Question

Steelcase has received orders of 2400, 2200, 2700, and 2500 units of a special-purpose panel for each of the next four months. SC can meet these demands by producing the panel, by drawing from its inventory, or by using any combination of the two alternatives. The production costs during each of the next four months are projected to be $74, $75, $76, and $76.5 per unit. Because costs are rising each month, SC might be better off producing more panel than it needs in a given month and storing the excess. Production capacity, though, cannot exceed 4000 units in any one month. The monthly production is finished at the end of the month at which time the demand is met. Any remaining panel is then stored in inventory at a cost of $1.2 per panel for each month that it remains there. If production level is increased from one month to the next, then the company incurs a cost of $0.5 per unit of increased production to cover the additional labor and/or overtime. Each unit of decreased production incurs a cost of $0.3 to cover the benefits of unused employees. The production level during the previous month was 1800 units, and the beginning inventory is 1000 units. Inventory at the end of the fourth month must be at least 1500 units to cover anticipated demand. Formulate a production plan for SC that minimizes the total costs over the next four months

Answer 1

Answer is explained in the explanation section below.

Step-by-step explanation:

Solution:

The Production Planning of the Steel Case:

Demand (units):

Month 1: 2400

Month 2: 2200

Month 3: 2700

Month 4: 2500

Production Cost ($/unit):

Month 1: 74

Month 2: 75

Month 3: 76

Month 4: 76.5

Inventory cost ($/panel):

Month 1: 1.2

Month 2: 1.2

Month 3: 1.2

Month 4: 1.2

Starting Inventory = 1000

Ending Inventory at the end of 4th month = 1500

Starting Production Level = 1800

Cost of changing production level = $0.5/unit (increase)

= $0.3/unit (decrease)

Decision Variables:

Let Pn be the production in the month n.

`I_(n)` be the inventory at the end of month n.

`I_(o)` be the initial inventory at the start of month 1.

Objective Function:

The production cost is given below:

74
`P_(1)` + 75
`P_(2)` + 76
`P_(3)` + 76.5
`P_(4)`

And the holding cost is given below:

1.2(
`I_(1) + I_(2) + I_(3)` )

And,

Constraints:

In order to meet the demand, we have the following constraints:

`P_(1)` +
`I_(o)`
`\geq` 2400

`P_(2)` +
`I_(1)`
`\geq` 2200

`P_(3)` +
`I_(2)`
`\geq` 2700

`P_(4)` +
`I_(3)`
`\geq` 2500

Now, considering the inventory level at the end of each month:

`I_(o)` = 1000

`I_(1)` =
`P_(1)` -
`I_(o)` - 2400

`I_(2)` =
`P_(2)` -
`I_(1)` - 2200

`I_(3)` =
`P_(3)` -
`I_(2)` - 2700

`I_(4)` =
`P_(4)` -
`I_(3)` - 2500

`I_(4)`
`\geq` 1500

It is given that, at maximum 4000 units can be produced each month

So, we have the following constraints:

`P_(n)`
`\leq` 4000 for n = 1, 2, 3 ,4

Also,

`P_(n)`
`\geq` 0

`I_(n)`
`\geq` 0