Question

Assignment LP Model

The Shang-ho Project Development and Construction Company has just been awarded a chemical plant construction contract. The contract terms require that at least five (5) other smaller companies be awarded subcontracts for portions of the total work. Shang-ho therefore requested bids from five (5) shortlisted small companies to do subcontract work in five (5) areas. Because of the production capacity, each company can work only in one area.

The bids submitted by the companies, in thousands of dollars, are as follows:

Company	1	2	3	4	5
A	$10	$6	$4	$8	$3
B	2	9	7	7	6
C	6	11	12	5	9
D	5	4	2	1	4
E	5	9	8	5	9

Required:

a. Use the Hungarian (manual) method, to determine which bid Shang-ho should accept in order to minimize overall costs. [12 marks]

b. Write an LP formulation that could be used to solve this problem. [13 marks]

Answer 1

Answer:

Hungarian (manual) method

Step 1: Subtract row minimums from each row

Step 2: Subtract column minimums from each column

Step 3. Cover all the zeros of the matrix with the minimum number of horizontal or vertical lines

Step 4. Note that the number of these lines = 4 which is less than the no. of resources (= 5), so we yet have to achieve optimality.

Step 5. Subtract smallest uncovered no. (= 1, in this case) from all uncovered elements, and add it to all elements that are covered twice (horizontal and vertical lines). Then repeat Step 3.

Note that since the number of lines is equal to that of the resources (= 5), we have reached optimality. The optimal assignment is as follows:

LP formulation

Let Xij be the binary integer such that Xij=1 when i-th company is assigned to the jth task.

Minimize Z =
10 X11 + 06 X12 + 04 X13 + 08 X14 + 03 X15 +
02 X21 + 09 X22 + 07 X23 + 07 X24 + 06 X25 +
06 X31 + 11 X32 + 12 X33 + 05 X34 + 09 X35 +
05 X41 + 04 X42 + 02 X43 + 01 X44 + 04 X45 +
05 X51 + 09 X52 + 08 X53 + 05 X54 + 09 X55

Subject to,

Xi1 + Xi2 + Xi3 + Xi4 + Xi5 = 1; i=1,2,..,5

X1j + X2j + X3j + X4j + X5j = 1; j=1,2,..,5

Xij = {0,1}

---------------------

Solution

NOTE:: I HOPE YOUR HAPPY WITH MY ANSWER....***PLEASE SUPPORT ME WITH YOUR RATING..

THANK YOU