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Find initial basic feasible solution for given problem by using: (a)North – West corner rule (b) Least cost method and (c) Vogel’s approximation method If the objective is to minimize the total transportation cost. Q. 2 A Company has factories at F1, F2, and F3 which supply to warehouses at W1, W2, and W3. Weekly factory capacities are 200,160,and 90 units, respectively. Weekly warehouse requirement are 180,120and 150 units respectively. Unit shipping costs (in rupees) are as follows: Ware house W1 W2 W3 SUPPLY F1 16 20 12 200 F2 14 8 18 160 F3 26 24 16 90 DEMAND 180 120 150 450

Determined the optimal distribution for this company to minimize total shipping cost. Second Topic Transportation Model Problems for Practice Q. 1 A company has four manufacturing plants and five warehouses. Each plant manufactures the same product which is sold at different prices in each warehouse area. The cost of manufacturing and cost of raw material are different in each plant to various factors. The capacities of the plants are also different. The data are given in the following table: PLANT ITEM Manufacturing cost (RS) per unit Raw material cost (RS)per unit Capacity per unit time 1 12 08 100 2 10 07 200 3 08 07 120 4 08 05 80

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The company has five warehouses. The sales prices, transportation costs and demands are given in the following table: ware house transportation cost per unit sale price demand per unit(Rs. ) 1 2 3 4 A 4 7 4 3 30 80 B 8 9 7 8 32 120 C 2 7 6 10 28 150 D 10 7 5 8 34 70 E 2 5 8 9 30 90 (a) Formulate this problem as a transportation problem to maximize profit. (b) Find the solution using VAM method. Q. 2 Obtain an optimal solution to the transportation problem by UV method. Use VAM method for obtaining initial feasible solution D1 19 70 40 5 D2 30 30 8 8 D3 50 40 70 7 D4 10 60 20 14 CAPACITY 7 9 18 34

S1 S2 S3 DEMAND Q. 3 Consider a firm having 2 factories. The firm is to ship its products from the factories to three- retail stores. The number of units available at factories X and Y are 200 and 300, respectively while those demanded at retail stores A ,B and C are 100,150 and 250, respectively. Rather than shipping directly from factories to retail stores, it is asked to investigate the possibility of Trans- shipment. The transportation cost (in Rupees) per unit is given in the table . FACTORY X Factory X Y 0 6 y 8 0 RETAIL STORE A 7 11 B 8 9 C 9 10 Retail store A B C 7 1 8 2 5 9 0 1 7 5 0 8 1 4 0

Find out the optimal shipping schedule. Q. 4 ABC limited has three production shops supplying a product to five warehouses. The cost of production varies from shop to shop and cost of transportation from one shop to a warehouse also varies. Each shop has a specific production capacity and each warehouse has certain amount of requirement. The cost of transportation are given below: Ware house III 4 7 6 85 A B C DEMAN D I 6 5 3 60 II 4 6 4 80 IV 7 4 3 105 V 5 8 4 70 SUPPLY 100 125 175 400 The Cost Of Manufacturing The Product At Different Production Shop Is shop A B C Variable cost 14 16 15 Fixed cost 7000 4000 5000

Find the optimum quantity to be supplied from each shop to different warehouses at minimum total cost. Third Topic Assignment Model Problems discussed in the class Q. 1 A job production unit has four jobs A,B,C,D which can be manufactured on each of the four machines P,Q,R and S. The processing cost of each job on each machine is given in the table below: Jobs P A B C D 31 25 19 38 Machine Q R Processing cost (Rs. ) 25 33 24 23 21 23 36 34 S 29 21 24 40 To achieve minimum processing cost, which job will you process on which machine? Q. 2 A workshop has four machines and four tasks for completion.

Each of the machines can perform each of four tasks. Time taken at each of the machines to complete each task is given in the matrix below: How should the tasks be assigned to machines requirement of machine hours? Tasks A I II III IV 51 32 37 55 Machine B C Processing times (Hrs. ) 77 49 34 59 44 70 55 58 D 55 68 54 55 Q. 3 A pharmaceutical company has four branches, one each at city A, B, C and D. A branch manager is to be appointed one at each city, out of four candidates P, Q, R and S. The monthly business depends upon the city and the effectiveness of the branch manager in that city.

