11:03 Subject: Operations Research (Assignment) Course: M.com (Semester 2) Max Marks: 10 1) A firm uses three machines in the manufacturing of three products. Each unit of Product A requires 3 hours on machine I, 2 hours on machine II and 1 hour on machine III respectively. Each unit of product B requires 4 hours on machine I, 1 hour on machine II and 3 hours on machine III, while each unit of product \( \mathrm{C} \) requires 2 hours on each of the three machines. The contribution margin of the three products is Rs 30, Rs 40, Rs 35 per unit respectively. The machine hours available on three machines are 90, 54 and 93 respectively. I. Formulate the above problem as a linear programming problem. II. Obtain optimal solution to the problem by using the simplex method. Which of the three products shall not by produced by the firm? Why? III. Calculate the percentage of capacity utilization in the optimal solution. IV. What are the shadow prices of the machine hours? V. Is the optimal solution degenerate? (5 Marks) 2) What is 'game' in game theory? What are the properties of a game? Explain the "best strategy" on the basis of minimax criterion of optimality. (5 Marks)
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Q1 (a) Four products are processed successively on two machines. The manufacturing times in hours per unit of each product are tabulated below for the two machines. The total cost of producing 1 unit of each product is based directly on the machine time. Assume that the costs per hour for machines 1 and 2 are $10 and $15, respectively. The total hours budgeted for all the products on machines 1 and 2 are 500 and 380. If the sales prices per unit for products 1, 2, 3 and 4 are $65, $70, $55 and $45 respectively, formulate the problem as a linear programming model to maximize total net profit. (b) (i) Find the optimal solution for the following Linear Programming problem, using Simplex method. Maximize 2 x1 + x2 + x3 subject to x1 + 5 x2 + x3 ≤ 5 3 x1 + 2 x2 + x3 ≤ 9 5 x1 + 4 x2 + 2 x3 ≤ 10 x1, x2, x3 ≥ 0 (ii) Which of the constraints (1) - (3) are binding, and which of them are non-binding? (iii) Write down the shadow prices of constraints (1) - (3). (iv) Write down the reduced costs for x1, x2 and x3.
Sri K.
The Pinewood Furniture Company produces chairs and tables from two resources - labor and wood. The company has 80 hours of labor and 36 board-ft. of wood available each day. Demand for chairs is limited to 6 per day. Each chair requires 8 hours of labor and 2 board-ft. of wood, whereas a table requires 10 hours of labor and 6 board-ft. of wood. The profit derived from each chair is $400 and from each table, $100. The company wants to determine the number of chairs and tables to produce each day in order to maximize profit. Solve this model by using linear programming. The total number of constraints in this problem, including non-negativity constraints is: 4 7 8 6 5 7. If tables (T) sell for $50 profit and chairs (C) sell for $30 profit, then which of the following represents the objective function? Minimize: Z = 50C + 30T Maximize: Z = 50C + 30T Maximize: Z = 30C + 50T Maximize: Z = 30T - 50C None of the above 8. Cerebro Manufacturing produces four types of structural support fittings - plugs, rails, rivets, and clips - which are machined on two CNC machining centers. The machining centers have a capacity of 250,000 minutes per year. The gross margin per unit and machining requirements are shown in the spreadsheet below. A B C D E F 1 Cerebro Manufacturing Model 2 3 Product Plugs Rails Rivets Clips Machine Capacity (mins./year) 4 Gross margin/unit $0.40 $1.20 $0.80 $1.10 5 Minutes/unit 1 2 3 1.5 6 Gross margin/minute 7 Maximum production 8 Profit Assuming that one half of the machine capacity is used for the production of plugs. What is the maximum possible production of plugs based on this capacity? 83,333.33 125,000 166,666.67 250,000 None of the above
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