# Silver–Meal heuristic

The Silver–Meal heuristic method was composed in 1975 by E.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.

## 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

• 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:

• C(2) = (K + (h*r2))/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:

• C(3) = (K + (h*r2)+(2hr3))/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,

• C(j) = (K + hr2 + 2hr3 + ... + (j − 1)hrj) / 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.

For numerical example, see Malakooti (2013).

## Extensions

There is another heuristic algorithm similar to Silver-Meal algorithm, called least-unit-cost (LUC) heuristic. In the LUC method, the purpose is to minimize the total holding cost by finding the average cost per part, as compared to the average cost per period of the Silver–Meal method (Malakooti,2013): C(j) = (K + hr2 + 2hr3 + ... + (j − 1)hrj) / (r1+r2+...+rj) After the first lot size is determined, the procedure is repeated using the next available net requirement.

## References

• Production and Operations Analysis by S. Nahmias, McGraw-Hill
• Malakooti, B. (2013). Operations and Production Systems with Multiple Objectives. John Wiley & Sons.