OPERATIONS RESEARCH ASSIGNMENT


Theory
Introduction
The mathematical models which tells to optimise (minimize or maximise) the objective function Z subject to certain condition on the variables is called a Linear programming problem (LPP).
During World War II, the military managements in the U.K and the USA engaged a team of scientists to study the limited military resources and form a plan of action or programme to utilise them in the most effective manner. This was done under the name 'Operation Research' (OR) because the team was dealing with research on military operation.
Linear Programming Problems (LPP)
The standard form of the linear programming problem is used to develop the procedure for solving a general programming problem.
A general LPP is of the form
Max (or min) Z = c1x1 + c2x2 + … +cnxn
x1, x2, ....xn are called decision variable.
Application Areas of Linear Programming
The Application Areas of Linear Programming are:
1. Transportation Problem
2. Military Applications
3. Operation of System Of Dams
4. Personnel Assignment Problem
5. Other Applications: (a). manufacturing plants, (b). distribution centres, ©. production management and manpower management.
Basic Concept of Linear Programming Problem
Objective Function: The Objective Function is a linear function of variables which is to be optimised i.e., maximised or minimised. e.g., profit function, cost function etc. The objective function may be expressed as a linear expression.
Constraints: A linear equation represents a straight line. Limited time, labour etc. may be expressed as linear inequations or equations and are called constraints.
Optimisation: A decision which is considered the best one, taking into consideration all the circumstances is called an optimal decision. The process of getting the best possible outcome is called optimisation.
Solution of a LPP: A set of values of the variables x1, x2,….xn which satisfy all the constraints is called the solution of the LPP..
Feasible Solution: A set of values of the variables x1, x2, x3,….,xn which satisfy all the constraints and also the non-negativity conditions is called the feasible solution of the LPP.
Optimal Solution: The feasible solution, which optimises (i.e., maximizes or minimizes as the case may be) the objective function is called the optimal solution. Important terms Convex Region and Non-convex Sets.
Mathematical Formulation of Linear Programming Problems
There are mainly four steps in the mathematical formulation of linear programming problem as a mathematical model. We will discuss formulation of those problems which involve only two variables.
1. Identify the decision variables and assign symbols x and y to them. These decision variables are those quantities whose values we wish to determine.
2. Identify the set of constraints and express them as linear equations/inequations in terms of the decision variables. These constraints are the given conditions.
3. Identify the objective function and express it as a linear function of decision variables. It might take the form of maximizing profit or production or minimizing cost.
4. Add the non-negativity restrictions on the decision variables, as in the physical problems, negative values of decision variables have no valid interpretation.
Advantages of Linear Programming
i. The linear programming technique helps to make the best possible use of available productive resources (such as time, labour, machines etc.)
ii. In a production process, bottle necks may occur. For example, in a factory some machines may be in great demand while others may lie idle for some time. A significant advantage of linear programming is highlighting of such bottle necks.
Limitations of Linear Programming
(a). Linear programming is applicable only to problems where the constraints and objective function are linear i.e., where they can be expressed as equations which represent straight lines. In real life situations, when constraints or objective functions are not linear, this technique cannot be used.
(b). Factors such as uncertainty, weather conditions etc. are not taken into consideration.






















ANALYSIS-
Objective Function – Maximization of profit by selling new flavors with
Answer Report –
 The demand is not fully met for large, small, family packets as they are not binding. We can come to this conclusion through answer reports.
 We can conclude that the storage capacity is not fully utilized and we can store more ice creams as we have more capacity as the storage for all the ice cream packs are non binding and shows slack of 10,10 & 5 respectively.
 Quantity all type of packs are binding thus hence we conclude that the company is possessing optimum quantity of ice cream and also no extra cost is incurred.
SENSITIVITY REPORT –
 Demand for small, large, family packs show a shadow price of “0” The increase in quantity demanded will not change any other variables i.e factors as the demand for these packs indicates a shadow price of “0” and same is the case with storage.
• Procurement of small, large, family packs show a shadow price of 40, 60 & 150 respectively which indicates that increase in the quantity of small cup by one unit will indicate an increase the total cost by 40.