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TUTORIAL 4 1. A Head of Department has five jobs, A, B, C, D, E and five people V, W, X, Y, Z who are capable of carrying out the jobs. He assesses the number of hours each person would take to perform each job: V W X Y Z A 3 5 10 15 8 JOB B 4 7 15 18 8 C 8 12 20 20 12 D 5 5 8 10 6 E 10 10 15 25 10 How should the jobs be allocated to the men to minimise the total time taken? 2. Glenaff Ltd have appointed five new clerks A, B, C, D, E to fill the five positions of cashier, invoice clerk, payroll clerk, cost clerk and general ledger clerk. In order to find the optimal assignment of people to posts all the clerks were given a test that covered the five areas of work involved. Each section of the test was marked out of ten and the scores were as follows: Cashier Invoicing Payroll Costing General Ledger A 3 3 1 3 5 B 5 8 9 4 4 C 1 9 10 2 7 D 5 5 9 6 7 E 5 5 4 5 6 What is the optimum assignment of people to posts? 3. A market research firm has three clients who have each requested that the firm conduct a sample survey. Four available statisticians can be assigned to these three projects; however, all four statisticians are busy, and therefore each can handle only one of the clients. The following data show the number of hours required for each statistician to complete each job; the differences in time are based on experience and ability of the statisticians. Tutorial 4.doc Client Statistician A B C 1 150 210 270 2 170 230 220 3 180 230 225 4 160 240 230 a. Formulate a linear programming model for this assignment problem. b. Solve the assignment problem using the Hungarian method. b. Suppose that the time it takes statistician 4 to complete the job for client A is increased from 160 to 165 hours. What effect will this have on the solution? c. Suppose that the time it takes statistician 4 to complete the job for client A is decreased to 140 hours. What effect will this have on the solution? d. Suppose that the time it takes statistician 3 to complete the job for client B increases to 250 hours. What effect will this have on the solution? (ASW : Ch 7, Qn 21 – min - sensitivity) 4. The Department head of a management science department at a major university will be scheduling faculty staff to teach courses during the coming Autumn term. Four core courses need to be covered. The four courses are at the UG, MBA, MS and PhD levels. Four professors will be assigned to the courses, with each professor receiving one of the courses. Student evaluations of professors are available from previous terms. Based on a rating scale of 4 (excellent), 3 (very good), 2 (average), 1 (fair) and 0 (poor), the average student evaluations for each professor are shown. Professor D does not have a PhD, and cannot be assigned to teach the PhD level course. If the department head makes teaching assignments based on maximising the student evaluation ratings over all four course, what staffing assignments should be made? Course Professor UG MBA MS PhD A 2.8 2.2 3.3 3.0 B 3.2 3.0 3.6 3.6 C 3.3 3.2 3.5 3.5 D 3.2 2.8 2.5 - (ASW : Ch 7, Qn 20 – max - prohibited) 5. Steel mills in three cities produce the following amounts of steel: Location Weekly Production (tons) A. Bethlehem 150 B. Birmingham 210 C. Gary 320 Tutorial 4.doc 680 These mills supply steel to four cities where manufacturing plants have the following demand: Location Weekly Demand (tons) 1. Detroit 130 2. St Louis 70 3. Chicago 180 4. Norfolk 240 620 Shipping costs (£) per ton of steel are as follows: To From 1 2 3 4 A 14 9 16 18 B 11 8 7 16 C 16 12 10 22 Because of a lorry drivers’ strike, shipments are presently prohibited from Birmingham to Chicago. a. Set up a transportation tableau for this problem and determine the initial solution using minimum cost method. b. Solve this problem using the MODI method. c. Are there any alternate optimum solutions? Explain. If so, identify them. d. Formulate this problem as a general linear programming model. (Taylor : Ch 6, Qn 9 – min – unbalanced - alternative) Tutorial 4.doc 6. Coal is mined and processed at the following four mines in Kentucky, West Virginia and Virginia: Location Capacity (tons) A. Cabin Creek 90 B. Surry 50 C. Old Fort 80 D. McCoy 60 280 These mines supply the following amount of coal to utility power plants in three cities: Plant Demand (tons) 1. Richmond 120 2. Winston-Salem 100 3. Durham 110 330 The railroad shipping costs (£1000s) per ton of coal are shown in the following table. Because of railroad construction, shipments are presently prohibited from Cabin Creek to Richmond: To From 1 2 3 A 7 10 5 B 12 9 4 C 7 3 11 D 9 5 7 a. Set up the transportation tableau for this problem, determine the initial solution using the least cost method, and compute total cost. b. Solve using the MODI method. c. Are there multiple optimal solutions? Explain. If there are alternative optimum solutions, identify them. (Taylor : Ch 6, Qn 11 – min – prohibited - unbalanced) 7. A manufacturing firm produces diesel engines in four cities - Phoenix, Seattle, St Louis and Detroit. The company is able to produce the following numbers of engines per month: Plant Production 1. Phoenix 5 2. Seattle 25 3. St Louis 20 4. Detroit 25 Tutorial 4.doc Three trucking firms purchase the following numbers of engines for their plants in three cities: Firm Demand A. Greensboro 10 B. Charlotte 20 C. Louisville 15 The transportation costs per engine (£’00) from sources to destinations are shown in the following table. However, the Charlotte firm will not accept engines made in Seattle, and the Louisville firm will not accept engines from Detroit; therefore, these routes are prohibited. To From A B C 1 7 8 5 2 6 10 6 3 10 4 5 4 3 9 11 a. Set up the transportation tableau for this problem. Find the initial solution using VAM. b. Solve for the optimal solution using the stepping-stone method. Compute the total minimum cost. c. Formulate this problem as a linear programming model. (Taylor: Ch 6, Qn 13 – min – unbalanced - prohibited) Tutorial 4.doc