Lab 3 of sem 2 M tech egg

 

 

JHARKHAND UNIVERSITY OF TECHNOLOGY, RANCHI

Department of Mechanical Engineering

LABORATORY MANUAL

PEML3001

Decision Making and Optimization Laboratory

Lab-III  |  Semester-II  |  Credit: 2 (0-0-4)

M.Tech — Project Engineering and Management (PEM)

Comprehensive Reference & Laboratory Guide

 

Table of Contents

 

 

Course Information

Field	Details
University	Jharkhand University of Technology (JUT), Ranchi
Department	Mechanical Engineering
Course Code	PEML3001
Course Title	Decision Making and Optimization Laboratory
Lab Designation	Lab-III
Semester	Semester-II
Credit Structure	2 Credits (0-0-4)

 

List of Experiments

The laboratory spans twelve experiments covering the two principal domains of decision science: Multi-Criteria Decision Making (MCDM) for ranking discrete alternatives, and metaheuristic Optimization Algorithms for tuning continuous engineering parameters.

No.	Experiment / Topic	Category	Main Purpose
1	Simple Additive Weighting (SAW)	MCDM	Rank alternatives using weighted sums
2	Weighted Product Method (WPM)	MCDM	Rank alternatives using weighted multiplication
3	Analytic Hierarchy Process (AHP)	MCDM	Determine criteria weights and rank alternatives
4	TOPSIS	MCDM	Choose the option closest to the ideal solution
5	Modified TOPSIS	MCDM	Improved TOPSIS for special decision problems
6	VIKOR	MCDM	Compromise ranking among conflicting criteria
7	Graph Theory & Matrix Approach (GTMA)	Decision Analysis	Analyze complex systems using graphs and matrices
8	ELECTRE	Outranking Method	Compare alternatives through dominance relations
9	PROMETHEE	Outranking Method	Preference ranking using pairwise comparisons
10	Genetic Algorithm (GA)	Optimization	Evolution-based optimization
11	Simulated Annealing (SA)	Optimization	Global optimization inspired by metallurgy
12	Particle Swarm Optimization (PSO)	Optimization	Swarm intelligence optimization

 

A.  Course Outcomes

On successful completion of this laboratory, students will be able to:

1.   Perform multi-criteria decision-making (MCDM) using SAW, WPM, AHP, TOPSIS, Modified TOPSIS, VIKOR, and Graph Theory & Matrix Approach (GTMA / Digraph).

2.   Apply advanced outranking methods — ELECTRE and PROMETHEE — to resolve conflicting-criteria decision problems.

3.   Optimize engineering design and process parameters using metaheuristic algorithms: Genetic Algorithm (GA), Simulated Annealing (SA), and Particle Swarm Optimization (PSO).

4.   Select an appropriate decision-making or optimization technique based on problem structure, data type, and computational complexity.

5.   Validate and compare rankings/solutions across multiple methods to arrive at robust, defensible engineering decisions.

B.  Suggested Learning Progression

This laboratory is structured to move from basic compensatory MCDM techniques through hierarchy-based and ideal-solution methods, into outranking approaches, and finally into metaheuristic optimization algorithms — building conceptual depth step by step.

Stage	Method	Learning Focus
1	Simple Additive Weighting (SAW)	Foundational compensatory scoring
2	Weighted Product Method (WPM)	Multiplicative aggregation, unit-free comparison
3	Analytic Hierarchy Process (AHP)	Pairwise comparison & subjective weighting
4	TOPSIS	Geometric distance-based ranking
5	Modified TOPSIS	Handling uncertainty/variants of TOPSIS
6	VIKOR	Compromise solutions under conflicting criteria
7	Graph Theory & Matrix Approach	Structural / interrelationship analysis
8	ELECTRE	Outranking via concordance-discordance
9	PROMETHEE	Preference-function-based outranking flows
10	Genetic Algorithm (GA)	Evolutionary parameter optimization
11	Simulated Annealing (SA)	Probabilistic global optimization
12	Particle Swarm Optimization (PSO)	Swarm-intelligence optimization

 

Rationale: SAW and WPM establish the foundation of weighted scoring. AHP introduces structured, pairwise-derived weighting. TOPSIS, Modified TOPSIS, and VIKOR extend the concept to geometric and compromise-based ranking. ELECTRE and PROMETHEE address situations where compensatory aggregation is inappropriate. GA, SA, and PSO then shift the focus from ranking discrete alternatives to optimizing continuous engineering parameters.

