(SEM VII) THEORY EXAMINATION 2022-23 OPERATIONS RESEARCH

SECTION A – Short Answers (2 Marks Each)

(a) Steps involved in Operations Research

The steps include problem formulation, construction of mathematical model, solution of the model, testing the model, establishing controls, and implementation of the solution.

(b) Limitations of Linear Programming

LP assumes linearity, certainty, divisibility, and non-negativity. It cannot handle non-linear relationships, qualitative factors, or uncertain environments.

(c) Optimality test in transportation problem

Optimality test checks whether the current basic feasible solution is optimal by computing opportunity cost (Δij). If all Δij ≥ 0, the solution is optimal.

(d) Difference between transportation and assignment problem

Transportation problem deals with shipping goods from sources to destinations at minimum cost, while assignment problem assigns one job to one person with a one-to-one relationship.

(e) Critical path

Critical path is the longest duration path in a network diagram which determines the minimum project completion time.

(f) Meaning of crashing

Crashing is the process of reducing project duration by shortening activity times at an additional cost.

(g) Limitations of Game Theory

It assumes rational players, limited strategies, complete information, and does not handle uncertainty or large real-life situations well.

(h) Queuing system (M/M/C):(∞/FIFO)

It represents a system with Poisson arrivals, exponential service time, multiple servers, infinite queue capacity, and First-In-First-Out discipline.

(i) Costs in inventory problems

Costs include ordering cost, carrying (holding) cost, shortage cost, and purchase cost.

(j) Group replacement

Group replacement refers to replacing all items together at a fixed interval instead of individual replacement upon failure.

SECTION B – Long Answers (10 Marks Each)

(a) Advantages, limitations and applications of Linear Programming

Advantages:
Optimal utilization of resources, improved decision-making, cost minimization, and profit maximization.

Limitations:
Linear relationships, certainty assumption, inability to include qualitative factors.

Applications:
Production planning, transportation, diet problems, scheduling, and resource allocation.

(b) Transportation problem using VAM & optimality test

Using Vogel’s Approximation Method (VAM), an initial basic feasible solution is obtained by calculating penalties and allocating minimum cost cells.
Optimality is tested using the Stepping Stone Method. If all opportunity costs are non-negative, the solution is optimal.

(c) Types of floats in network analysis

Total Float: Maximum delay allowed without delaying project completion.
Independent Float: Delay allowed without affecting preceding or succeeding activities.
Interference Float: Portion of total float affecting successor activities.
Free Float: Delay without affecting earliest start of succeeding activities.

(d) Game theory problem (Prachi vs Pavitra)

By analyzing the payoff matrix, saddle point is checked.
If no saddle point exists, mixed strategy is applied to compute the value of the game.
Prachi should adopt the strategy that maximizes minimum gain based on available information.

(e) EOQ – Manufacturing model (Numerical – Method)

Given production rate, demand rate, setup cost, and carrying cost:
Optimal number of cycles per year is calculated using manufacturing EOQ formula.
Time between two orders = 1 / number of cycles.
Batch cycle time = production quantity / production rate.

SECTION C – Long Answers (10 Marks Each)

3(a) Airlines ticket problem (Linear Programming)

Let x₁ = first class tickets, x₂ = economy class tickets.
Formulate constraints based on capacity, minimum reservation, and preference ratio.
Solve graphically to find optimal values of x₁ and x₂.
Maximum profit is obtained at the feasible corner point.

3(b) Simplex method problem

The problem is converted into standard form.
Slack and surplus variables are introduced.
Simplex table is constructed and iterated until optimality condition is satisfied.
Final table gives optimal values of X₁, X₂ and Z.

4(a) Transportation problem & VAM steps

Transportation problem minimizes cost of shipping goods.
Steps in VAM include:

Calculate row and column penalties

Select highest penalty

Allocate minimum cost cell

Adjust supply and demand

Repeat until completion

4(b) Assignment problem (Numerical)

The problem is solved using Hungarian Method.
Row and column reductions are performed, zeros are assigned, and minimum cost assignment is obtained.

5(a) Network diagram & critical path

Network diagram is drawn using given activities.
Forward and backward pass calculations determine earliest and latest times.
Critical path is identified where total float is zero.

5(b) PERT and time estimates

PERT is a probabilistic network technique.
Optimistic time (to), pessimistic time (tp), and most likely time (tm) are defined.
Expected time:

te=to+4tm+tp6t_e = \frac{t_o + 4t_m + t_p}{6}te​=6to​+4tm​+tp​​ 

6(a) Graphical method for 2×n and m×2 games

Payoff lines are plotted.
Upper and lower envelopes are identified.
Intersection point gives the value of the game and optimal mixed strategies.

6(b) Queuing system problem (Numerical)

Arrival rate λ = 6/hr, service time = 1.5 min.
Average waiting time, service time, and most probable service time are calculated using queuing formulas.

7(a) EOQ derivation (Manufacturing model)

EOQ is derived by minimizing total cost (ordering + carrying cost).
Manufacturing EOQ considers finite production rate and zero shortages.

7(b) Replacement problem

Average annual cost is calculated for each year considering time value of money.
The year with minimum average cost gives the optimal replacement policy.