# Wilson Formula Economic Order Quantity

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The Economic Order Quantity Wilson Formula or Wilson Formula (created in 1934) allows calculation of order optimal quantity and the time between two orders of a product for a given entity (plant, logistic centre)

Introduced for the first time in 1913 by Harris, it is sometimes known as the Harrys-Wilson Formula.

The hypotheses on which this model is based are the followings:

1. Inventory management horizon is unlimited; consequently, we consider that the process is constant in time.
2. The demand is continuous, known and homogeneous in time; according to this concept, we suppose that the annual consumption is X units/year.
3. The supply lead-time, L, is constant and known.
4. Inventory shortages are not accepted.
5. The acquisition cost variable seems constant, CA \$/unit.
6. The order entrance into the system is immediate once the supply lead-time is over.
7. We consider a launching cost of CL \$/order and a stock ownership cost of CP \$/unit.
8. The order size will always be the same, to keep constant the model parameters.

The most economic, in these conditions, is that an order enters the system when the inventory level is at zero. It supposes that the order must be made at a moment where the inventory level is sufficient to cope with the demand during the supply lead-time.

This very inventory level is called control point Pc

The control point Pc is calculated the following way:

Pc = X*L

Q is the quantity of every order.

The number of necessary annual orders to satisfy the demand, the frequency of resupplying (N), will be:

N=X / Q

The opposite of N is the time which passes by between two orders; it is the lead-time of a cycle which repeats throughout the horizon of management.

This lead-time is called supply cycle time (TC)

The economic lot is the lot associated to the minimal costs relative to inventory which means the lot which minimizes the function of the total annual inventory cost.

To calculate this lot, we must know at first which function we have to minimize.

The total annual cost is constituted by the following costs:

1. Annual launching cost: KL=CL*N=CL*X / Q \$/year
2. Annual variable cost of acquisition: KA=CA*X \$/year
3. Annual cost of inventory ownership : KP=CP*Q / 2 \$/year

The annual total cost will be K=KL+KA+KP

But, according to the model, the annual variable cost of acquisition KA doesn’t depend on the lot size nor on dates at which orders are emitted; for this reason, to find the lot minimizing the function of total cost, we shall disregard the KA cost.

K=KL+KP=CL*X / Q+CP*Q / 2 \$/year

To minimize the function we have to derivate by Q and equal to zero:

DK / dQ =-CL*X / Q2+CP / 2=0

The economic order quantity of Harris-Wilson, that is the optimum order quantity will be:

Q + = ( 2*CL*X / CP ) units/order

The optimal lead-time of resupplying between two orders will be:

T + = Q + / right = ( 2*CL / CP*X )