SECTION A
(Attempt all questions in brief – 2 × 7 = 14 marks)
(a) Why is optimization required?
Optimization is required to obtain the best possible solution under given constraints. It helps in minimizing cost, time, or resources and maximizing profit, efficiency, or performance in engineering and industrial problems.
(b) Mathematical formulation of a problem
Mathematical formulation is the process of expressing a real-life problem in mathematical terms using decision variables, objective function, and constraints so that optimization techniques can be applied.
(c) CPM (Critical Path Method)
CPM is a network-based project scheduling technique used to determine the longest path of activities in a project. It identifies critical activities that directly affect project completion time.
(d) Dynamic Programming
Dynamic programming is an optimization technique that solves complex problems by breaking them into smaller overlapping sub-problems and solving them stage by stage.
(e) Queueing model
A queueing model studies situations where customers wait for service. It analyzes arrival rate, service rate, waiting time, and system capacity to improve service efficiency.
(f) Simulation
Simulation is a technique that imitates the behavior of a real system using a model. It is used when analytical solutions are difficult or impossible to obtain.
(g) Network logic
Network logic represents the sequence and dependency of activities in a project network, showing how activities are logically connected.
SECTION B
(Attempt any THREE – 7 × 3 = 21 marks)
2(a) Mathematical formulation of design problems
Design problems are formulated as mathematical programming problems by identifying design variables, defining an objective function (such as minimum cost or weight), and specifying constraints like strength, capacity, or safety limits.
For example, minimizing material cost of a beam subject to stress and deflection constraints can be formulated as a linear or nonlinear programming problem.
2(b) Sequencing and its relevance in engineering
Sequencing determines the optimal order of jobs on machines to minimize total processing time or idle time.
In engineering industries, proper sequencing improves productivity and reduces machine downtime.
In 2 jobs through m machines, the objective is to find the order of processing that minimizes total elapsed time using sequencing rules.
2(c) Principle of dominance
The principle of dominance states that a solution or strategy is dominated if there exists another solution that performs better under all conditions.
Dominated strategies are eliminated to reduce problem size and simplify decision-making, especially in game theory and optimization problems.
2(d) Monte Carlo simulation and its applications
Monte Carlo simulation uses random numbers to solve problems involving uncertainty.
It is widely used in engineering for reliability analysis, risk assessment, inventory control, and queuing problems where probabilistic behavior is involved.
2(e) Individual and group replacement policies
Individual replacement replaces items as they fail, while group replacement replaces all items at fixed intervals.
These policies are applied in engineering for maintenance of machines, bulbs, tools, and equipment to minimize total replacement and downtime costs.
SECTION C
3(a) Simplex method and Dual (Duplex) method
The Simplex method is an iterative technique used to solve linear programming problems by moving from one feasible solution to another until the optimal solution is reached.
The Dual (Duplex) method solves the dual of a linear programming problem and is useful when constraints are violated but optimality conditions are satisfied.
Both methods are widely applied in production planning, transportation, and resource allocation.
3(b) Historical development of optimization
Optimization developed from classical mathematics, calculus, and economics. Early work focused on maxima and minima problems. Later, linear programming, game theory, dynamic programming, and computer-based optimization techniques evolved, making optimization a vital tool in modern engineering.
4(a) Transportation model
The transportation model determines the optimal distribution of goods from multiple sources to multiple destinations at minimum cost while satisfying supply and demand constraints.
It is solved using methods like North-West Corner Rule, Least Cost Method, and MODI method.
4(b) Travelling Salesman Problem (TSP)
The Travelling Salesman Problem finds the shortest possible route that visits each city exactly once and returns to the starting point.
Applications include route planning, logistics, circuit design, and scheduling.
5(a) Forward and backward computation in PERT
Forward computation determines earliest start and finish times, while backward computation determines latest start and finish times.
These computations help identify critical activities and project slack.
5(b) Single server model
In a single server model, one server provides service to arriving customers.
Applications include bank counters, machine repair systems, and call centers, where waiting time and queue length are analyzed.
6(a) Capital budgeting and cargo-loading problems
Capital budgeting involves selecting projects that maximize return under limited capital.
Cargo-loading problems determine optimal loading of items to maximize value without exceeding weight or volume limits.
6(b) Types of simulation
Simulation types include deterministic simulation, stochastic simulation, continuous simulation, and discrete event simulation.
They are used in manufacturing systems, traffic flow, and inventory management.
7(a) Deterministic and probabilistic inventory models
Deterministic models assume known demand and lead time, while probabilistic models consider uncertainty.
Both are applied in engineering to control inventory cost and avoid shortages.
7(b) Equipment renewal problem
The equipment renewal problem determines the optimal replacement time of equipment considering maintenance cost, efficiency loss, and replacement cost to minimize total cost.
