A) by (B) to determine the score for each alternative Select the alternative having the highest score

Political Approaches Actions and decisions result from bargaining among players To predict decision, find out: who the players are what are the players’ interests or stands? what are the players’ relative influence? How does the combined dynamics of the above affect the decisions

Anarchic Theory of Decision-Making
Decision-making in organizations are random and disjointed e.g. Lindblom: “muddling-through” concept of decision theory; avoid comprehensive analysis and concentrate on marginal gains An organization is a collection of: choices looking for problems issues and feelings waiting for decision situations solutions looking for issues they apply to; or decision makers looking for work so-called organized anarchy.

Major categories of MCDM methods
A normative approach the disregards decision maker Statement of decision maker’s preference prior to analysis Interactive responses between analyst and decision maker Generation of noninferior (nondominated) solutions that decision maker selects from.

Multiple Criteria Decision Making
Multiple Objective Decision Making MODM: used primarily for designing Multiple Attribute Decision Making MADM: used primarily for choosing an alternative

Multiple‑Objective Decision Making
Dr. Shahram Yazdani

Multi‑Objective Decision Making
Some decisions involve more than one objective. Utility theory provides a methodology that allows a subjective tradeoff among valued attributes.

Approach is demonstrated with a decision table:

EV(Alternative j) The expected effectiveness of each decision alternative is the weighted average of the probabilities with the utilities as weights. EV(Aj) = i=1,n Ui pij j=1,…m

Relative Utilities Every objective must has a utility value
The most important objective has the highest utility value 3. The utility value of achieving 2 objectives is the sum of the individual utility values

Clearly shows interrelationships among objectives and alternatives Allows non-quantifiable objectives

Problems Difficult to get consistent utilities, meaningful probabilities, realistic objectives and feasible alternatives 2. Must generate new utilities in every situation for every individual or group

Multiple Attribute Decision Making
Dr. Shahram Yazdani

continuous discrete quantitative Qualitative/ mixed Nature of
alternatives continuous discrete Linear programming Goal programming Nature of Criteria/ objective quantitative Qualitative/ mixed Multi-attribute utility theory Weighted summation Ideal point method Concordance analysis AHP Regime Method Evamix Method ELECTRE

Generic process for MCDM
Identify objectives Develop Criteria/ attributes Identify alternatives Weight Criteria/ attributes Rank alternatives Choose alternative

MADM Matrix Attributes X1 X2 X3 Xn A1 A2 A3 Am Alternatives

Give Numerical Values to Attributes of Each Alternative
Consider simple measures in simple quantitative attributes Consider decision tree analysis in complex quantitative attributes Consider pair wise comparison in qualitative attributes

MADM Matrix X1 X2 X3 Xn A1 A2 A3 Am Attributes Alternatives
v1n A2 v21 v22 v23 v2n A3 v31 v32 v33 v3n Am vm1 vm2 vm3 vmn Alternatives vij is the specific value of attribute Xj for alternative Ai

Standardizing the attribute values
Normalization Linear Fuzzy j rij = vij – min vij maxvij – min vij For positive attributes Where more is better j rij = max vij – vij maxvij – min vij For negative attributes Where less is better

Standardized attributes
X1 X2 X3 Xn A1 r11 r12 r13 r1n A2 r21 r22 r23 r2n A3 r31 r32 r33 r3n Am rm1 rm2 rm3 rmn Alternatives rij is the standardized value of attribute Xj for alternative Ai

Weight of each attribute
Wn X1 X2 X3 Xn A1 r11 r12 r13 r1n A2 r21 r22 r23 r2n A3 r31 r32 r33 r3n Am rm1 rm2 rm3 rmn Alternatives

Weighting methods for attributes
Fixed Point Scoring Paired Comparisons Judgment Analysis

Give to each attribute a weight
Fixed Point Scoring Give to each attribute a weight (<1) that sum up in 1 Attribute 1 Attribute 2 Attribute 3 Attribute n W1 W2 W3 Wn 1