 

Detailed Experiment Guide

The following sections provide, for each experiment, the historical background, governing formulae, procedural steps, diagram description, worked example, and application context necessary for laboratory execution and report writing.

Experiment 1.  Simple Additive Weighting (SAW) / Weighted Sum Model

Historical Background

One of the oldest MCDM methods, with roots tracing to Fishburn (1967). SAW relies on simple linear (additive) compensation between criteria and remains the most widely taught baseline method in decision science.

Governing Formulae

Normalization (benefit criteria): r_ij = x_ij / max(x_ij)

Normalization (cost criteria): r_ij = min(x_ij) / x_ij

Weighted score: S_i = Σ w_j r_ij   (j = 1 to n; higher S_i is better)

Procedure / Steps

1.   Construct the decision matrix of alternatives (rows) against criteria (columns).

2.   Classify each criterion as benefit-type or cost-type.

3.   Normalize the matrix using the appropriate formula.

4.   Assign weights wj to each criterion such that Σwj = 1.

5.   Compute the weighted sum Si for every alternative.

6.   Rank alternatives in descending order of Si.

Diagram / Schematic Description

A tabular layout is used: alternatives (A1, A2, …) as rows, criteria (C1, C2, …) as columns, with normalized values populating the matrix and a final 'Score' column on the right for ranking.

Worked Example

In a car-selection problem with criteria such as purchase cost and mileage, alternatives A2 and A3 frequently tie or lead depending on how mileage (benefit) and cost (cost-type) are weighted — illustrating the sensitivity of SAW rankings to weight allocation.

Applications & Evidence Base

Widely applied in engineering component selection and supplier ranking due to its computational simplicity and ease of interpretation by non-specialist stakeholders.

Experiment 2.  Weighted Product Method (WPM)

Historical Background

WPM was developed to address a key limitation of SAW — its sensitivity to measurement units — by replacing additive aggregation with multiplicative aggregation.

Governing Formulae

Weighted product score: P_i = Π (x_ij)^wj   (j = 1 to n)

Pairwise ratio form: R(A/B) = Π (x_Aj / x_Bj)^wj

Procedure / Steps

1.   Build the decision matrix as in SAW.

2.   Raise each normalized value to the power of its criterion weight.

3.   Multiply the powered values across all criteria for each alternative to obtain Pi.

4.   Rank alternatives by descending Pi (or use pairwise ratios for direct comparison).

Diagram / Schematic Description

Same tabular structure as SAW, but the final column shows a multiplicative product score instead of an additive sum.

Worked Example

In job-candidate selection using CGPA and stipend expectation as criteria, WPM ranks candidates via product scores, which naturally penalizes any single very low criterion value more heavily than SAW would.

Cause – Effect – Solution

Because multiplication is unit-free and non-compensatory in extreme cases, WPM avoids the hidden compensation bias that can occur in SAW when criteria are measured on very different scales or conflict strongly.

Experiment 3.  Analytic Hierarchy Process (AHP)

Historical Background

Developed by Thomas Saaty in the 1970s, AHP structures a decision as a hierarchy (Goal → Criteria → Alternatives) and derives weights from pairwise comparisons rather than direct assignment, making it well suited to qualitative and expert-judgment-based decisions.

Governing Formulae

Pairwise comparison scale: a_ij ∈ {1, 2, … 9} (Saaty scale)

Consistency Index: CI = (λ_max − n) / (n − 1)

Consistency Ratio: CR = CI / RI   (acceptable when CR < 0.1)

Procedure / Steps

1.   Structure the hierarchy: overall goal at the top, criteria/sub-criteria in the middle, alternatives at the base.

2.   Construct pairwise comparison matrices at each level using the 1–9 Saaty scale.

3.   Compute priority (weight) vectors via the eigenvector method or normalized geometric mean.

4.   Calculate λmax, CI, and CR to verify judgment consistency (CR < 0.1 required).

5.   Synthesize local priorities into global priorities across the hierarchy.

6.   Rank alternatives by their global priority scores.

Diagram / Schematic Description

A hierarchical tree: the decision goal occupies the top node; criteria (and sub-criteria, if any) branch below it; alternatives form the leaf nodes connected to every criterion.

