Question

A company employs plumbers to service, repair and replace pipes and other fittings for customers throughout the country. These plumbers are based at different locations. Four requests for service have been received and the company finds that four plumbers are available. The distances each of the plumbers is from the various customers are given in the following table. The company wishes to assign plumbers to customers to minimise the total distance to be travelled.

PLUMBERS		CUSTOMERS
	1	2	3	4
1	25	18	23	14
2	38	15	53	23
3	15	17	41	30
4	26	28	36	29

1. Using the Hungarian (manual) method, determine the optimal assignment of plumbers to customers in order to minimize the total distance travelled. Indicate this overall distance and show all working.                                                                              [12 marks]
2. Write an LP formulation that could be used to solve this problem with the relevant LP software packages.       [13 marks]

[TOTAL 25 MARKS]

QUESTION 4

A manufacturer of car transmissions wants to develop a cost estimate for a customer’s order for 25 transmissions. It is estimated that the first transmission will take 100 hours of shop time and an 80% learning curve is expected. Using the learning curve tables:-

1. How many labour-hours should the 25^th transmission require?                                 [5 marks]
2. How many labour-hours should the whole order for 25 transmission require?    [5 marks]
3. If the labour hour rate is $50 per hour and the pricing policy of the company is to double the labour costs of the order, what is the customer price for each transmission?     [6 marks]
4. Total Quality Management (TQM) can be an excellent quality management technique if properly implemente Discuss the obstacles associated with the implementation of TQM. [9 marks]

Answer 1

Hungarian method starts with identifying the lowest values at each row. Then it notes the difference between the lowest value on a row and other values on the same row. This is where step 1 is performed. The step 2 is performed in a similar way but with the columns. In step 3 we attempt to cover all the areas with 0 using straight lines. Since the number of lines is not equal to number of dimension, we move ahead to step 4. In this step we identify the area not covered by lines. We note the smallest value of from these cells and subtract it from other cells (that are not covered by line). In addition to this, we add the value to the intersection of lines. We try to cover all the 0s again. This time the number of lines needed is equal to the dimension of the table. This gives up step 5 matrix. In step 5, we identify the rows that have unique 0s and then assign the rest to the others. This completes the assignment and brings us to finish.

The workings are shown below. The assignment is highlighted in green, for example plumber 3 to customer 1, plumber 2 to customer 2, plumber 4 to customer 3 and plumber 1 to customer 4

An LP formulation that can be used on computer software such as Lindo is shown below

Minimize 25X11 + 18X12 + 23X13 + 14X14 + 28X21 + 15X22 + 53X23 + 23X24 + 15X31 + 17X32 + 41X33 + 30X34 + 26X41 + 28X42 + 36X43 + 29X44

ST

X11 + X12 + X13 + X14 = 1

X21 + X22 + X23 + X24 = 1

X31 + X32 + X33 + X34 = 1

X41 + X42 + X43 + X44 = 1

X11 + X21 + X31 + X41 = 1

X12 + X22 + X32 + X42 = 1

X13 + X23 + X33 + X43 = 1

X14 + X24 + X34 + X44 = 1

END

INT X11

INT X12

INT X13

INT X14

INT X21

INT X22

INT X23

INT X24

INT X31

INT X32

INT X33

INT X34

INT X41

INT X42

INT X43

INT X44

[This model when run on Lindo will provide the solution]