Paired Comparisons of Attributes Importance

Paired Comparisons attributes X1 X2 X3 X4 X1 X2 X3 X4 n×n AHP matrix

Paired Comparisons Bipolar Scale for positive attributes
Equally Important Slightly More Moderately Very Bipolar Scale for positive attributes Less

Paired Comparisons Perform Pairwise Comparison attributes X1 X2 X3 X4
5 5 7 X2 1 3 3 X3 1 9 X4 1

Perform Pairwise Comparison
Paired Comparisons Perform Pairwise Comparison Using reciprocals attributes X1 X2 X3 X4 X1 1 5 5 7 X2 0.2 1 3 3 X3 0.2 0.33 1 9 X4 0.14 0.33 0.11 1

Paired Comparisons Sum the columns attributes X1 X2 X3 X4 X1 1 5 5 7
0.2 1 3 3 X3 0.2 0.33 1 9 X4 0.14 0.33 0.11 1 Totals 1.54 6.66 9.11 20

Normalize the values in each column
Paired Comparisons Normalize the values in each column attributes X1 X2 X3 X4 X1 0.65 1 5 0.75 0.55 5 0.35 7 X2 0.2 0.13 0.15 1 3 0.33 3 0.15 X3 0.2 0.13 0.05 0.33 1 0.11 9 0.45 X4 0.14 0.09 0.33 0.05 0.01 0.11 1 0.05 Totals 1.54 1 6.66 1 9.11 1 20 1

Calculate sum of normalized values for
Paired Comparisons Calculate sum of normalized values for Each row attributes X1 X2 X3 X4 Sum X1 1 0.65 0.75 5 5 0.55 7 0.35 2.3 X2 0.13 0.2 0.15 1 0.33 3 0.15 3 0.76 X3 0.2 0.13 0.33 0.05 1 0.11 0.45 9 0.74 X4 0.14 0.09 0.33 0.05 0.01 0.11 0.05 1 0.2 Totals 1.54 6.66 9.11 20

Calculate the Average (weight) for
Paired Comparisons Calculate the Average (weight) for Each Row attributes X1 X2 X3 X4 Sum Average (W) X1 1 0.65 0.75 5 0.55 5 7 0.35 2.3 0.575 X2 0.13 0.2 0.15 1 3 0.33 3 0.15 0.76 0.19 X3 0.2 0.13 0.05 0.33 0.11 1 0.45 9 0.74 0.185 X4 0.14 0.09 0.05 0.33 0.01 0.11 0.05 1 0.2 0.05 Totals 1.54 6.66 9.11 20 4.00 1.00

Analytical hierarchic process

Analytical hierarchic process
Information is decomposed into a hierarchy of alternatives and criteria Information is then synthesized to determine relative ranking of alternatives Both qualitative and quantitative information can be compared using informed judgements to derive weights and priorities

Analytical hierarchic process
Classifying attributes in a hierarchic model No end branch should contain more than 10 (preferably 8) attribute Begin cross sectional weighting at root side of three and progress to the end branch side of three At each level if the number of items are less than 6 (preferably 4) use fixed point scoring otherwise use paired comparison through AHP matrix Combine cross sectional weights into hierarchical weights which must sum up to 1 for end branches

Hierarchic Organization of Attributes

End Branch Attributes

Determine simple weight of attributes

Determine simple weight of attributes
1 w1 w2 w3 1 w11 w12 w13 w14 w31 w32 1 w111 w112 w141 w142 w143 w321 w322 w323

Determine final weight of attributes
w11 w12 w13 w14 w31 w32 w111 w112 w141 w142 w143 w321 w322 w323 Wf1

Determine final weight of attributes
w11 w12 w13 w14 w31 w32 Wf4 w111 w112 w141 w142 w143 w321 w322 w323

 Wfi = 1 Determine final weight of attributes Wf3 Wf4 Wf8 Wf1 Wf2 Wf5

Analytical hierarchic process
Application areas strategic planning resource allocation source selection, program selection business policy etc., etc., etc.. AHP software (ExpertChoice) computations sensitivity analysis graphs, tables Group AHP