Worked Example

Supplier selection or facility-location problems commonly use AHP, where expert pairwise judgments on cost, quality, delivery reliability, and service are synthesized into a single ranked priority list.

Applications & Evidence Base

Extensively used in engineering, manufacturing, and business decisions where qualitative expert judgment must be systematically quantified; valued for its transparency and built-in consistency check.

Experiment 4.  TOPSIS & Modified TOPSIS

Historical Background

Introduced by Hwang and Yoon in 1981, TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) ranks alternatives by their geometric distance from an ideal best and ideal worst solution.

Governing Formulae

Vector normalization: r_ij = x_ij / √(Σ x_kj²)

Weighted normalized value: v_ij = w_j r_ij

Separation from ideal: d_i⁺ , d_i⁻  (Euclidean distance to A⁺ and A⁻)

Relative closeness: C_i = d_i⁻ / (d_i⁺ + d_i⁻)   (rank descending)

Procedure / Steps

1.   Build and vector-normalize the decision matrix.

2.   Compute the weighted normalized matrix vij.

3.   Determine the ideal best (A+) and ideal worst (A−) solutions for each criterion.

4.   Calculate Euclidean distances di+ and di− of each alternative from A+ and A−.

5.   Compute relative closeness Ci and rank alternatives in descending order.

6.   For Modified TOPSIS, apply adjusted normalization/weighting to better handle uncertainty or data variability before repeating the above steps.

Diagram / Schematic Description

The decision matrix is transformed into an n-dimensional space; the ideal (A+) and negative-ideal (A−) points are plotted, and each alternative's Euclidean distance to both is measured geometrically.

Worked Example

In mobile-phone ranking using RAM, battery life, and price as criteria, TOPSIS identifies the option positioned geometrically closest to the ideal specification profile and farthest from the worst profile.

Experiment 5.  Compromise Ranking Method (VIKOR)

Historical Background

Proposed by Opricovic in 1998, VIKOR focuses on ranking and selecting from a set of alternatives with conflicting criteria, providing a compromise solution that balances maximum group utility with minimum individual regret.

Governing Formulae

Group utility measure: S_i = Σ w_j (f_j* − f_ij) / (f_j* − f_j−)

Individual regret measure: R_i = max_j [w_j (f_j* − f_ij) / (f_j* − f_j−)]

Compromise index: Q_i = v(S_i − S*)/(S− − S*) + (1−v)(R_i − R*)/(R− − R*)

Procedure / Steps

1.   Determine best (fj*) and worst (fj−) values for each criterion.

2.   Compute Si (utility) and Ri (regret) for each alternative.

3.   Compute Qi using the weighting parameter v (commonly v = 0.5).

4.   Rank alternatives by S, R, and Q.

5.   Propose a compromise solution, verifying acceptable advantage and stability conditions.

Worked Example

VIKOR is commonly applied to project or construction-material selection where cost and performance criteria genuinely conflict and no single alternative dominates on all fronts.

Experiment 6.  Graph Theory and Matrix Approach (GTMA / Digraph)

Historical Background

GTMA represents interrelationships between decision factors using a directed graph (digraph), then converts this structural representation into a permanent-function matrix to derive a single decision index. It is especially effective for problems involving structural or reliability interdependencies.

Procedure / Steps

1.   Identify the relevant factors/attributes and their mutual influence relationships.

2.   Construct the digraph with nodes as factors and directed edges as influence relationships.

3.   Convert the digraph into an equivalent variable permanent-function matrix.

4.   Compute the matrix permanent to obtain a single composite decision index.

5.   Compare indices across alternatives to rank them.

Diagram / Schematic Description

A digraph with nodes representing decision factors and directed, weighted edges representing the strength/direction of influence between factors.

