MCDM BOOK Notes

 

MULTI-CRITERIA DECISION MAKING (MCDM)

 Study Notes

Methods, Formulas, Step-by-Step Procedures & Applications

Source: J.J. Thakkar, "Multi-Criteria Decision Making", Studies in Systems, Decision and Control 336, Springer Nature Singapore, 2021

Prepared for: Vimal Noble

 JUT Ranchi — M.Tech (Project Engineering & Management)

 

Table of Contents

●    Chapter 1 — Introduction to MCDM

●    Chapter 2 — Simple Additive Weightage (SAW)

●    Chapter 3 — Analytic Hierarchy Process (AHP), incl. Fuzzy AHP

●    Chapter 5 — TOPSIS

●    Chapter 6 — ELECTRE

●    Chapter 7 — PROMETHEE

●    Chapter 8 — VIKOR

●    Quick-Revision Comparison Table

Note: The source book's chapter numbering skips '4' in this edition — Chapter 3 (AHP) is directly followed by Chapter 5 (TOPSIS) in the scanned copy.

 

Chapter 1 — Introduction to MCDM

1.1 What is Decision Making?

●    Decision making is a systematic, formal process with four steps: (1) identifying the problem, (2) deriving preferences, (3) evaluating alternatives, (4) identifying the system.

●    Three kinds of decision analysis:

○    Descriptive analysis — involves graphical/tabular presentation of data to draw inferences.

○    Prescriptive analysis — enables decision makers to draw inferences and recommendations from data.

○    Normative analysis — deals with cross-examination of opinions where consensus must be built to reach the 'right' decision.

●    MCDM techniques primarily fall under descriptive and normative analysis, both nested within Operations Research (OR).

●    OR does not always seek an optimal solution — it also generates alternative scenarios for 'what-if' analysis.

1.2 What is MCDM?

●    MCDM = evaluation of multiple, often conflicting criteria and alternatives; also called Multi-Criteria Decision Analysis (MCDA).

●    Two broad sub-domains:

○    MADM (Multiple Attribute Decision Making) — limited, predetermined discrete alternatives (selection-type problems).

○    MODM (Multiple Objective Decision Making) — design/planning problems seeking an optimal solution across a set of goals/constraints.

●    Three governing steps of MCDM analysis:

○    1. Identify relevant criteria and alternatives from theory & practice.

○    2. Assign numerical values (weights) to criteria reflecting relative importance and alternative impacts.

○    3. Apply a formal mathematical procedure to rank/prioritize alternatives.

●    Typical decision-making process flow: Problem identification → Problem definition (objective, criteria, alternatives, constraints) → Decision matrix → Analytical model → Ranking → Group consensus → Select top alternative(s) → Implementation guidelines.

1.3 Why MCDM Matters

●    Real decisions (e.g., choosing a hospital) involve criteria that conflict — cost vs. quality vs. convenience — needing a systematic trade-off method rather than pure intuition.

●    MCDM has grown rapidly across engineering, science, management, psychology, law and politics over the last three decades.

●    MCDM is especially useful where no single 'optimal' answer exists, but decision makers must resolve conflicting objectives through trade-offs.

●    Concept of a non-dominated (Pareto) solution: one that cannot be improved on any criterion without sacrificing another.

●    MCDM helps build organizational/group consensus and ensures the chosen solution does not optimize one part of a system at another part's expense.

1.4 Single-Criteria vs. Multi-Criteria Decision Making

●    Single-criteria problems (e.g., maximize profit / minimize cost) are handled with classical OR tools: linear programming, integer programming, nonlinear optimization.

●    Multi-criteria problems require MCDM methods that can weigh and trade off several, often conflicting, criteria simultaneously.

1.5 Screening Approaches for Alternatives

●    Pareto-optimality approach

●    Sequential approach

●    Distance-based approach

●    Trade-off based on weightages

 

Chapter 2 — Simple Additive Weightage (SAW)

2.1 Overview

●    Also known as the weighted linear combination method or scoring method.

●    First applied historically to a portfolio-selection problem.

●    Best suited to simple, less complex decision environments.

●    Core idea: multiply each alternative's scaled/normalized value by the criterion's weight, then sum the products across all criteria to get a final score.

2.2 Step-by-Step Procedure

●    Step 1: Construct a pairwise comparison matrix for criteria; assign scores based on relative importance.

●    Step 2: Compute the weighted-sum matrix (pairwise matrix × priority vector). The priority vector = row averages; comparison-matrix totals = column totals.