Weight of each attribute
Wn X1 X2 X3 Xn A1 r11 r12 r13 r1n A2 r21 r22 r23 r2n A3 r31 r32 r33 r3n Am rm1 rm2 rm3 rmn Alternatives

Weighted value of attributes for alternatives
X1 X2 X3 Xn A1 R11 R12 R13 R1n A2 R21 R22 R23 R2n A3 R31 R32 R33 R3n Am Rm1 Rm2 Rm3 rmn Alternatives Rij = rij × Wj

Scoring Alternatives: Weighted summation
Attributes X1 X2 X3 Xn Score A1 R11 R12 R13 R1n S1 A2 R21 R22 R23 R2n S2 A3 R31 R32 R33 R3n S3 Am Rm1 Rm2 Rm3 rmn Sm Alternatives Si = ΣRij Si = Σrij×wj J= n J= n

Judgement Analysis through Virtual Portfolios
Dr. Shahram Yazdani

Judgement analysis through virtual portfolios
Choosing the right people Defining objectives and options Determining ranking attributes Defining ranking attributes Scaling ranking attributes Constructing random virtual portfolios Ranking or scoring virtual portfolios by panel performing stepwise regression analysis Finding independent attributes and their weight (regression coefficient) Formulating ranking equation Assessing the validity and reliability of equation on a separate set of portfolios

Level 1 Level 2 Level 3 Level 4 Level 5 Attribute 1 S11 S12 S13 S14 S15 Attribute 2 S21 S22 S23 S24 S25 Attribute 3 S31 S32 S33 S34 S35 Attribute 4 S41 S42 S43 S44 S45 Attribute 5 S51 S52 S53 S54 S55 Attribute 6 S61 S62 S63 S64 S65 Attribute 7 S71 S72 S73 S74 S75

Random Virtual Portfolio 1
Level 1 Level 2 Level 3 Level 4 Level 5 Attribute 1 S11 S12 S13 S14 S15 Attribute 2 S21 S22 S23 S24 S25 Attribute 3 S31 S32 S33 S34 S35 Attribute 4 S41 S42 S43 S44 S45 Attribute 5 S51 S52 S53 S54 S55 Attribute 6 S61 S62 S63 S64 S65 Random Virtual Portfolio 1 Attribute 7 S71 S72 S73 S74 S75 S11 S25 S32 S54 S61 S74 S74

Ranking of virtual portfolios
5 3 Random Virtual Portfolio 1 8 Random Virtual Portfolio 1 6 Random Virtual Portfolio 1 1 Random Virtual Portfolio 1 2 Random Virtual Portfolio 1 7 Random Virtual Portfolio 1 3 Random Virtual Portfolio 1 4 Random Virtual Portfolio 1 9 Random Virtual Portfolio 1 Random Virtual Portfolio 1 Random Virtual Portfolio 1 Random Virtual Portfolio 10

Perform Stepwise Regression Analysis
Portfolios Rank = iAi Where Ai = ith Attribute i = Regression Coefficient (weight) of ith Attribute Results in the minimal set of independent attributes contributing in the judgment of stakeholders about the topic

Problem: Diagnosis of malignant tumors of breast
Positive Biopsy rate is 10%-31% for cancer Total cost of percutaneous large core biopsy of a breast nodule is \$1000 The total cost of excisional biopsy of a breast lump is between \$3000 and \$4500

Alternatives Magnetic Resonance Imaging Mammography Ultrasonography
Positron Emission Tomography

Attributes Sensitivity (SE) Specificity (SP)
Positive Predictive Value (PPV) Negative Predictive Value (NPV) Complexity of Interaction with Patients (CIP) Includes time spent, degree of discomfort, Invasiveness Complexity of Interaction with Doctors (CID) Includes Time spent, Level of necessary training and experience, Complexity of protocol Cost (C)

SE SP PPV NPV CIP CID C MRI 96 69 75 97 9 5 500 Mammography 89 45 57 83 2 Ultrasonography 48 94 65 90 6 200 PET 80 73 41 4 7 350

Thank You ! Any Question ?

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