The Patricia Garcia Company is producing seven new medical p

The Patricia Garcia Company is producing seven new medical products. Each of Garcia\'s eight plants can add one more product to its current line of medical devices. The unit manufacturing costs for producing the different parts at the eight plants are shown below. How should Garcia assign the new products to the plants in order to minimize manufacturing costs?

Solution

Since the number of plants and number of products are not equal so the problem is not balanced, in this case we add a fictitious product row with zero cost.

1

2

3

4

5

6

7

8

C53

0.10

0.12

0.13

0.11

0.10

0.06

0.16

0.12

C81

0.05

0.06

0.04

0.08

0.04

0.09

0.06

0.06

D5

0.32

0.40

0.31

0.30

0.42

0.35

0.36

0.49

D44

0.17

0.14

0.19

0.15

0.10

0.16

0.19

0.1

E2

0.06

0.07

0.10

0.05

0.08

0.10

0.11

0.05

E35

0.08

0.10

0.12

0.08

0.09

0.10

0.09

0.06

G99

0.55

0.62

0.61

0.70

0.62

0.63

0.65

0.59

X

0

0

0

0

0

0

0

0

First we find the minimum element in each row and from the minimum element we subtract all the other element.

The new table is as follows :

1

2

3

4

5

6

7

8

C53

0.04

0.06

0.07

0.05

0.04

0

0.1

0.06

C81

0.01

0.02

0

0.04

0

0.05

0.02

0.02

D5

0.02

0.1

0.01

0

0.12

0.05

0.06

0.19

D44

0.07

0.04

0.09

0.05

0

0.06

0.09

0

E2

0.01

0.02

0.05

0

0.03

0.05

0.06

0

E35

0.02

0.04

0.06

0.02

0.03

0.04

0.03

0

G99

0

0.07

0.06

0.15

0.07

0.08

0.1

0.04

X

0

0

0

0

0

0

0

0

Now we draw horizontal and vertical lines to cover all the zeroes

1

2

3

4

5

6

7

8

C53

0.04

0.06

0.07

0.05

0.04

0

0.1

0.06

C81

0.01

0.02

0

0.04

0

0.05

0.02

0.02

D5

0.02

0.1

0.01

0

0.12

0.05

0.06

0.19

D44

0.07

0.04

0.09

0.05

0

0.06

0.09

0

E2

0.01

0.02

0.05

0

0.03

0.05

0.06

0

E35

0.02

0.04

0.06

0.02

0.03

0.04

0.03

0

G99

0

0.07

0.06

0.15

0.07

0.08

0.1

0.04

X

0

0

0

0

0

0

0

0

The minimum number of lines required to cover all the zeroes is 7 < order of the matrix

We subtract the minimum element 0.01 among the uncovered element, from the uncovered element and add 0.01 to covered element at the intersection of the horizontal and vertical lines.

1

2

3

4

5

6

7

8

C53

0.04

0.05

0.06

0.05

0.04

0

0.09

0.06

C81

0.02

0.02

0

0.05

0.01

0.06

0.02

0.03

D5

0.02

0.09

0

0

0.12

0.05

0.05

0.19

D44

0.07

0.03

0.08

0.05

0

0.06

0.08

0

E2

0.01

0.01

0.04

0

0.03

0.05

0.05

0

E35

0.02

0.03

0.05

0.02

0.03

0.04

0.02

0

G99

0

0.06

0.05

0.15

0.07

0.08

0.09

0.04

X

0.01

0

0

0.01

0.01

0.01

0

0.01

The minimum number of lines required to cover all the zeroes is 7 < order of the matrix

We subtract the minimum element 0.01 among the uncovered element, from the uncovered element and add 0.01 to covered element at the intersection of the horizontal and vertical lines.

1

2

3

4

5

6

7

8

C53

0.04

0.04

0.07

0.05

0.04

0

0.08

0.06

C81

0.01

0.01

0

0.04

0

0.05

0.01

0.02

D5

0.02

0.08

0.01

0

0.12

0.05

0.04

0.19

D44

0.07

0.02

0.09

0.05

0

0.06

0.07

0

E2

0.01

0

0.05

0

0.03

0.05

0.04

0

E35

0.02

0.02

0.06

0.02

0.03

0.04

0.01

0

G99

0

0.05

0.06

0.15

0.07

0.08

0.08

0.04

X

0.02

0

0.01

0.02

0.02

0.02

0.01

0.02

Column Subtraction

1

2

3

4

5

6

7

8

C53

0.04

0.04

0.07

0.05

0.04

0

0.07

0.06

C81

0.01

0.01

0

0.04

0

0.05

0

0.02

D5

0.02

0.08

0.01

0

0.12

0.05

0.03

0.19

D44

0.07

0.02

0.09

0.05

0

0.06

0.06

0

E2

0.01

0

0.05

0

0.03

0.05

0.03

0

E35

0.02

0.02

0.06

0.02

0.03

0.04

0

0

G99

0

0.05

0.06

0.15

0.07

0.08

0.07

0.04

X

0.02

0

0.01

0.02

0.02

0.02

0

0.02

The minimum number of lines required to cover all the zeroes is 8 = order of the matrix

Products

Plants

Cost

C53

6

$0.06

C81

3

$0.04

D5

4

$0.03

D44

5

$0.10

E2

2

$0.10

E35

8

$0.06

G99

1

$0.55

X

7

$0.00

Total Cost                                                                                                                            $0.94

	1	2	3	4	5	6	7	8
C53	0.10	0.12	0.13	0.11	0.10	0.06	0.16	0.12
C81	0.05	0.06	0.04	0.08	0.04	0.09	0.06	0.06
D5	0.32	0.40	0.31	0.30	0.42	0.35	0.36	0.49
D44	0.17	0.14	0.19	0.15	0.10	0.16	0.19	0.1
E2	0.06	0.07	0.10	0.05	0.08	0.10	0.11	0.05
E35	0.08	0.10	0.12	0.08	0.09	0.10	0.09	0.06
G99	0.55	0.62	0.61	0.70	0.62	0.63	0.65	0.59
X	0	0	0	0	0	0	0	0