Problems

CHAPTER 4 – TRANSPORTATION & ASSIGNMENT PROBLEMS PROBLEMS 4.1

Transport Consortium has a contract to move 115 truckloads of sand per week between three areas I, II and III and three destinations, project X, Y and Z. Cost and volume information is given below: Requirement per week Available per week Project Area (truckloads) (truckloads) X 45 I 35 Y 50 II 40 Z 20 III 40 Transportation cost per truckload from areas to destinations ($) is as follows: From To Project X To project Y To Project Z I 5 10 10 II 20 30 20 III 5 8 12 (a) Find a starting solution using least-cost method. (b) Solve the problem using MODI method if the objective is to minimize total transportation cost. (c) Find a starting solution using Northwest-corner method, and then compare the result with the least-cost starting solution obtained in (a). (d) Find a starting solution using Vogel approximation method, and then compare the result with the starting solutions obtained in (a) and (c).

4.2

A manufacturer of hair shampoo and conditioner has three plants located in Georgetown, Subang, and Pasir Gudang. Major warehouses are located at Kuantan, Seremban, Johor Bharu, Ipoh, and Klang. Sales requirements for the next year at each warehouse are given in the table below: Warehouse Requirements: Warehouse Location Annual Sales (000s of cases) Kuantan 50 Seremban 10 Johor Bahru 60 Ipoh 30 Klang 20 170 There are some concerns in the company about which factory should supply each warehouse. The factory capacity at each location is limited. Georgetown has an annual capacity of 100,000 cases. Subang plant has a capacity of 60,000 cases, and Pasir Gudang plant has a capacity of 50,000 cases. The shipping cost from each factory to each warehouse is given in the following table.

Cost of Shipping 1,000 Cases of Hair Shampoo: To Kuantan Seremban Johor Bharu Ipoh Klang From Georgetown $500 $420 $480 $160 $340 Subang $300 $200 $340 $240 $100 Pasir Gudang $440 $300 $100 $320 $300

(a) (b) (c) (d)

Formulate the problem as a transportation problem.

Find a starting solution using least-cost method.

Find a starting solution using Northwest-corner method, and then compare the result with the least-cost starting solution obtained in (b).

Find a starting solution using Vogel approximation method, and then compare the result with the starting solutions obtained in (b) and (c).

4.3

A company produces a product at each of its three plants located at A, B and C. The weekly capacity of the plant at A, B and C are 100 units, 220 units and 150 units respectively. The plants supply four customers W, X, Y and Z. The weekly requirement of W, X, Y and Z are 80 units, 160 units, 80 units and 150 units respectively. The distribution costs (in $) for each unit transported from the plants to the customers are given in the table below.

To W X Y Z A 5 3 7 2 From B 1 5 9 7 C 3 4 9 6 (a) Using the transportation algorithm, determine the distribution that minimizes the transportation cost and the transportation cost associated with this policy. (b) As a result of a fire, the plant at B has its output reduced to 180 units per week.

What effect does this fire have on the supply of the product to customers if the company still wishes to minimize transportation costs? (c) If based entirely on transportation cost, the distribution plan in (a) will reduce the supply of some customers drastically. The company now wishes to restrict the shortage to 15 units per week for any particular customer. What is the new allocation? What is the extra cost of operating this restriction?

4.4

Seni Batik Sdn. Bhd. produces batik clothes for three outlets at each of its three factories. The unit production cost varies with the location due to the differences in the overhead costs. The unit production cost and monthly capacities are shown below.

Factory Unit production cost ($) Monthly production capacity (units) Pasir Mas 50 1500 Wakaf Bharu 45 2500 Pasir Pekan 55 2500 These clothes would be transported to the three outlets. The demand from each outlet and the unit transportation cost (in $) are given in the following table.

Factory Kuala Lumpur Penang Johor Bharu Pasir Mas 7 4 7 Wakaf Bharu 6 5 8 Pasir Pekan 4 4 6 Demand (units) 3000 2300 1200 (a) Set up a transportation table and obtain the initial solution that will minimize the total production and transportation costs. (b) Use the transportation algorithm to determine the optimum distribution plan of the batik clothes.

4.5

A trader wishes to purchase the following quantities of uniform of a special design for sales to one of its regular customers.

