Inventory Control: Dynamic Programming

Example

Spitzen Ltd. has to supply the following number of items at the end of each month.

Production during a month is available for supply at the end of the month. The stock holding cost per month is Re. 1 per item. The setup cost is Rs. 900 per setup and Rs. 2 per item. Find the optimal policy so that total cost may be minimum.

Solution.

The production cost Rs. 2 per item is always incurred, whether items are produced in the beginning or at any other time.
Fixed cost = 1000 X 2 = Rs. 2000

Case I - April

Requirement = 400
Cost = Rs. 900 (setup cost)

Case II - March

We have the following alternatives.

1. Produce 700 (demand for March & April) items in the beginning of March.
Cost = 900 + (400 X 1) = Rs. 1300

2. Produce 300 items in March & 400 in April.
Cost = 900 + 900 = Rs. 1800

Hence, the optimal sub-policy for March is: Produce 700 items in March.

Case III - February

We have the following alternatives.

1. Produce 900 items in the beginning of February.
Cost = 900 + 700 X 1 + 400 X 1 = Rs. 2000

2. Produce 500 items now & 400 in April.
Cost = 900 + 300 X 1 + 900 = Rs. 2100

3. Produce 200 items now & 700 in March.
Cost = 900 + 1300 = Rs. 2200

The optimal sub-policy for March was to produce 700 items. Therefore, in February we have not considered the following case: Produce 200 items now, 300 items in March & 400 in April.
Hence, the optimal sub-policy for February is: Produce 900 items in February.

Case IV - January

We have the following alternatives.

1. Produce 1000 items in the starting of January.
Cost = 900 + (900 X 1 + 700 X 1 + 400 X 1) = Rs. 2900

2. Produce 600 items now & 400 in April.
Cost = 900 + 500 X 1 + 300 X 1+ 900 = Rs. 2600

3. Produce 300 items now & 700 in March.
Cost = 900 + 200 X 1 + 1300 = Rs. 2400

4. Produce 100 items now & 900 in February.
Cost = 900 + 2000 = Rs. 2900

The minimum cost is Rs. 2400. Hence, the best policy is: Produce 300 items in January & 700 in March.