(SEM VII) THEORY EXAMINATION 2021-22 OPERATIONS RESEARCH

OPERATIONS RESEARCH – KOE 075

B.Tech (Semester VII) – Detailed Answers

SECTION A – Descriptive Answers (2 Marks Each)

a) Slack and Surplus Variables

Slack and surplus variables are introduced in linear programming problems to convert inequalities into equations. A slack variable is added to a “≤” constraint to represent unused resources, while a surplus variable is subtracted from a “≥” constraint to represent excess usage over the minimum requirement.

b) Basic Feasible Solution (BFS) of LPP

A basic feasible solution of a linear programming problem is a solution obtained by setting n − m variables equal to zero and solving the remaining m equations, where n is the number of variables and m is the number of constraints. A BFS must satisfy all constraints and non-negativity conditions.

c) Unbalanced Assignment Problem

An assignment problem is said to be unbalanced when the number of jobs is not equal to the number of machines. To solve it, a dummy row or dummy column with zero cost is added to balance the problem.

d) Objective of Transportation Problems

The main objective of a transportation problem is to determine the most economical way of transporting goods from multiple sources to multiple destinations while satisfying supply and demand constraints at minimum total transportation cost.

e) PERT

PERT (Program Evaluation and Review Technique) is a network-based project management tool used to plan, schedule, and control uncertain projects. It uses probabilistic time estimates to determine project duration.

f) Rules for Drawing Network Diagram

The rules include:                                                  Each activity must have a unique start and end node

No looping is allowed                                           Only one arrow should represent one activity

Dummy activities are used when required to maintain logical relationships

g) Saddle Point and Optimal Strategy

A saddle point is a position in a payoff matrix where the maximum of row minimum equals the minimum of column maximum. An optimal strategy is the strategy that guarantees the best outcome regardless of the opponent’s action.

h) Customer Behaviors

Customer behaviors in queuing theory include arrival pattern, service pattern, waiting discipline (FIFO, LIFO, priority), and customer impatience such as balking and reneging.

i) Costs on which EOQ Depends

EOQ depends on ordering cost, carrying (holding) cost, purchase cost, and shortage cost.

j) Deterministic vs Stochastic Inventory Models

SECTION B – Long Answers (10 Marks Each)

2(a) OR Models and Graphical Solution of LPP

Classification of OR Models

Operations Research models are classified as:                   Deterministic models

Probabilistic models                                                           Static models

Dynamic models                                                                 Simulation models

Graphical Solution

Minimize                                                                             Z = 2x₁ + 3x₂

Subject to:                                                                           x₁ + 2x₂ ≥ 40
2x₁ + x₂ ≥ 50                                                                       x₁, x₂ ≥ 0

By plotting the constraints and identifying the feasible region, corner points are evaluated. The minimum value of Z occurs at the intersection point satisfying both constraints.

2(b) Transportation as a Special LPP & Least Cost Method

A transportation problem is a special type of LPP because it has a linear objective function and linear constraints.
Using the Least Cost Method, allocations are made starting from the cell with the lowest cost until all supply and demand constraints are satisfied, resulting in an initial basic feasible solution.

2(c) Network Analysis and Its Significance

Network analysis is a technique used for project planning and scheduling using network diagrams. It helps in identifying critical activities, minimizing project duration, and efficient resource utilization.

Differences:

2(d) Game Theory

Game theory is useful in competitive business situations such as pricing, advertising, and bidding.
Using graphical method, optimal strategies for both players are determined by plotting payoff lines and finding equilibrium.

2(e) Inventory Types and Costs

Types of inventory include raw materials, work-in-progress, finished goods, and spare parts.
Inventory is maintained to meet demand, avoid shortages, and take advantage of bulk purchasing.

Inventory costs include:

Ordering cost                                                                Holding cost

Shortage cost                                                                Purchase cost

Economic parameters include demand rate, lead time, order quantity, and cost factors.

SECTION C – Very Long Answers (10 Marks Each)

3(a) Simplex Method Solution

Maximize                                                                       Z = 3x + 2y

Subject to:                                                                     2x + y ≤ 18
2x + 3y ≤ 42                                                                  3x + y ≤ 24
x, y ≥ 0

By converting inequalities into equations using slack variables and applying simplex iterations, the optimal solution is obtained at:
x = 6, y = 6                                                                   Z = 30

3(b) Infeasibility in LPP & Dual Problem

Infeasibility occurs when no solution satisfies all constraints simultaneously. Indicators include contradictory constraints and artificial variables not leaving the basis.

The dual of the given primal problem is formulated by converting constraints into dual variables and reversing inequality signs.

4(a) MODI Method

The MODI (Modified Distribution) method is used to test optimality of a transportation problem. After computing uᵢ and vⱼ values, opportunity costs (Δij) are evaluated. If all Δij ≥ 0, the solution is optimal; otherwise, further iterations are performed.