●    Step 3: Compute the Consistency Index (CI):

CI = (λmax − n) / (n − 1),  where n = number of criteria, λmax = average of the weighted sum / priority vector ratios

●    Step 4: Compute the Consistency Ratio:

CR = CI / RI   (RI = Random Index, from Saaty's standard table, depends on number of criteria n)

●    Step 5: Build the decision matrix (m alternatives × n criteria) and normalize it for benefit/positive criteria.

●    Step 6: Multiply each normalized value by its criterion's weight.

●    Step 7: Sum each row (alternative) to get the final priority score; rank alternatives by score (higher = better).

2.3 Illustrative Application — Hotel Prioritization

●    Example criteria used in the book: C1 Location, C2 Price, C3 Rating, C4 Type of hotel, C5 Amenities.

●    Procedure: build decision matrix → normalize → apply criterion weights → compute weighted score per hotel → rank hotels by final score.

2.4 Strengths & Limitations

●    Strength: simple, transparent, easy to compute and explain to non-technical stakeholders.

●    Limitation: works best with a small number of criteria/alternatives and assumes criteria are independent and linearly additive.

 

Chapter 3 — Analytic Hierarchy Process (AHP)

3.1 Background

●    Developed by Thomas L. Saaty; one of the most widely applied MCDM techniques across engineering, management, humanities and psychology.

●    Best suited to problems that have a linear hierarchical structure of goal → criteria → sub-criteria → alternatives.

●    Limitation: cannot natively accommodate non-linear / interdependent relationships among criteria (that gap is addressed by the Analytic Network Process, ANP).

3.2 Step-by-Step Procedure

●    Step 1: Structure the problem as a hierarchy — Goal at top, Criteria in the middle, Alternatives at the bottom.

●    Step 2: Construct pairwise comparison matrices for criteria (and sub-criteria) using Saaty's 1–9 fundamental scale of relative importance.

●    Step 3: Normalize the pairwise comparison matrix (divide each entry by its column sum) and compute the priority vector (row averages) = criteria weights.

●    Step 4: Check consistency —

λmax = average of (weighted-sum vector element ÷ priority-vector element)

CI = (λmax − n) / (n − 1)

CR = CI / RI

●    A CR ≤ 0.10 is generally considered acceptable; if CR > 0.10 the pairwise judgments should be revised.

●    Step 5: Repeat pairwise comparison for alternatives against each criterion to get local priority scores.

●    Step 6: Aggregate — multiply each alternative's local score under a criterion by that criterion's weight, then sum across all criteria to get the overall priority/rating.

●    Step 7: Rank alternatives by their overall aggregated score.

3.3 Random Index (RI) Table (Saaty)

n	1	2	3	4	5	6	7	8	9	10
RI	0.00	0.00	0.58	0.90	1.12	1.24	1.32	1.41	1.45	1.49

 

3.4 Illustrative Application

●    Book example: ranking varieties of tea by overall quality using Total Criteria Composite Measure (TCCM) × Relative Variety Value (RVV) — the same AHP weighting-and-aggregation logic applied to a real product-comparison problem.

●    Also demonstrated for Healthcare Supply Chain Management (HSCM) criteria prioritization, walking through the full pairwise → weight → CR-check → aggregate ranking cycle.

3.5 Fuzzy AHP (Extension)

●    Motivation: classical AHP relies on crisp 1–9 judgments, but human judgments are often vague/imprecise — Fuzzy AHP replaces crisp numbers with triangular (or trapezoidal) fuzzy numbers to better capture linguistic uncertainty (e.g., 'moderately more important').

●    Pairwise comparisons are expressed as fuzzy numbers (l, m, u) — lower, middle, upper bounds — instead of single crisp values.

●    Fuzzy synthetic extent / extent analysis is used to defuzzify and derive crisp priority weights for ranking, after which the standard AHP aggregation and consistency logic still applies.

●    Preferred when the decision-making group faces high subjectivity/ambiguity in expert judgments (common in emerging technology, policy, or qualitative-heavy evaluations).

3.6 Strengths & Limitations

●    Strength: intuitive hierarchical structuring; built-in consistency check (CR) is a unique advantage over most other MCDM methods.

●    Limitation: number of pairwise comparisons grows quickly with more criteria/alternatives; assumes a strictly hierarchical (non-networked) criteria structure — use ANP when criteria interact.

 

Chapter 5 — TOPSIS — Technique for Order Preference by Similarity to Ideal Solution

5.1 Background

●    Developed by Hwang and Yoon (1981).

●    Core logic: the best alternative should have the shortest distance from the Positive Ideal Solution (PIS) and the longest distance from the Negative Ideal Solution (NIS).

●    A compensatory method — a poor score on one criterion can be offset/compensated by a good score on another.