Uniform size small medium large extra large Quantity 200 300 450 150 Three manufacturers are willing to supply the uniforms in accordance with the customer’s specification of material and design. From the prices the customer pays and the quotation from the manufactures, the trader estimates that the profit ($) per uniform he expects to be as follows: Manufacturer small medium large extra large A 2.50 3.50 5.00 2.00 B 3.00 3.50 5.50 1.50 C 2.00 4.00 4.50 3.00 (a) What is the maximum profit the trader will earn if each manufacturer is able to supply any number of uniforms of whatever size? (b) If manufacturer A is only able to supply a maximum of 500 uniforms, B a maximum of 300 uniforms and C a maximum of 300 uniforms of any combination of sizes, use the transportation technique to solve how many uniforms of each size should trader placed on the manufacturers to maximize profit. Determine the maximum profit. (c) State whether or not your solution is unique. If not, give one other solution.

4.6

Dugong-Rubber produces tires for four automobile assemblers at each of its three factories. The unit production cost varies with the location due to the differences in the production equipment, cost of materials and labour availability. The unit production cost and monthly capacities are shown below.

Factory Unit production cost Monthly production capacity Melaka 17 800 Kuala Pilah 18 600 Dungun 20 600 The cost of supplying to the automobile assemblers varies from each factory. The demand for each assembler and unit transportation cost in $ are given in the table below. The selling price of the tire is $100 per unit for each assembler.

From / To Shah Alam Pekan Petaling Jaya Butterworth Melaka 4 4 6 7 Kuala Pilah 6 4 8 5 Dungun 9 3 6 5 Demand 600 450 450 500 (a) Find the optimal production and distribution plan that will maximize the company’s profit. (b) Calculate the maximum profit for the optimal solution in (a).

4.7

ABC Semiconductor company produces a particular solid-state module that it supplies to four different television manufacturers, I, II, III and IV. The module can be produced at each of the company's three plants, although the costs vary because of differing production efficiencies at the plants. It costs $1.10 to produce a module at plant A, $0.95 at plant B and $1.03 at plant C. Monthly production capacities of the plants are 7,500, 10,000 and 8,100 modules respectively. The prices that the television manufacturers offer and their monthly requirements are as follows:

Monthly Demand Price Per Module Manufacturer (No. of Modules) $ I 4,200 1.90 II 8,300 1.70 III 6,300 1.80 IV 2,700 1.90 The cost ($) for shipping a module from a factory to a manufacturer is shown below.

Manufacturer I II III IV A 0.11 0.13 0.09 0.19 Factory B 0.12 0.16 0.10 0.14 C 0.14 0.13 0.12 0.15 (a) Explain how the transportation algorithm can be modified for profit maximization rather than cost minimization. (b) Use the transportation algorithm to determine a delivery schedule that will maximize its total contribution to profit from this module. (c) Determine the contribution to profit for the optimal allocation. (d) If the optimal solution in (b) is not unique, describe all other production schedules with the same maximum contribution to profit.

4.8

A legal firm, Al-Ali & Co., has accepted four new cases, each of which can be handled adequately by any one of its five junior partners. Because of differences in experience and expertise, however, the junior partners would spend varying amount of time in the cases. The managing partner has estimated the following time requirements (in hours) for each case and lawyer.

Case 1 Case 2 Case 3 Case 4 Lawyer 1 80 120 60 104 Lawyer 2 72 144 48 110 Lawyer 3 96 148 72 120 Lawyer 4 60 108 52 92 Lawyer 5 64 140 60 96 As the four cases have equal priority, the manager partner would like to assign the lawyers in a way that would minimize the total time necessary to complete all four cases. Determine an optimal assignment of lawyers to cases and hence determine the total number of hours allocated to these four cases.

4.9

The coach of a swimming team needs to assign swimmers to a 200-meter medley relay to prepare for Junior SEA Games. Since most of his best swimmers are very fast in more than one stroke, it is not clear which swimmer should be assigned to each of the four strokes. The four fastest swimmers and their best times (in seconds) achieved in each of the stroke (for 50 meters) are: Stroke Julia Suzana Dianne Elizabeth Backstroke 38 33 34 37 Breaststroke 43 33 40 42 Butterfly 33 28 39 30 Freestyle 29 26 30 31 Help the coach to do the optimal assignment in order to minimize total time.