Linear programming A large number of decision problems faced by a business manager involve allocation of limited resources to different activities. Linear programming has been successfully applied to a variety of problems of management, such as production, advertising, transportation, refinery operation, investment analysis, etc. Over the years, linear programming has been found useful not only in business and industry but also in non-profit organizations such as government, hospitals, libraries, education, etc. Actually, linear programming improves the quality of decisions by amplifying the analytic abilities of a decision maker. Please note that the result of the mathematical models that you will study cannot substitute for the decision maker's experience and intuition, but they provide the comprehensive data needed to apply his knowledge effectively
What is the meaning of Linear Programming? The word 'linear' means that the relationships are represented by straight lines, i.e., the relationships are of the form k = p + qx. In other words, it is used to describe the relationships among two or more variables, which are directly proportional. The word 'programming' is concerned with optimal allocation of limited resources.
Some Important terms:- Solution of LPP:- A set of values of the variables x 1, x 2,..x n satisfying the constraints of a L.P.P. is called a solution of a L.P.P. Linear Function of L.P.P. A linear function contains terms each of which is composed of only a single, continuous variable raised to (and only to) the power of 1. Objective Function The linear programming problem must have a well defined objective function to optimize i.e. it is a linear function of decision variables expressing the objective of the decisionmaker which is to be optimize. For example maximization of profit, minimization of cost.
Constraints There must be limitations on resources which are to be allocated among various competing activities or These are linear equations arising out of practical limitations. The mathematical forms of the constraints are: f(x) b or f(x) b or f(x) = b. Feasible Solution of L.P.P. Any non-negative solution which satisfies all the constraints is known as a feasible solution. Optimal Solution The solution where the objective function is maximized or minimized is known as optimal solution.
Linear programming deals with the optimization (maximization or minimization) of a function of variables known as objective functions. It is subject to a set of linear equalities and or/ inequalities known as constraints. Linear programming is a technique which involves the allocation of limited resources in an optimal manner. LP Formulation Graphical Method Simplex Method Max Min
Egg contains 6 units of vitamin A per gram and 7 units of vitamin B per gram and cost 12 paise per gram.milk contains 8 units of vitamin A per gram and 12 units of vitamin B per gram and costs 20 paise per gram.the daily min requirement of vitamin A and vitamin B are 100 units and 120 units respectively.find the optimal product mix.
Maximization Case:- Example 1:- A manufacturing company in producing two products A and B. each unit of product A requires 2 kg of raw material and 4 labour hours for processing. Whereas each unit of product B requires 3 kg of raw material and 3 hours of labour of the same type. Every week, the firm has an availability of 60 kg of raw material and 96 labour hours. One unit of product A sold yields Rs. 40 and one unit of product B sold gives Rs. 35 as profit. Formulate this problem as a linear programming problem to determine as to how many units of each of the products should be produced per week so that the firm can earn the maximum constraint so that all that is produced can be sold.
Solution: In this problem, our goal is the maximization of profit, which would be obtained by producing (and selling) products A and B. if we let x 1 and x 2 represent the number of units produced. Per week products A and B respectively. Total profit X = 40x 1 + 35x 2..Objective Function Or Max Z = 40x 1 + 35x 2 Notice that function is linear one. The constraints:- Another requirements of linear programming is that the resources must be limited supply. The limitation itself is known as a constraints. Each unit of Product A requires 2 kg of raw material while each unit of Product B needs 3 kg. The total consumption would be 2x 1 + 3x 2 which cannot exceed the total availability of 60 kg every week. 2x 1 + 3x 2 < 60 Similarly it is given that a unit of A requires 4 labour hours for its production and one unit of B requires 3 hours with an availability 96 hours a week, we have 4x 1 + 2x 2 < 96
Non Negativity Condition: - X 1 and x 2 being the number of units produced, can not have negative values. X 1 > 0, x 2 > 0 Maximize CZ = 40x 1 + 35x 2 Sub to constraint 2x 1 + 3x 2 < 60 Raw material constraints 4x 1 + 3x 2 < 96 Labour hours constraints x 1, x 2 > 0 Non Negative constraints restriction
The Minimization Case: Example 2:- In an agricultural Research Institute suggested to a farmer to spread out at least 4800 kg of a special phosphate fertilizer and not less than 7200 kg of a special nitrogen fertilizer to raise productivity of crops in his fields. There are two sources for obtaining these mixtures A and B. Both of these are available in bags weighing 100 kg each and they cost Rs 40 and Rs 24 respectively. Mixture A contains phosphate and nitrogen equivalent of 20 kg and 80 kg respectively while mixture B contains these ingredients equivalent of 50 kg each. Write this as a linear programming problem and determine how many bags of each type the farmer should buy in order to obtain the required fertilizer at minimum cost.