Applications & Evidence Base

Strong for structural analysis in mechanical systems, e.g., evaluating machine-tool reliability or manufacturing system flexibility where factor interdependency matters more than isolated scoring.

Experiment 7.  ELECTRE (I / II / III)

Historical Background

Developed by Bernard Roy in the 1960s, ELECTRE (ELimination Et Choix Traduisant la REalité) is an outranking method that determines whether one alternative sufficiently outranks another using concordance and discordance analysis, rather than compensatory aggregation.

Governing Formulae

Concordance index: C(a,b) — strength of agreement that a is at least as good as b

Discordance index: D(a,b) — strongest opposing evidence / possible veto

Threshold parameters: q (indifference), p (preference), v (veto)

Procedure / Steps

1.   Normalize the decision matrix and apply weights.

2.   Compute pairwise concordance sets and the concordance matrix.

3.   Compute pairwise discordance sets and the discordance matrix.

4.   Apply concordance and discordance thresholds to build the outranking relation/graph.

5.   Derive the kernel (non-dominated set) and final ranking from the outranking graph.

Diagram / Schematic Description

An outranking graph in which nodes represent alternatives and a directed edge from A to B indicates that A outranks B according to the concordance-discordance test.

Applications & Evidence Base

Frequently applied to infrastructure planning and environmental decision-making, where non-compensatory trade-offs (a very poor score on one criterion cannot always be offset by strong scores elsewhere) must be respected.

Experiment 8.  PROMETHEE (I / II)

Historical Background

Developed by Brans and Vincke in the 1980s, PROMETHEE (Preference Ranking Organization METHod for Enrichment of Evaluations) builds outranking flows from pairwise preference functions, offering both partial (PROMETHEE I) and complete (PROMETHEE II) rankings.

Governing Formulae

Preference function: P_j(a,b) — linear, Gaussian, etc., shaped by thresholds q and p

Aggregated preference index: π(a,b) = Σ w_j P_j(a,b)

Positive outranking flow: ϕ⁺(a) = [1/(m−1)] Σ π(a,x)

Negative outranking flow: ϕ⁻(a) = [1/(m−1)] Σ π(x,a)

Net flow: ϕ(a) = ϕ⁺(a) − ϕ⁻(a)

Procedure / Steps

1.   Compute pairwise criterion-wise differences between alternatives.

2.   Select and apply an appropriate preference function per criterion (linear, Gaussian, etc.).

3.   Aggregate preference indices π(a,b) using criterion weights.

4.   Compute positive and negative outranking flows for each alternative.

5.   Derive net flow φ(a) for a complete ranking (PROMETHEE II), or use φ+/φ− separately for a partial ranking (PROMETHEE I).

Diagram / Schematic Description

Preference-function shapes (e.g., linear with indifference/preference thresholds q and p) plotted against criterion differences; and an outranking flow network summarizing net positive/negative flows per alternative.

Worked Example

Applied to machinery or supplier ranking where preference intensity — not just direction — matters, such as when the degree of cost difference should influence the strength of preference.

Experiment 9.  Genetic Algorithm (GA)

Historical Background

Inspired by natural selection and introduced by John Holland in the 1970s, Genetic Algorithms are evolutionary metaheuristics that iteratively evolve a population of candidate solutions toward an optimum.

Procedure / Steps

1.   Initialize a population of candidate solutions (chromosomes) representing possible parameter sets.

2.   Evaluate the fitness of each chromosome against the objective function.

3.   Select high-fitness individuals as parents (e.g., via roulette-wheel or tournament selection).

4.   Apply crossover to generate offspring, recombining parent chromosomes.

5.   Apply mutation with a small probability to maintain diversity.

6.   Form the new generation and repeat evaluation–selection–crossover–mutation until convergence.

Diagram / Schematic Description

Flowchart: Initialize Population → Evaluate Fitness → Selection → Crossover/Mutation → New Population → (loop until convergence).

Worked Example

GA is commonly used to optimize machining parameters (e.g., cutting speed, feed rate, depth of cut) to minimize production cost or surface roughness subject to machine and tolerance constraints.

