Question
An agriculturist has a 125-acre farm. He produces radish, muttar (Peas) and potato. Whatever he raises is fully sold in the market. He gets Rs. 5 for radish per kg, Rs 4 for muttar (Peas) and Rs 5 for potato per kg. The average per acre yield is 1500 kg of radish, 1800 kg of muttar (Peas) and 1200 kg of potato. To produce each 100 kg of radish and muttar (Peas) and 80 kg of potato, a sum of Rs 12.50 has to be used for manure. Labour required for each acre to raise the crop is 6 man-days for radish and potato each and 5 man-days for muttar (Peas). A total of 500 man-days for labour at a rate of Rs 40 per man-day are available. Formulate this as a linear programming model to maximize the agriculturist's total profit.
Answer :
Word Count : 921
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Skip in 10
Operations Research provides a systematic and quantitative framework for solving optimization problems involving limited resources. In the given situation, the agriculturist must decide how to allocate his limited land and labour among three crops—radish, muttar (peas), and potato—in such a way that total profit is maximized. Since the relationships between inputs and outputs are assumed to be linear and all production is sold in the market, the problem can be appropriately formulated as a linear programming model. The first step in formulating a linear programming problem is to define the decision variables. Let the number of acres devoted to radish be denoted by (x_1), the number of acres devoted to muttar (peas) by (x_2), and the number of acres devoted to potato by (x_3). These variables represent the decisions under the control of the agriculturist. Since acreage cannot be negative, all decision variables must be non-negative. The next step is to express the objective of the agriculturist in mathematical terms. The objective is to maximize total profit from the farm. Profit is obtained as the difference between total revenue and total variable costs, which in this case consist of manure costs and labour costs. Revenue depends on the yield per acre and the market price per kilogram for each crop. For radish, the average yield per acre is 1500 kg, and the selling price is Rs. 5 per kg. Therefore,
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