# Silver–Meal heuristic

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The Silver–Meal heuristic method was composed in 1973[1] by Edward A. Silver and H.C. Meal. It refers to production planning in manufacturing and its purpose is to determine production quantities to meet the requirement of operations at minimum cost.

The method is an approximate heuristic for the dynamic lot-size model, perceived as computationally too complex.

## Definition

The Silver–Meal heuristic is a forward method that requires determining the average cost per period as a function of the number of periods the current order is to span and stopping the computation when this function first increases.

## Procedure

Define :

K: the setup cost per lot produced.

h: holding cost per unit per period.

C(T) : the average holding and setup cost per period if the current order spans the next T periods. Let (r1, r2, r3, …….,rn) be the requirements over the n-period horizon.

To satisfy the demand for period 1

• ${\displaystyle C(1)=K}$

The average cost = only the setup cost and there is no inventory holding cost.

To satisfy the demand for period 1, 2 Producing lot 1 and 2 in one setup give us an average cost:

• ${\displaystyle C(2)={\frac {K+hr_{2}}{2}}}$

The average cost = (the setup cost + the inventory holding cost of the lot required in period 2.) divided by 2 periods.

To satisfy the demand for period 1, 2, 3 Producing lot 1, 2 and 3 in one setup give us an average cost:

• ${\displaystyle C(3)={\frac {K+hr_{2}+2hr_{3}}{3}}}$

The average cost =( the setup cost + the inventory holding cost of the lot required in period 2+ the inventory holding cost of the lot required in period 3) divided by 3 periods.

In general,

• ${\displaystyle C(j)={\frac {K+hr_{2}+2hr_{3}+...+(j-1)hr_{j}}{j}}}$

The search for the optimal T continues until C(T) > C(T − 1).

Once C(j) > C(j − 1), stop and produce r1 + r2 + r3 + ... + rj − 1 And, begin the process again starting from period j.

## References

1. ^ EA Silver, HC Meal, A heuristic for selecting lot size quantities for the case of a deterministic time-varying demand rate and discrete opportunities for replenishment, Production and inventory management, 1973
• Production and Operations Analysis by S. Nahmias, McGraw-Hill