Silver–Meal heuristic

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.

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, …….,r_n) 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 + ... + r_{j − 1} And, begin the process again starting from period j.

• Infinite fill rate for the part being produced: Economic order quantity
• Constant fill rate for the part being produced: Economic production quantity
• Demand is random: classical Newsvendor model
• Demand varies over time: Dynamic lot size model

• Production and Operations Analysis by S. Nahmias, McGraw-Hill