Branch manager P 11 City B C Monthly business (Rs. lakhs) 11 9 A D 9 Q R S 13 12 16 16 17 14 11 13 16 10 8 12 Which manager should be appointed at which city so as to get maximum total monthly business? Q. 4 The production cost or products P1, P2, P3, P4 and P5 per unit made on machines M1, M2, M3, M4 and M5 are tabulated below: P1 50 60 40 35 40 P2 80 30 40 40 45 P3 30 40 50 30 50 P4 40 40 45 35 45 P5 45 50 35 50 45 M1 M2 M3 M4 M5 Selling prices per unit are as follows: P1= Rs. 80, P2=Rs. 90, P3=Rs. 105, P4= Rs. 70 and P5 = Rs. 5 Decide which product should be made on which machine to realize maximum profit. Q. 5 Darda oil mills have four plants each of which can manufacture anyone of the four products. The manufacturing costs differ from plant to plant and so do the sales revenues. The revenue and cost details are as given below: Sales revenue (Rs. Lakhs) Plants A B C D I 70 80 75 78 II 88 90 87 85 Manufacturing cost (Rs. Lakhs) Plants A B C D I 59 65 62 65 II 70 73 72 74 Products III 55 55 59 58 IV 71 79 68 76 Products III 69 71 73 74 IV 82 94 80 89 Suggest which plant should produce which product to maximize profit? Q. A company has four territories open, and four salesman available for an assignment. The territories are not equally rich in its sales potential. It is estimated that, a typical salesman, operating in each territory would bring in the following annual sales. Territory Annual Sales(Rs. ) I 126000 II 105000 III 84000 IV 63000 The four salesmen also differ in their ability. It is estimated that, working under the same conditions, their yearly sales would be proportionately as follows: Salesman Proportion A 7 B 5 C 5 D 4 Assign the salesmen to each territory if the criterion is maximum expected total sales.

Third Topic Assignment Model Problems for Practice Q. 1 A departmental head has four subordinates and four tasks for completion. The subordinates differ in their capabilities and tasks differ in their work contents and intrinsic difficulties. His estimate of time for each subordinate and each task is given in matrix below: Tasks I A B C D 17 28 20 28 Subordinates II III Processing cost (Rs. ) 25 26 27 23 18 17 25 23 IV 20 25 14 19 How should the tasks be assigned to minimize requirements of man-hours? Q. 2 A departmental head has three subordinates and four tasks for completion.

The employees differ in their capabilities and the tasks differ in their work contents. With the performance matrix given below, which three of four tasks should be assigned to subordinates? Tasks Subordinates A B C D I 9 8 20 21 II 12 13 12 15 III 11 17 13 17 Q. 3 A gear manufacturer requires 2000 numbers per month of each of the six types of gears. Six hobbing machines are available to process these gears. The gears differ in their work contentsgear with, number of teeth, module etc-and machine differ in their capabilities-speeds, feeds and ability to take depth of cut.

The production control department has prepared the machine wise cost matrix as shown in the matrix below: Gear I II III IV V VI M1 15 20 19 30 6 13 M2 18 16 16 8 12 Hobbing machines M3 M4 13 10 12 14 15 42 38 10 12 16 14 M5 18 19 35 9 15 M6 14 15 20 36 10 18 Gear I can be assigned to machine M5 because of steep helix angle. Gear III can not be assigned to machine M4 as it is not within the capacity of this machine. And gear IV can not be loaded on machine M2 because of limitations of process capability of the machine. Find the optimum assignment schedule.

Q. 4 A salesman has to visit five cities. He wishes to start from a particular city, visit each city once and return to starting city. The cost of going from a city to another in Rs. is given below: From City A B C D E 0 16 18 21 11 To city A 12 0 17 14 13 B 15 13 0 18 12 C 17 18 14 0 18 D 11 12 17 16 0 E Determine the least cost route. Q. 1 A departmental store purchases Christmas trees, which can be ordered only in lots of 100. Each tree selling price Rs. 40 each. Unsold trees, however, have no salvage value. The purchase price of the trees is Rs. 5 each The probability distribution obtained from analysis of past data is given below: Trees sold 100 200 300 400 500 probabilities 0. 20 0. 35 0. 25 0. 15 0. 05 (a) Setup a payoff table (b) How much quantity should the departmental store buy to maximize its profit? Q. 2 A manufacturer of sewing machines is faced with the problem of selecting one of the two models for manufacturing. The profit depends on the market acceptability of the model which are present is uncertain but is had been broadly classified into four categories: excellent, good, fair and poor.

