Find the optimal solution.

Problem 3 The Lock and Co Hatters produces two types of hats, the panama hat and baseball cap. Each first type of hat which is the panama hat requires twice as much as labor time as the baseball cap. If all hats are baseball only, the company can produce a total of 500 hats a day. The market limits daily sales of the panama hat and baseball cap to 150 and 350 hats respectively. Assuming that the profits per hat are Rs. 8 for panama hat and Rs. 5 for baseball cap. Formulate the problem as a linear programming model in order to determine the number of hats to be produced of each type so as to maximize the profit. Solution: Data Summary Chart Variable Labor Market Demand Profit per Hat Resource Unit Time Hats (Rs.) Limit Panama Hat 2 150 8 150 (max Baseball Cap 1 250 250 (max) Decision Variables: r = Panama Hat x = Baseball Cap . Objective Functions: Maximize (z) = 8x, + 5x2 Constraints: Market Constraints: 2x, + x, $ 500 Panama Hat Sales Constraints: x, $ 150 Baseball Cap Sales Constraints: * $ 250 Non- Negativity: where: *,. x, 20 3 / 4 ::Problem 1 The Brilliant Inc. produces three products sunscreen, moisturizer and facial mist. It uses two types of raw material which are A and B of which 500 and 700 respectively are available. The raw material requirement per unit of product is given below. Raw Material Requirements Sunscreen Moisturizer Facial Mists A 3 5 5 3 5 The labor unit of sunscreen is twice that of the moisturizers and three times that of facial mist. The entire labor force of the firm can produce the equivalent of 3,000 units. The minimum demand of the product is 600, 650 and 500 units respectively. Assuming the profits per unit of sunscreen, moisturizer and facial mist as Rs. 50, 50 and 80. Formulate the LPP and determine the number of units of each product with maximum profit. Solution: . Decision Variable: x1 = Sunscreen * = Moisturizer x, = Facial Mist From the statement, it says that the labor time of sunscreen is twice of moisturizer and thrice of the facial mist. If the x, = 1 than x, = 3 must be twice of x2. x, = 1.5. Which is equivalent to 3,000 unit that is 9,000 unit of all the three products. . Objective Function Maximize (z) = 50x, + 50x, + 80x, Constraints Raw Materials Constraints: A = 3x, + 4x, + 5x, $ 5.000 B = 5x, + 3x, + 5x, $ 7,000 Labor Time Constraints: 6x, + 3x2 + 2x) = $ 1,800 Demand Constraints: x, 2 600 x, 2 650 x 2 500 Non-Negativity: where x1, *2. x, 20Problem 2 Woods products Lid. currently produces two major products, tables and chairs, When sold, each chair yields a profit of Rs. 35 and table Rs. 45. An analysis of the production work sheets reveals the following manufacturing data: Product Man hrs. per unit Machine hrs. per unit Chair 5 0.8 Table 8 1.2 Time available during the year 800 Man Hours 485 Machine Hours The company has a minimum demand for 50 chairs and maximum demand for 25 tables during year 2003. Construct an appropriate linear program for maximizing the profit of Woods Product Lid. Solution: Data Summary Chart Variable per Man hrs. per Machine hrs. Demand Profit Unit Unit per Unit (Units) Chairs 5 0.8 50 (min) Rs.35 Tables B 1.2 25 (max Rs.45 Resource Limits 800 (max) 485 (max) . Decision Variable: x1 = Chairs *2 = Tables . Objective Function: Maximize (z) = 35x, + 45x2 Constraints: Man Hour Constraints: 3 /4 5x, + 8x2 5 800 Machine Hour Constraints: 0.8x1 + 1.2x2 $ 485 Demand Constraints: x, 2 50 x2 2 25 Non-Negativity: where x1, x2 20