1 10 points Formulate the following problem Clearly identify

1) (10 points) Formulate the following problem: Clearly identify the decision variables, the objective function and the relevant constraints. The production manager of cereal must plan for the next 6 months. The expected demand in boxes of cereal and expected cost per box of cereal is shown below: The inventory cost of holding a box of cereal per month is $0.75. The production capacity per month 80 boxes. The current inventory is 50 boxes. Write a mathematical model using an appropriate objective function.

Solution

Let the production for each of the months be as follows

Month

Production

1

X1

2

X2

3

X3

4

X4

5

X5

6

X6

Therefore, the decision making variables for the problem will be X1, X2, X3, X4, X5, X6               

Month

(1)

Opening Inventory (2)

Production

(3)

Demand

(4)

Closing Inventory

(5) = (2)+(3)-(4)

Production Cost / Unit

(6)

Total Production Cost

(7)

0

50

1

50

X1

120

X1-70

3

3X1

2

X1-70

X2

100

X1+X2-170

2.5

2.5X2

3

X1+X2-170

X3

95

X1+X2+X3-265

3

3X3

4

X1+X2+X3-265

X4

105

X1+X2+X3+X4-370

2.25

2.25X4

5

X1+X2+X3+X4-370

X5

125

X1+X2+X3+X4+X5-495

2.5

2.5X5

6

X1+X2+X3+X4+X5-495

X6

90

X1+X2+X3+X4+X5+X6-585

3

3X6

Total

635

6X1+5X2+4X3+3X4+2X5+X6-1955

3X1+2.X2+3X3+2.25X4+2.5X5+3X6

Hence the total Cost incurred is as follows

Total Production Cost = 3X1+2.5X2+3X3+2.25X4+2.5X5+3X6

Total Inventory Carrying Cost           = (6X1+5X2+4X3+3X4+2X5+X6-1955)*0.75

= 4.5X1 + 3.75X2 + 3X3 + 2.25X4 + 1.5X5 + 0.75X6 – 1466.25

Total Cost = (3X1+2.5X2+3X3+2.25X4+2.5X5+3X6) + (4.5X1 + 3.75X2 + 3X3 + 2.25X4 + 1.5X5 + 0.75X6 – 1466.25)

= 7.5X1 + 6.25X2 + 6X3 + 4.5X4 + 4X5 + 3.75X6 – 1466.25

The Objective function would be to minimize the total Cost. Hence the Objective function shall be

7.5X1 + 6.25X2 + 6X3 + 4.5X4 + 4X5 + 3.75X6 – 1466.25

Constraints:

X1, X2, X3, X4, X5, X6, 80

Month	Production
1	X1
2	X2
3	X3
4	X4
5	X5
6	X6