Solution: Let x 1 and x 2 are number of bags of mixture A and B respectively. Objective Function Minimize Z = 40x 1 + 24x 2 cost The constraints A B Total Phosphate 20 50 4800 Nitrogen 80 50 7200 Constraints Min Z = 40x 1 + 24x 2 Cost requirement 20x 1 + 50x 2 > 4800 Phosphate requirement 80x 1 + 50x 2 > 7200 Nitrogen restriction x 1, x 2 > 0 Non Negative constraints
Formulation of L.P.P. In general terms a linear programming problem can be written as Consider the following Optimize (maximize or minimize) z = c 1 x 1 + c 2 x 2 + c 3 x 3 +...+ c n x n subject to Constraints :- a 11 x 1 + a 12 x 2 + a 13 x 3 +...+ a 1n x n ( = ) b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 +...+ a 2n x n ( = ) b 2...... a m1 x 1 + a m2 x 2 + a m3 x 3 +...+ a mn x n ( = )b m x 1, x 2,..., x n ³ 0
Maximization Problem Maximize Z = C X Subject to ax < b X > 0 Minimization Problem Minimization Z = C X Subject to ax > b X > 0
Application of Linear Programming in different areas:- Application in Petroleum Industry:- This industrial area has furnished a great many important and interesting linearprogramming applications. The one of them is the problem of blending gasonlines into required products for maximum profits. Other may be the problems of optimum crude allocation to several refineries, and the optimum inventory and production rate for a seasonal product. Application in Commercial Airlines:- The applications is related to the problems of routing air craft and of airline management. One L.P. problem may be to determine the pattern and timing of flights. Application in Communication Industry: This application is another area in which problems involving facilities for transmission, switching, relaying etc. are solved by linear programming.
Application in Paper Industry:- The application of linear programming in the pulp and paper industry has been in the transportation problem. The problem is how to assign the various orders to the mills so as to reduce the total company freight bill to a minimum. Application in Farm Management:- In this field the limited resources such as labour, acreage, water supply, working capital etc., are allocated in such a way that maximize net revenue. Application in Chemical Industry:- In chemical industry, the application is mostly related to production and inventory control. Our purpose is to find the optimal scheduling of different machines of various capacities, power limitations and other restrictions.
Limitations of Linear Programming:- 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 can not be used. Factors such as uncertainty, weather conditions etc are not taken into consideration. There may not be an integer as the solution e.g. the number of men required may be a fraction and the nearest integer may not be the optimal solution. i.e. linear programming technique may give practical valued answer which is not desirable. One single objective is dealt with while in real life situations, problems come with multi-objectives. Parameters are assumed to be constants but in reality.
Example 1: A firm makes two types of locks A and B. the contributions for each product as calculated by the accounting department are Rs. 30 per lock A and Rs. 20 per lock B. both locks are processed on three machines M 1, M 2, and M 3. The time required by each lock and total time available per week on each machine are as follows: How should the manufacturer schedule his production in order to maximize contribution? Solution: let x = number of locks A to be produced y = number of locks B to be produced here the object is to maximize the profit, thus objective function is given by Maximize z = 30x + 20y Subject to constraints: 3x +3y < 26 (Total tome of machine M 1 ) 5x + 2y < 30 (Total time of machine M 2 ) 2x + 6y < 40 (Total time of machine M 3 )
A paper mill produces two grades of paper namely X and Y.Because of raw material restrictions, It cannot produce more than 400 tons of grade X and 300 tons of grade Y in a week.there are 160 production hours in a week.it requires 0.2 and 0.4 hours to produce a ton of products X and Y respectively with corresponding profits of 200 and rs 500 per ton.formulate the above as a LPP to maximize profit and find the optimum product mix.
An electric appliance company produces two products refrigerators and ranges. Production takes place in two separate departments I and II.Refrigerators are produced in department I and ranges in department II.the company s two products are sold on a weekly basis. The weekly production can not exceed 25 refrigerators and 35 ranges.the company regularly employ a total of 60 workers in two departmenta A refrigerator requires 2 man-weeks labour while a range requires 1 man week labour.a refrigerator contributes a profit of Rs 60 and a range contributes a profit of Rs 40 how many units of refrigerators and ranges shouls the company produce to realize the maximum profit? Formulate the above as a LPP.
Example 3:A manufacturer produces two type of models- P 1 and P 2. Each model of the type P 1 requires 4 hrs of grinding and 2 hrs of polishing where as each model of the type P 2 requires 2 hrs of grinding and 5 hrs of polishing. The manufacturers have 2 grinders and 3 polishers. Each grinder works 40 hrs a week and each polisher works for 60 hours a week. Profit on P 1 model is Rs 3.00 and on model P 2 is Rs 4.00 Whatever is produced in a week is sold in a market. How should the manufacturer allocate his production capacity to the two types of models,so that he may make the maximum profit in a week?