FORMULATION OF LINEAR PROGRAMMING PROBLEM (LPP): Formulation of a Linear Programming Problem involves constructing a mathematical model from the given data. This can be done only if the following requirements are met: There should be a clearly identifiable objective and it should be measurable in quantitative terms. E.g. In a manufacturing problem the objective can be maximisation of profit or minimisation of cost. The resources to be allocated in the problem should be identifiable and quantitatively measurable. E.g. The use of labour time, or raw material in the manufacturing process should be clearly stated. The relationships representing the objective function and the constraints equations must be linear. There should be a series of feasible alternative courses of action available to the decision maker. These are determined by the resource constraints. When all the above mentioned conditions are satisfied the problem can be expressed as L.P. problem. Then solve it for Continue reading
Management Science
Duality in linear programming
Corresponding to every linear programming problem, there is another linear programming problem. The given problem is called the primal and the other its dual. Although the idea of duality is essentially mathematical, it has important interpretations. This can help managers in answering questions about alternative courses of action and their effect on values of the objective function. When the primal problem is of the maximisation type the dual is of the minimisation type and vice versa. It is an interesting feature of the simplex method that we can use it to solve either the original problem – the primal – or the dual. Whichever problem we start out to solve, it will also give us the solution to the other problem. Consider the following general linear programming problem. Maximise Z = c1x1 + c2x2, Subject to a11x1 + a12x2
Operations Research approach of problem solving
Operations Research approach of problem solving Optimization is the act of obtaining the best result under any given circumstance. In various practical problems we may have to take many technical or managerial decisions at several stages. The ultimate goal of all such decisions is to either maximize the desired benefit or minimize the effort required. We make decisions in our every day life without even noticing them. Decision-making is one of the main activity of a manager or executive. In simple situations decisions are taken simply by common sense, sound judgment and expertise without using any mathematics. But here the decisions we are concerned with are rather complex and heavily loaded with responsibility. Examples of such decision are finding the appropriate product mix when there are large numbers of products with different profit contributions and production requirement or planning public transportation network in a town having its own layout of Continue reading
Introduction to Linear Programming (L.P)
The mathematical definition of linear programming (L.P.) can be stated as – “It is the analysis of problems in which a linear function of a number of variables is to be maximized (minimized), when those variables are subject to a number of restraints in the form of linear inequalities”. Linear programming models thus belong to a class of mathematical programming models concerned with efficient allocation of resources to known activities with the objective of meeting a desired goal. Organizations can have many goals. Hence, a wide variety of problems can be efficiently solved using L.P. technique. Here are a few examples: A product mix problem: Decide the combination of various product quantities to maximise profit or to minimise production cost. Allocation of bank funds: To achieve highest possible returns. This should be achieved within liquidity limits set by RBI and maintaining flexibility to meet the customers demand for loans. Manufacturing Continue reading
Economic Interpretation of Linear Programming Duality
We see that the primal and the dual of linear programming are related mathematically, we can now show that they are also related in economic sense. Consider the economic interpretation of the duality of linear programming – first for a maximization problem and then for a minimization problem. The maximization problem: Consider the following linear programming problem. The optimal solution to this problem dives production of 18 units of Xi and 8 units of x2 per week. It yields the maximum prof of a Rs. 1000, Maximize Z = 40×1 + 35×2, Subject to 2×1 + 3X2 < or = 60, Raw materials constraint per week. 4×1 + 3X2 < or = 96, Capacity constraint per week. x1,x2 > or = 0 The optimal solution to this problem gives production of 18 units of x1 and 8 units of x2 per week. It yields the maximum profit of a Rs. Continue reading
Model of Quantitative Analysis
Quantitative analysis is a scientific approach to decision making. As a first step to decision making, decision model has to be evolved. The decision model depends on two factors, namely the problem and the problem environment. Defining the Problem The first step in decisions making is defining the problem. The problem (i.e. the threat, opportunity, etc.) must be fully understood as to its nature, dimensions, intensity and so on. Take labor absenteeism. It becomes a problem when it is rampant affecting work schedules. (one or two absentees, here and there is no problem). What is its nature? Deliberate absenteeism, absenteeism due to unavoidable causes, absenteeism as a mark of protest to certain managements attitudes/actions, absenteeism due to healthy/family/social reasons, etc. are indicative of the nature of the problem of absenteeism. Specific nature must be understood, since each type of absenteeism needs a Continue reading