# Optimization of the production run-length

Stocks in companies are created due to the purchase of materials and goods and the production of series of specific products. Companies bear the costs of storage and run batches (series) of materials, products or goods.

These costs depend on the size of the batch of production. When choosing optimal batch size company must take into account the many factors that require different decision. Increasing the batch causes an increase in storage costs and decrease the costs of preparation. In such case it becomes important to determine the optimal production quantity and an optimal order size.

## Optimization of production series

Costs relevant to the determination of this value:

- Startup production costs of one batch:
- cost of equipment and transposition devices adjustment,
- costs for the preparation and design of production schedule,
- costs of procurement of materials for the production of new batch.

- Costs of storage:
- rents, insurance, taxes,
- cost of credit for the financing of stocks,
- cost of maintaining storage facilities.

The total cost of storage and production relevant to decision about the length of the series:

K = Kp + Km

- where:
- K - total costs of production and of the contract,
- Kp - total costs of production,
- Km - the total costs of the contract.

Km= km * D/2

- km - the storage costs of the product in a single period,
- D/2 – average number of product in storage in a single period.

KP = kp * Q/D

- kp - unitary cost of launching a series of production,
- Q/D - number of production series, which must be run within a set period (e.g. in the course of one year).

Total costs of storage and production, relevant to the decision about the length of production series, can thus be presented as follows:

K = km * D/2 + kp * Q/D

This function has a minimum at the point:

D0 = (2 * kp * Q/km)

This formula allows to determine the optimal length of production series, where the total cost of the production and storage of the product is the smallest (in given period).

## References

- Kim, C. H., & Hong, Y. (1999).
*An optimal production run length in deteriorating production processes*. International Journal of Production Economics, 58(2), 183-189.