Chapter 09 - Practice Problems Problem 1: The Bay City Parks and Recreation Department has received a federal grant of $600,000 to expand its public recreation facilities. City council representatives have demanded four different types of facilities gymnasiums, athletic fields, tennis courts, and swimming pools. In fact, the demand by various communities in the city has been for 7 gyms, 10 athletic fields, 8 tennis courts, and 12 swimming pools. Each facility costs a certain amount, requires a certain number of acres, and is expected to be used a certain amount, as follows: Facility Gymnasium Athletic field Tennis court Swimming pool Cost Required Acres $80,000 24,000 15,000 40,000 4 8 3 5 Expected Usage (people/week) 1,500 3,000 500 1,000 The Parks and Recreation Department has located 50 acres of land for construction (although more land could be located, if necessary). The department has established the following goals, listed in order of their priority: (1) The department wants to spend the total grant because any amount not spent must be returned to the government. (2) The department wants the facilities to be used by a total of at least 20,000 people each week. (3) The department wants to avoid having to secure more than the 50 acres of land already located. (4) The department would like to meet the demands of the city council for new facilities. However, this goal should be weighted according to the number of people expected to use each facility. a. Formulate a goal programming model to determine how many of each type of facility should be constructed to best achieve the city's goals. Problem 2: Computers Unlimited sells microcomputers and distributes them from three warehouses to four universities. The available supply at the three warehouses, demand at the four universities, and shipping costs are shown in the following table: Warehouse Richmond Atlanta Washington Demand University Tech A&M State $22 17 30 15 35 20 28 21 16 520 250 400 Central 18 25 14 380 Supply 420 610 340 Instead of its original objective of cost minimization, Computers Unlimited has indicated the following goals, arranged in order of their importance: 1 2 3 4 5 A B A&M has been one of its better long-term customers, so Computers Unlimited wants to meet all of A&M's demands. Because of recent problems with a trucking union, it wants to ship at least 80 units from the Washington warehouse to Central University. To maintain the best possible relations with all its customers, Computers Unlimited would like to meet no less than 80% of each customer's demand. It would like to keep total transportation costs to no more than 110% of the $22,470 total cost achieved with the optimal allocation, using the transportation solution method. Because of dissatisfaction with the trucking firm it uses for the Atlanta-to-State deliveries, it would like to minimize the number of units shipped over this route. Formulate a goal programming model for this problem to determine the number of microcomputers to ship on each route to achieve the goals. Solve this model by using the computer. Chapter 09 - ANSWERS Problem1: Minimize P1d1, P2d2, P3d3+, 3P4d4 + 6P4d5 + P4d6 + 2P4d7 subject to 80,000x1 + 24,000x2 + 15,000x3 + 40,000x4 + d1 = 600,000 1,500x1 + 3,000x2 + 500x3 + 1,000x4 + d2 d2+ = 20,000 4x1 + 8x2 + 3x3 + 5x4 + d3 d3+ = 50 x1 + d4 d4+ = 7 x2 + d5 d5+ = 10 x3 + d6 d6+ = 8 x4 + d7 d7+ = 12 where x1 = no. of gymnasiums, x2 = no. of athletic fields, x3 = no. of tennis courts, x4 = no. of pools Problem 2: a) Minimize Z = P1 (d1 + d2 + d3), P2d5, P3d8, P4 (d9 + + d10 d11 + d12 ), P5 d13 , P6 d14 subject to x1A + x1B + x1C + x1D + d1 = 420 x2A + x2B + x2C + x2D + d2 = 610 x3A + x3B + x3C + x3D + d3 = 340 x1A + x2A + x3A + d4 = 520 x1B + x2B + x3B + d5 = 250 x1C + x2C + x3C + d6 = 400 x1D + x2D + x3D + d7 = 380 x3D + d8 d8+ = 80 + x1A + x2A + x3A + d9 d9 = 416 x1B + x2B + x3B + = 200 d10 d10 x1C + x2C + x3C + d11 d11 = 320 x1D + x2D + x3D + d12 d12 = 304 22x1A + 17x1B + 30x1C + 18x1D + 15x2A + 35x2B + 20x2C + 25x2D + 28x3A + 21x3B + 16x3C + 14x3D + = $24,717 d13 d13 x2C d14 = 0 Note that the negative deviational variables must be at the highestpriority level to force all the supply to be used. b) x1B = 215, x1C = 205, x2A = 520, x2D = 90, x3C = 50, x3D = 290 A 1 2 B C 215 205 520 3 Demand Achieved 520 215 D 420 90 610 50 290 340 255 380 The noninteger solution values were rounded to integer values (making sure no rim requirements were violated)