5.2 Step-by-Step Procedure

●    Step 1: Normalize the decision matrix (typically vector normalization) so criteria become comparable/dimensionless.

●    Step 2: Construct the weighted normalized decision matrix by multiplying each normalized value by its criterion weight.

●    Step 3: Determine the Ideal (best, A+) and Negative-Ideal (worst, A−) solutions — taking the maximum for benefit criteria and minimum for cost criteria (and vice-versa for A−).

●    Step 4: Calculate the Euclidean distance of each alternative from the ideal solution (S+) and from the negative-ideal solution (S−).

●    Step 5: Compute the relative closeness coefficient to the ideal solution:

C*i = S−i / (S+i + S−i)      (0 ≤ C*i ≤ 1)

●    Step 6: Rank alternatives in descending order of C*i — the alternative with the highest closeness coefficient is the best choice.

5.3 Strengths & Limitations

●    Strength: simple, intuitive geometric logic (distance from 'ideal'); handles any number of criteria/alternatives efficiently; widely used in engineering and supply-chain selection problems.

●    Limitation: rank reversal can occur when alternatives are added/removed; assumes criteria weights are already known (often obtained via AHP or entropy method as a first step).

 

Chapter 6 — ELECTRE — Elimination et Choix Traduisant la Réalité

6.1 Background

●    Proposed by Bernard Roy and team at the SEMA consultancy; French for 'Elimination and Choice Expressing Reality.'

●    Family of methods that has evolved into several variants: ELECTRE I, II, III, IV, IS, TRI — each suited to different problem types (choice, ranking, sorting).

●    Classified as an outranking method — rather than compensating scores like SAW/TOPSIS, it builds pairwise 'outranks' relations.

●    Two major phases: (1) constructing outranking relations via pairwise comparison, (2) an exploitation phase that derives recommendations from those relations.

●    Because outranking relations may be incomplete, ELECTRE does not always identify a single best alternative — it often narrows down to a reduced, prioritized shortlist (useful combined with another MCDM method for the final pick).

6.2 Step-by-Step Procedure

●    Step 1: Normalize the decision matrix:

xij = aij / √(Σ aij²)

●    Step 2: Weight the normalized matrix — multiply matrix X by criterion weights (Σ wi = 1) to get weighted matrix Y.

●    Step 3: Determine Concordance and Discordance sets for every alternative pair (Ak, Al):

○    Concordance set Ckl = criteria j where Ak is preferred/equal to Al (ykj ≥ ylj).

○    Discordance set Dkl = criteria j where Ak is worse than Al (ykj < ylj).

●    Step 4: Construct the Concordance Matrix (C) and Discordance Matrix (D) summarizing agreement/disagreement strength for every alternative pair.

●    Step 5: Determine concordance and discordance indices/thresholds — a pair (Ak, Al) is validated as 'Ak outranks Al' when the concordance index ≥ concordance threshold C* AND the discordance index ≤ discordance threshold D*.

●    Step 6: Build concordance dominance matrix and discordance dominance matrix, then the aggregate dominance matrix — combining both to identify which alternatives are outranked/dominated.

●    Step 7: Eliminate dominated alternatives; the surviving, non-dominated alternatives form the prioritized shortlist (partial ranking).

6.3 Strengths & Limitations

●    Strength: does not force full compensation between criteria — a very poor score on one criterion can veto an otherwise strong alternative (useful for risk-averse decisions).

●    Limitation: more complex to compute; thresholds (C*, D*) are somewhat subjective; may yield an incomplete ranking rather than one clear winner.

 

Chapter 7 — PROMETHEE — Preference Ranking Organization Method for Enrichment Evaluations

7.1 Background

●    Proposed by Jean-Pierre Brans (1982); further extended by Vincke and Brans (1985).

●    An outranking method based on pairwise comparison of alternatives using preference functions for each criterion.

●    Two main variants:

○    PROMETHEE I — partial ranking of alternatives.

○    PROMETHEE II — complete ranking of all alternatives (most commonly used variant).

●    Key strengths cited in the book: simplicity, clarity and balance in handling conflicting criteria.

7.2 Step-by-Step Procedure (PROMETHEE II)

●    Step 1: Normalize the decision matrix:

Rij = (xij − min xij) / (max xij − min xij)

●    Step 2: Pairwise-compute the evaluative difference for every alternative vs. every other alternative, criterion by criterion.

