ANNA UNIVERSITY - M.C.A IV SEMESTER Problem Set MC1752/MC9242/ Resource Management Techniques

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1 Sou rce S r i V e n k a t e s w a r a C o l l e g e o f E n g i n e e r i n g, S r i p e r u m b u d u r D e p a r t m e n t o f C o m p u t e r A p p l i c a t i o n s ANNA UNIVERSITY - M.C.A IV SEMESTER Problem Set MC1752/MC9242/ Resource Management Techniques U N I T I I T R A N S P O R T A T I O N A N D A S S I G N M E N T M O D E L S 1. Transportation Model 1. A company has three production facilities S1, S2, and S3 with production capacity 7, 9 and 18 units ( 00s) per week of a product respectively. These units are to be shipped to four warehouses D1, D2, D3 and D4 with requirements 5, 6, 7 and 14 units ( 00s) per week, respectively. The transportation costs, in rupees, per unit between factories and warehouses are given in the table below. Formulate this problem as an LP model to minimize the total transportation cost. D 1 D 2 D 3 D 4 S S S A dairy firm has three plants located in a state. The daily milk production at each plant is as follows: P1 -> 6 million litres, P2 ->1 m.l, P3 ->10 m.l. Each day, the firm must fulfill the needs of four distribution centers. A minimum requirement at each center is as follows: D1 -> 7 m.l, D2 -> 5 m.l, D3 ->3 m.l, and D4 ->2 m.l. Transportation costs of 1 m.l. from a plant to a distribution center (in hundreds of rupees) are shown below. Find the initial solution by (i) North-West corner rule (ii) Least cost method (iii) Row minimum method (iv) Column Minimum method and (v) Vogel s Approximation Method (VAM). D 1 D 2 D 3 D 4 P P P Find the initial solution to the following transportation problem. D 1 D 2 D 3 D 4 Supply S S

2 Source Origin Factory Origin 2 S Demand Find the initial solution to the following transportation problem. D 1 D 2 D 3 D 4 Supply O O O Demand Find the initial solution to the following transportation problem. Distribution Centers D 1 D 2 D 3 D 4 Supply F F F Demand Find the initial solution to the following transportation problem. D 1 D 2 D 3 D 4 Supply O O O Demand Obtain the optimum solution of the following transportation problem. (Problem 1) D 1 D 2 D 3 D 4 Supply S S S Demand (Solution: x11 = 5, x14 = 2, x22 = 2, x23 = 7, x32 = 6 and x34 = 12; Total Cost = Rs. 743)

3 Factory Warehouse s Factory 3 8. Obtain the optimum solution of the following transportation problem. Warehouse W 1 W 2 W 3 Supply F F F Demand (Solution: x11 = 140, x13 = 60, x21 = 40, x22 = 120, and x33 = 90; Total Cost = Rs. 5,920) 9. Obtain the optimum solution of the following transportation problem. Markets M 1 M 2 M 3 M 4 Supply W W W Demand (Solution: x12 = 12, x13 = 2, x14 = 8, x23 = 15, x31 = 7 and x34 = 1; Total Cost = Rs. 149) 2. Variants of Transportation Model - Unbalanced 10. Obtain the optimum solution of the following transportation problem. (Unbalanced) Warehouse W 1 W 2 W 3 Availability F F F Requirement (Solution: x12 = 76, x22 = 21, x23 = 41, x24 = 20, x31 = 72 and x32 = 5; Total Cost = Rs. 2,424) 11. Obtain the optimum solution of the following transportation problem, considering transportation cost and production costs given below. The production costs per unit of product in each of factories F 1, F 2, F 3, and F 4 are Rs. 2, Rs. 3, Re. 1 and Rs. 5 Respectively. (Hint: Form New cost matrix by adding production cost with transportation costs. Solution: x11 = 25, x12 = 5, x13 = 20, x23 = 70, x32 = 30, x43 = 15, x44 = 20, x45 = 15; Total Cost = Rs. 1,465)

