Paper Example on Mathematical Modeling Techniques

Linear Programming Model (Maximization)

Carrefour Company is involved in the production of two items namely dining tables and dining chairs. The resources needed to produce these two items are twofold, namely machine time for automatic processing and craftsman time for hand completion. The table below gives the amount of time in minutes for each item: (Let X represents Chairs and Y represents Tables)

	Machine time	Craftsman time
X	13	20
Y	19	29

Carrefour Company has 40 hours of machine time and 35 hours of crafts time in the next working week. The cost of machine time is AED 10 per hour worked and craftsman time costs AED 2 per hour worked. The machine and craftsman not working times will incur no costs. The revenue received for each item produced with the assumption that all production is sold is AED 20 for chairs and AED 30 for tables.

The company has won a contract to produce 10 chairs per week for a particular customer.

Formulate the problem of deciding how much to produce per week as a linear program.

Solve this linear program graphically.

The company has a specific contract to produce 10 items of X per week for a particular customer.

Solution

Let:

• x be the number of items of Chairs
• y be the number of items of Tables

Then the Linear Programming model is:

Maximize

20x + 30y - 10(machine time worked) - 2(craftsman time worked)

Subject to:

13x + 19y <= 40(60) machine time

20x + 29y <= 35(60) craftsman time

x >= 10 contract

x, y >= 0

So that the objective function will become:

Maximize

20x + 30y - 10(13x + 19y)/60 - 2(20x + 29y)/60

That is, maximize

17.1667x + 25.8667y

Subject to:

13x + 19y <= 2400

20x + 29y <= 2100

x >= 10

x, y >= 0

Solving simultaneously, we have that x=10 and y=65.52 with the value of the objective function being AED1866.5

So we can say production of 10 chairs and 66 tables will bring a maximum profit of AED 1866.5

Solving it graphically, it is clear from the diagram above that the maximum occurs at the intersection of x=10 and 20x + 29y <= 2100.

Transportation Model

Carrefour would assign the most efficient transportation company, for 4 different towns in the country, 4 transportation companies are chosen. The assignment model below will determine which company transport to which town.

The first step is to find the biggest number.

Stage	A	B	C	D
1	39	27	30	37
2	46	40	43	27
3	37	34	35	27
4	30	27	29	42

Subtract it from each and every cell

Stage	A	B	C	D
1	7	19	16	9
2	0	6	3	19
3	9	12	11	19
4	16	19	17	4

Subtracting the smallest number of each row with the whole row

Stage	A	B	C	D
1	0	12	9	2
2	0	6	3	19
3	0	3	2	10
4	12	15	13	0

The next step is column deduction; cover all zeros with less than 4 lines?

Stage	A	B	C	D
1	40640-5080000	12	9	29210-508002
2	0	6	3	19
3	0	3	2*	10
4	12	15	13	0

The smallest number is subtracted from all the cells

Stage	A	B	C	D
1	501652032000	10	361952032007	482602032002
2	0	4	1	19
3	0	1*	0	10
4	12	13	11	0

The smallest number is subtracted from all the cells

Stage	A	B	C	D
1	501652159000	9	7	482602159002
2	0	3	1*	19
3	495300698503	0	0	10
4	12	12	11	0

The smallest number is subtracted from all the cells

Stage	A	B	C	D
1	50165444500	24130444508	36195444506	48260444502
2	0	2	0	19
3	1	0	0	11
4	12	11	10	0

From the table above all the zeros are covered by four lines. Therefore select row with one zero and cross its column.

Stage	A	B	C	D
1	0*	8	6	2
2	0	2	0*	19
3	1	0*	0	11
4	12	11	10	0*

Choosing the cell from the original table

Stage	A	B	C	D
1	39*	27	30	37
2	46	40	43*	27
3	37	34*	35	27
4	30	27	29	42*

The optimal assignment cost is 39+34+43+42=158

References

Winston, W. L., & Goldberg, J. B. (2004). Operations research: applications and algorithms (Vol. 3). Belmont^ eCalif Calif: Thomson/Brooks/Cole.