●    Step 3: Apply a preference function Pj(i, i′) to each pairwise difference. Brans identified six standard preference-function types; a simplified (usual/linear) form used in the book:

Pj(i,i′) = 0            if Rij ≤ Ri′j

Pj(i,i′) = (Rij − Ri′j)  if Rij > Ri′j

●    Step 4: Compute the aggregated (weighted) preference function across all criteria:

π(i,i′) = [ Σ wj·Pj(i,i′) ] / Σ wj

●    Step 5: Compute outranking flows for each alternative:

○    Positive (leaving) flow φ+(i) — how much alternative i outranks all others.

○    Negative (entering) flow φ−(i) — how much all other alternatives outrank i.

●    Step 6: Compute the Net Outranking Flow (PROMETHEE II complete ranking):

φ(i) = φ+(i) − φ−(i)

●    Step 7: Rank alternatives by φ(i) — the highest net flow is the best alternative.

7.3 Illustrative Application

●    Book example: selecting the best e-retailer among alternatives for a consumer's product purchase, applying the full PROMETHEE II pipeline (normalize → pairwise preference → aggregate → net flow → rank).

7.4 Strengths & Limitations

●    Strength: flexible choice of preference function per criterion; provides both partial (PROMETHEE I) and complete (PROMETHEE II) rankings.

●    Limitation: choosing the 'right' preference function and thresholds per criterion can be difficult and somewhat subjective.

 

Chapter 8 — VIKOR — VseKriterijumska Optimizacija I Kompromisno Rešenje

8.1 Background

●    Proposed by S. Opricovic (1979/1980); named VIKOR in 1990 — Serbian for 'Multi-criteria Optimization and Compromise Solution.'

●    Designed to find a compromise ranking/solution for problems with conflicting criteria, useful across social, economic and environmental decision contexts.

●    Focuses on ranking and selecting from a set of alternatives, and determining compromise solutions for a problem with conflicting criteria — helping decision makers reach a final decision.

8.2 Step-by-Step Procedure

●    Step 1: Determine the best (f i*) and worst (f i−) value for every criterion i across all alternatives:

f i* = max fij   (or min, for cost criteria)      f i− = min fij   (or max, for cost criteria)

●    Step 2: Compute the group-utility measure Si and individual-regret measure Ri for each alternative j:

Sj = Σ wi (f i* − fij) / (f i* − f i−)

Rj = max i [ wi (f i* − fij) / (f i* − f i−) ]

●    Step 3: Compute the VIKOR index Qj for each alternative:

Qj = v·(Sj − S*)/(S− − S*) + (1−v)·(Rj − R*)/(R− − R*)

●    where S* = min Sj, S− = max Sj, R* = min Rj, R− = max Rj, and v = weight of the 'majority of criteria' strategy (commonly v = 0.5, representing consensus between maximum group utility and minimum individual regret).

●    Step 4: Rank alternatives by S, R and Q (ascending — lower is better).

●    Step 5: Propose the alternative with the smallest Q as the compromise solution, subject to two conditions:

○    Acceptable advantage — the gap between the top two Q-ranked alternatives is sufficiently large.

○    Acceptable stability — the top alternative must also be best-ranked (or close) by S and/or R.

●    If either condition fails, VIKOR proposes a set of compromise alternatives rather than a single winner.

8.3 Strengths & Limitations

●    Strength: explicitly balances 'majority/group utility' against 'minimum individual regret' via the v parameter — useful when a negotiated compromise (not a mathematically pure optimum) is what stakeholders actually need.

●    Limitation: choice of v is somewhat subjective; like TOPSIS, can be sensitive to the addition/removal of alternatives.

 

Quick-Revision Comparison Table

Method	Type	Core Logic	Best Suited For
SAW	Compensatory (additive)	Weighted sum of normalized scores	Simple problems, few criteria
AHP	Compensatory + hierarchical	Pairwise comparison, eigenvector weights, CR consistency check	Structuring hierarchical goals; weight derivation
Fuzzy AHP	Compensatory + fuzzy	AHP with triangular fuzzy judgments	High subjectivity/linguistic uncertainty
TOPSIS	Compensatory (distance-based)	Closeness to ideal vs. negative-ideal solution	Ranking with a clear best/worst benchmark
ELECTRE	Non-compensatory (outranking)	Concordance/discordance thresholds	Risk-averse choices; partial ranking/shortlisting
PROMETHEE	Outranking (preference functions)	Pairwise preference functions + net outranking flow	Flexible criterion-specific preference modelling
VIKOR	Compromise-based	Balances group utility (S) and individual regret (R) via Q	Negotiated compromise among conflicting stakeholders

 

Note: For exams/revision: AHP is most often used to derive criteria weights, which then feed into TOPSIS, ELECTRE, PROMETHEE or VIKOR for final ranking — a common hybrid MCDM pipeline.