4 Machines Source Sources Factories 4 Stores S 1 S 2 S 3 S 4 Production F F F F Demand The following matrix contains distance, in kilometers, between source and destination. The Shipping costs per load per Kilometer Rs. 10. Find the optimum shipping schedule. (Solution: x15 = 8, x21 = 4, x24 = 1, x32 = 4, x33 = 2, x34 = 3, x43 = 3; Total Cost = Rs. 920) D 1 D 2 D 3 D 4 D 5 Supply S S S Demand Variants of Transportation Model Degenerate Solution 13. Solve the following transportation problem. (Degenerate Solution) D 1 D 2 D 3 Availability S S S Requirement (Solution: x11 = 70, x13 = 50, x22 = 80, x31 = 80; Total Cost = Rs. 1900) 14. Obtain the optimum solution of the following transportation problem. M/c 4 cannot be assigned to task 4. Tasks T 1 T 2 T 3 T 4 T 5 Supply M M M M

5 Manufacturer Plants Factory 5 Demand Obtain the optimum solution of the following transportation problem. Warehouses W 1 W 2 W 3 W 4 W 5 Production F F F F Demand Obtain the optimum solution of the following transportation problem. Warehouses W 1 W 2 W 3 W 4 Supply P P P Demand Variants of Transportation Model Maximization Type 17. Obtain the optimum solution of the following transportation problem. Entry in the matrix shows the profit per saree. Types of Sarees T 1 T 2 T 3 T 4 T 5 Supply M M M M Purchase (Solution: x12 = 25, x14 = 50, x15 = 200, x21 = 150, x32 = 75, x33 = 75, x44 = 200; Total Profit = Rs)

6 Men Jobs Program s 6 5. Assignment Model 1. Find the optimum assignment to the following assignment problem that minimizes the total computer time. Entry in the matrix denotes the time for developing the programs. (P1->R3, P2->R2 and P3->R1 and total time=280) Programmers R 1 R 2 R 3 P P P The following matrix represents the time taken by employee to perform the given jobs. Find the optimum assignment that minimizes the total man-hours. Employees E 1 E 2 E 3 E 4 E 5 J J J J J (Solution: J1 -> E2, J2 -> E1, J3 -> E5, J4 -> E3 and J5 -> E4 and Optimum Man-hours = 23) 3. Five Men are available to do five jobs. The following table represents the time in hours taken by men to do each jobs. Find the assignment of men to jobs that minimizes the total time. Jobs J 1 J 2 J 3 J 4 J 5 M M M M M (Solution: M1 ->J3, M2 ->J5, M3 ->J1, M4 ->J4 and M5 ->J2 and Optimum Time = 13 Hrs.) 4. A Pharmaceutical company wants to transport some medicines to 5 different deficit cities from 5 different surplus cities. The distance between these cities is shown in the

7 Repairman Subordinates Surplus City 7 table below. Determine optimum assignment and minimum distance. (Solution: S1 -> D5, S2 ->D3, S3 ->D2, S4 ->D1 and S5 ->D4 and Minimum Distance = 570 Km) Deficit City D 1 D 2 D 3 D 4 D 5 S S S S S A departmental head has four subordinates and four tasks to be performed. The subordinates differ in their efficiency and the tasks differ in the level of difficulty. His estimate of the times that each man would take to perform each task is given below in the matrix. How should the tasks be allocated to subordinates so as to minimize the total man-hours? (Solution: S1 ->T1, S2 ->T3, S3 ->T2, S4 ->T4 and Optimum Time = 41 Hrs.) Tasks T 1 T 2 T 3 T 4 S S S S In an automobile shop, four repairmen are to be assigned to four different jobs. Man ours required by each man to do each job is shown in the table. Find the Optimum assignment and total minimum man hours. Jobs J 1 J 2 J 3 J 4 R R R R (Solution: R1 ->J2, R2 ->J3, R3 ->J4, R4 ->J1 and Optimum time = 17 hours.) 7. A Construction company has requested bids for five different projects. Five companies have responded, their bid amounts are shown below. Determine the best bidders to