Experiment 10.  Simulated Annealing (SA)

Historical Background

Inspired by the metallurgical annealing process, Simulated Annealing probabilistically accepts worse solutions during search — with acceptance probability governed by a decreasing 'temperature' — to escape local optima and approach a global optimum.

Governing Formulae

Acceptance probability: P(accept) = exp(−ΔE / T)

Cooling schedule (geometric): T_k+1 = α · T_k   (0 < α < 1)

Procedure / Steps

1.   Initialize a solution and a starting temperature T.

2.   Generate a neighboring solution by small perturbation.

3.   Accept the neighbor if it improves the objective; otherwise accept probabilistically based on ΔE and T.

4.   Gradually reduce T according to a cooling schedule.

5.   Repeat until the temperature is sufficiently low or convergence criteria are met.

Applications & Evidence Base

Effective for combinatorial and continuous optimization problems in scheduling, layout design, and process-parameter tuning where the objective surface has many local optima.

Experiment 11.  Particle Swarm Optimization (PSO)

Historical Background

Introduced by Kennedy and Eberhart in 1995 and inspired by bird-flocking behavior, PSO optimizes a problem by having a population ('swarm') of candidate solutions ('particles') move through the search space guided by their own and the swarm's best-known positions.

Governing Formulae

Velocity update: v_id = w·v_id + c₁r₁(pbest − x_id) + c₂r₂(gbest − x_id)

Position update: x_id = x_id + v_id

Procedure / Steps

1.   Initialize a swarm of particles with random positions and velocities.

2.   Evaluate the objective function for each particle.

3.   Update each particle's personal best (pbest) and the swarm's global best (gbest).

4.   Update particle velocities and positions using the governing equations.

5.   Repeat evaluation and update steps until convergence or a maximum iteration count is reached.

Applications & Evidence Base

Applied to continuous engineering-parameter optimization problems, such as tuning process variables for minimum cost or maximum efficiency, and increasingly hybridized with other metaheuristics (e.g., GA-PSO) for improved convergence.

 

C.  Right Path Forward — Implementation Roadmap

1. Problem Identification

Clearly define the decision goal, the set of feasible alternatives, and the evaluation criteria (quantitative and/or qualitative), including whether each criterion is benefit-type or cost-type.

2. Data Collection

Assemble an evidence-based performance matrix from measured data, expert judgment, or a combination of both, ensuring consistent units and traceable sources.

3. Method Selection

•     SAW / WPM — for simple, transparent compensatory scoring.

•     AHP — when criteria weights must be derived from structured expert pairwise judgment.

•     TOPSIS / Modified TOPSIS / VIKOR — when ranking by distance to an ideal or compromise solution is appropriate.

•     ELECTRE / PROMETHEE — when non-compensatory outranking is needed due to conflicting criteria.

•     GA / SA / PSO — for continuous parameter optimization rather than discrete alternative ranking.

4. Implementation

Implement the chosen method(s) in Excel, Python, or MATLAB; verify pairwise-judgment consistency (AHP) and perform sensitivity analysis on weights.

5. Validation

Cross-compare rankings/solutions across multiple applicable methods; anchor conclusions in real-world mechanical-engineering applications such as material or process selection.

6. Cause – Effect – Solution

Conflicting criteria naturally arise in engineering trade-off decisions. Where compensation between criteria is acceptable, compensatory methods (SAW, WPM, TOPSIS) are appropriate; where it is not, non-compensatory outranking methods (ELECTRE, PROMETHEE) are preferable. Hybrid approaches combining both families often yield the most robust and defensible engineering decisions.

D.  Laboratory Delivery Recommendations

•     Implement one to two experiments per laboratory session using representative sample datasets (e.g., supplier selection, machining-parameter optimization).

•     Require students to generate structured reports containing decision/performance matrices, computed tables, sensitivity analysis, and appropriate visualizations.

•     Encourage cross-validation: apply at least two MCDM methods to the same dataset and compare resulting rankings.

•     For optimization experiments (GA, SA, PSO), require convergence plots and a comparison of final objective values across independent runs.