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| Mathematical Formulation of Linear Programming Problems |
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| 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. |
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 Identify the decision variables and assign symbols x and y to them. These decision variables are those quantities whose values we wish to determine. |
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Identify the set of constraints and express them as linear equations/inequations in terms of the decision variables. These constraints are the given conditions. |
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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. |
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Add the non-negativity restrictions on the decision variables, as in the physical problems, negative values of decision variables have no valid interpretation. |
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| There are many real life situations where an LPP may be formulated. The following examples will help to explain the mathematical formulation of an LPP. |
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| 01. A diet is to contain at least 4000 units of carbohydrates, 500 units of fat and 300 units of protein. Two foods A and B are available. Food A costs 2
dollars per unit and food B costs 4 dollars per unit. A unit of food A contains 10 units of carbohydrates, 20 units of fat and 15 units of protein. A unit of food B contains 25 units of carbohydrates, 10 units of fat and 20 units of protein. Formulate the problem as an LPP so as to find the minimum cost for a diet that consists of a mixture of these two foods and also meets the minimum requirements. |
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| Suggested answer: |
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| The above information can be represented as |
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| Let the diet contain x units of A and y units of B. |
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| \ Total cost = 2x + 4y |
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| The LPP formulated for the given diet problem is |
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| Minimize Z = 2x + 4y |
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| subject to the constraints |
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| 02. In the production of 2 types of toys, a factory uses 3 machines A, B and C. The time required to produce the first type of toy is 6 hours, 8 hours and 12 hours in machines A, B and C respectively. The time required to make the second type of toy is 8 hours, 4 hours and 4 hours in machines A, B and C respectively. The maximum available time (in hours) for the machines A, B, C are 380, 300 and 404 respectively. The profit on the first type of toy is 5
dollars while that on the second type of toy is 3 dollars. Find the number of toys of each type that should be produced to get maximum profit. |
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| Suggested answer: |
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| Mathematical Formulation |
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| The data given in the problem can be represented in a table as follows. |
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| Let x = number of toys of type-I to be produced |
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| y = number of toys of the type - II to be produced |
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| \ Total profit = 5x + 3y |
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| The LPP formulated for the given problem is: |
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| Maximise Z = 5x + 3y subject to the constraints |
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