The profits or losses (losses are indicated by negative sign) expected by the management from the different levels of market acceptability of the models are as follows: ____________________________________________________________ ______ Market Profit (Rs. ) for the model for the Indicated market acceptability __________________________ Deluxe Janata ____________________________________________________________ ______ Excellent 60,000 78,000 Good 28,000 38,000 Fair 18,000 8,000 Poor 8,000 -12,000 ____________________________________________________________ _____ Which product should the company select from the standpoint of maximin (gain) criterion? Q. 3 A company is making a large boiler installation. A certain automatic monitoring unit is critical for the operation of the whole system. At the time of original order, the spares for this unit can be purchased for Rs. 2,000 per unit. The probability distribution for the failure of the unit during the life time of installation is given as : __________________________________________________________ _______Failure_________________________Probability__________ _ 0 0. 35 1 0. 25 2 0. 20 3 0. 15 4 0. 05 ___________________________________________________________ If a spare is needed and is not available, the total cost of idle time and replacement cost will be Rs. 15,000. Unused spares have no salvage value. Determine the optimal no. of spares to be ordered. Q. 4 A newspaper boy is thinking of selling a special one time edition of a magazine to his regular newspaper customers. Based on his he believes that he can sell between 9 to 12 copies. sports knowledge of his customers, The magazines can be purchased at Rs. each and can be sold for Rs. 12 each. Magazine that are not sold can be returned to the publisher for a refund of 50%. (a) Construct the decision matrix for the above inventory problem indicating possible monetary consequences. (b) Determine the best decision from the stand point of (i) Maximin criteria (ii) Maximax criteria (iii)Hurwiez a-criterion assuming a=0. 40 (iv) Minimax regret criteria (v) Laplace criteria Q. 5 Agent Corner, an authorized dealer in domestic appliances find that the cost of holding refrigerator in stock for a week is Rs. 50.

Customers who cannot get a new refrigerator immediately wanted to go to another dealer for which expected profit is Rs. 350 per customer. Probability distribution of demand is as follows: No. of refrigerator: 0 1 2 3 4 5 6 Probability : 0. 05 0. 10 0. 20 0. 30 0. 20 0. 10 0. 05 Assuming that there is no time lag between ordering and delivery, how many refrigerators should we order per week? Q. 6 A departmental store buys Christmas tree at a landing cost of Rs 25 each and sells them at an average of Rs 40. Any tree left over after the selling season has no resale value.

The productivity distribution of sale of trees derived from analysis of pas t sales data is under: Tree (sale) Probability 100 200 300 400 500 600 700 0. 10 0. 15 0. 35 0. 20 0. 10 0. 05 0. 05 a) How many trees should be department store buy to maximize its profit? b) If trees left after the selling season cost Rs 5 each to remove ,does it affect the inventory decision? Q. 7 A newspaper boy is thinking of selling a special one time edition of a sports magazine to his regular newspaper customers. Based on his knowledge of his customer the copies of the magazine with probabilities estimated as under: No of copies 6 7 8 9 10 11 12 probability 0. 0 0. 15 0. 35 0. 20 0. 10 0. 05 0. 05 The magazine can be purchased at Rs 8 each and can be sold for Rs 12 Each. a) Magazines that are not sold can be required to the publisher for a refund of 50%. Determine optimum quantity to be purchased? b) If the publisher does not take back the unsold magazines and the boy is forced to sell them as scrap at rs 1. 50, what should be the order quantity. c) And if the boy gets magazines “on sale basis”,what quantity should be ordered? Q. 8 A Ship building company has launched a program for the construction of new class of ships.

Certain spare units like prime over, each costing 200000 have to be purchased. If these units are not available when needed, a serious loss is incurred which is in order of Rs 10000000 each instance requirements of spares with the corresponding probabilities are given below. Nos of spares: 0 1 2 3 4 5 Probability of 0. 876 0. 062 0. 041 0. 015 0. 005 0. 001 requirement How many spares should the company buy in order to optimize inventory decision? Fifth Topic Decision Analysis Problems for Practice Q. 1 A perishable item is ordered only once each demand period.

Acquisition cost is $3, selling price is $5, and salvage value is $1. 50. What is optimal order quantity? Given: Demand 100 110 120 130 140 150 Probability 0. 1 0. 2 0. 2 0. 3 0. 1 0. 1 Q. 2 A newspaper boy buys papers for Rs 1. 30 each and sells them for Rs 1. 40 each. He cannot return unsold newspapers. Daily demand has the following distribution. No. of customers: Probability: 23 0. 01 24 0. 03 25 0. 06 26 0. 10 27 0. 20 28 0. 25 30 0. 10 31 0. 05 32 0. 05 If each day’s demand is independent of the previous day’s, how many papers he should order each day?