(SEM VIII) THEORY EXAMINATION 2023-24 INDUSTRIAL OPTIMIZATION TECHNIQUES

SECTION A

(2 × 10 = 20 | Very Short Answers)

a. Differentiate between CPM and PERT
CPM (Critical Path Method) uses deterministic time estimates and is suitable for repetitive projects, while PERT (Program Evaluation and Review Technique) uses probabilistic time estimates and is suitable for uncertain projects.

b. Individual vs Group replacement policy
Individual replacement replaces items as they fail, whereas group replacement replaces all items together at fixed intervals to reduce overall cost.

c. Saddle point and optimal strategy
A saddle point exists when the maximin value equals the minimax value. The strategy corresponding to this value is the optimal strategy.

d. Slack and surplus variables
Slack variables are added to ≤ constraints, while surplus variables are subtracted from ≥ constraints to convert inequalities into equations.

e. Customer behaviors in a queue
Customer behaviors include balking (not joining queue), reneging (leaving queue), and jockeying (switching queues).

f. Application of Monte Carlo simulation in engineering
Monte Carlo simulation is used for risk analysis, reliability estimation, inventory control, and complex system modeling under uncertainty.

g. Principle of dominance
A strategy dominates another if it yields better or equal payoff under all conditions and strictly better under at least one condition.

h. Optimistic, pessimistic, and most likely time
Optimistic time is the shortest possible duration, pessimistic time is the longest possible duration, and most likely time is the normal expected duration.

i. Dual of the given primal
The dual will be a maximization problem with ≤ constraints corresponding to primal ≥ constraints and vice-versa.

j. Degeneracy in transportation problem
Degeneracy occurs when the number of occupied cells is less than (m + n − 1).

SECTION B

(Attempt any THREE | 3 × 10 = 30 Marks)

2(a) LPP solved using Simplex Method

The given problem is formulated by converting inequalities into equations using slack variables.
After applying simplex iterations, the optimal solution is obtained at the feasible corner point that maximizes the objective function value Z.

(Simplex table construction, pivot selection, and iteration steps must be shown in exam.)

2(b) CPM network analysis of residential project

The project network is constructed using precedence relationships.
By calculating ES, EF, LS, LF, and Total Float, the critical path is identified as the path with zero float.
Any delay in a critical activity delays the entire project, increasing total completion time.

2(c) Two-person zero-sum game & saddle point

Row minimums and column maximums are calculated.
If maximin equals minimax, a saddle point exists and optimal strategies are pure.
If not, mixed strategies or dominance principle is applied to determine optimal strategies.

2(d) Dynamic Programming (DP) applications

Dynamic Programming breaks complex problems into smaller sub-problems.
In Capital Budgeting, DP allocates funds optimally to maximize returns.
In Cargo Loading, DP selects items to maximize value without exceeding capacity.

2(e) EOQ with price discounts

EOQ is calculated using:

EOQ=2DSHEOQ = \sqrt{\frac{2DS}{H}}EOQ=H2DS​​

Total cost is calculated for each discount level, and the lot size with minimum total cost is selected as the optimal order quantity.

SECTION C

3(a) Inventory costs & inventory models

Holding cost includes storage and insurance costs, ordering cost includes procurement expenses, and shortage cost includes loss of goodwill.
Deterministic models assume known demand, while probabilistic models consider demand uncertainty.

3(b) Replacement of machine problem

Average annual cost is calculated for each year by adding depreciation and maintenance cost.
The year with minimum average cost is selected as the optimal replacement period.

4(a) Single-server queuing model (M/M/1)

It consists of arrival rate (λ), service rate (μ), queue length, waiting time, and system capacity.
It is widely used to analyze service systems like banks and counters.

4(b) Numerical queuing problem

Using M/M/1 formulas:

Average customers in system = λ / (μ − λ)

Average waiting time = 1 / (μ − λ)
(Full numerical substitution shown in exam.)

5(a) Monte Carlo simulation steps

Steps include problem definition, random number generation, simulation execution, result analysis, and decision making.
It is applied in reliability analysis, inventory control, and risk assessment.

5(b) Capital budgeting using DP

DP evaluates combinations of investments under budget constraints and selects the combination that maximizes expected return.

6(a) Network analysis fundamentals

Network diagrams represent activities and events using arrows and nodes.
Rules ensure logical sequencing.
Used in construction, manufacturing, and software project planning.

6(b) Job sequencing problem

Using Johnson’s rule, jobs are sequenced to minimize total processing time (makespan) across machines M1, M2, and M3.

7(a) Transportation model

Initial solution is obtained using methods like NWCM or VAM, followed by optimality test using MODI method to minimize transportation cost.

7(b) LPP formulation and graphical solution

The problem is formulated with objective function and constraints.
Feasible region is plotted and optimal solution lies at a corner point giving maximum profit.