8 Sales Equipments Bidders 8 each of the projects that minimize the total bid amount. (Solution: B1 ->A5, B2 ->A2, B3 - >A1, B4 ->A3 and B5 ->A4 and Minimum Cost = Rs. 1,55,000) Bid Amounts ( 000) A 1 A 2 A 3 A 4 A 5 B B B B B Variations of Assignment Model Unbalanced & 7. Variations of Assignment Model Multiple assignment 8. A hospital needs three types of medical equipment and five manufacturers are willing to supply them. The following table shows the price quoted by each manufacturer for the equipments. Obtain the manufacturer that offer equipments at lower price. (Solution: M1 -> E, M2 -> E and M3 -> E and Total Price =Rs. ) Manufacturers M 1 M 2 M 3 M 4 M 5 E E E Variations of Assignment Model Maximization Type 9. The following table gives sales amount of five different salesmen in five different districts. Find the optimum assignment that will maximize the total sales thereby getting maximum profit. (Solution 1: P1 ->D2, P2 ->D1, P3 ->D5, P4 ->D3 and P5 ->D4 and Maximum Sales = Rs. 191) and (Solution 2: P1 ->D2, P2 ->D5, P3 ->D1, P4 ->D3 and P5 ->D4 and Maximum Sales = Rs. 191). Districts D 1 D 2 D 3 D 4 D 5 P P Person P P

9 Staff Centers Machines 9 P Variations of Assignment Model Unbalanced and Restricted Assignment 10. Four new machines are to be installed at five different locations in a factory. Because of limited space, machine M2 and M3 cannot be installed at locations L3 and L1 respectively. The following table provides cost of installation at each location. Find the optimum assignment that gives minimum costs. Locations L 1 L 2 L 3 L 4 L 5 M M M M (Solution: M1 ->L1, M2 ->L2, M3 ->L5, M4 ->L4 and Minimum Installation Cost = Rs. 32) 11. Four different techniques are to be followed in order to increase the production in three different centers. The following matrix shows the increase in production in units when different techniques are assigned to each centers. Find the optimum production assignment. (Solution 1: C1 ->T4, C2 -> T3 and C3 -> T1 and total Increase in Production = 30 Units) and (Solution 2: C1 ->T1, C2 -> T3 and C3 -> T4 and total Increase in Production = 30 Units) Techniques T 1 T 2 T 3 T 4 C C C Four courses are to be assigned to four staff members. The table given shows the relative ratings (100 = best rating) of each staff to teach each course. Find the optimum rating and assignment of courses to staff that maximizes the educational quality. (Solution: S1 ->C3, S2 ->C2, S3 ->C4, S4 ->C1 and Maximum Rating = 210) and (Solution 2: S1 ->C1, S2 ->C2, S3 ->C4, S4 ->C3 and Maximum Rating = 210) Courses C 1 C 2 C 3 C 4 S S S

10 From City From City From City 10 S Variations of Assignment Model Traveling Salesman Problem 13. A Traveling Salesman has to visit five cities. He wishes to start from a particular city, visits each city only once and then returns to his starting point. The traveling cost (in Rs. 000) of each city from a particular city is given below. What is the sequence of visit of the salesman so that minimizes the total cost? (Solution: C1 -> C2 ->C3 -> C4 -> C5 -> C1 and Total Minimum Cost = Rs. 15,000) To City C 1 C 2 C 3 C 4 C 5 C C C C C Solve the following Traveling Salesman Problem. (Solution 1: C1 -> C4 ->C2 -> C3 -> C1 and Total Minimum Cost = Rs. 19) and (Solution 2: C1 -> C3 ->C2 -> C4 -> C1 and Total Minimum Cost = Rs. 19) To City C 1 C 2 C 3 C 4 C C C C A Traveling Salesman has to visit five cities. The distance between each city (in Km) is shown below. Find the optimum route that passes through each city only once, starting and ending at A with minimum distance. To City C 1 C 2 C 3 C 4 C 5 C C C C C (Solution: C1 -> C3 ->C4 -> C2 -> C5 -> C1 and Total Minimum Distance = 30 Km)

11 From City A Traveling Salesman has to visit five cities. The distance between each city (in Km) is shown below. Find the optimum route that passes through each city only once, starting and ending at his home city A, with minimum distance. To City C 1 C 2 C 3 C 4 C 5 C C C C C (Solution: C1 -> C4 ->C3 -> C2 -> C5 -> C1 and Total Minimum Distance = 1165 Km)