# Planned backorders

Russian: Задолженный заказ

## The Setting

Consider the Basic economic order quantity (EOQ) system, but relax the requirement that all demand be met from stock on hand[1]. All demand is ultimately filled, though perhaps after a delay. That is, demand not filled immediately is backordered. Any inventory on hand is used to fill demand; backorders accumulate only when a warehouse runs out of stock entirely.

## Reorder-Point/Order-Quantity Policies

Define some functions and redefine some of those used in the EOQ model:

$\,I(t)$ -- inventory at time $\,t$;

$\,B(t)$ -- backorders at time $\,t$;

$\,IN(t)$ -- net inventory at time $\,t$, $\,IN(t)=I(t)-B(t)$;

$\,IO(t)$ -- stock on order at time $\,t$;

$\,IP(t)$ -- inventory position at time $\,t$, $\,IP(t)=IN(t)+IO(t)$.

The new functions are $\,B(t)$ and $\,IN(t)$, and $\,IP(t)$ has a new meaning.

The net inventory $\,IN(t)$ captures the information in both $\,I(t)$ and $\,B(t)$: At any given time, at least one of those two functions is zero, since we use any available stock to fill demand. Therefore,

$\,IN(t)=\left\{\begin{array}{ll} {I(t)} & {when\, IN(t)\ge 0} \\ {-B(t)} & {when\, IN(t)<0} \end{array}\right.$

The definition of $\,IN(t)$ treats backorders as negative inventories, and indeed they function in this way: Between receipts of orders, $\,IN(t)$ decreases at the constant rate $\,\lambda$, regardless of whether $\,IN(t)$ is positive or negative.

When an order arrives, $\,IN(t)$ jumps up by precisely $\,q$ in all cases; some of the batch may be used to fill backorders, and the rest is added to inventory.

Thus, $\,IN(t)$ behaves much like $\,I(t)$ did before, when backorders were forbidden, while now $\,I(t)$ itself is more complex ($\,IN(t)$ is sometimes called the inventory level.), fig. 1.

Consider,

$\,IN(t+L)=IN(t)+IO(t)-D=IP(t)-D \, \, \, \, \, \, \, (1)$

It is the conservation-of-flow law for this system: Between $\,t$ and $\,t + L$, $\,IO(t)$ gets added to the net inventory, and $\,D$ gets subtracted.

Thus, $\,IP(t)$ summarizes all the information needed to predict the net inventory a leadtime into the future.

As in the EOQ model, assume that all orders are of the same size $\,q>0$. The issue of when to order is now more complex. We need a second policy variable in addition to $\,q$:

$\,r$ - reorder point (quantity-units)

This variable can take on any real value, positive or negative. Consider the following policy:

Monitor the inventory position $\,IP(t)$ constantly. When $\,IP(t^{-} )=r$, place a new order of size $\,q$ at time $\,t$.

(The EOQ model's policies are special cases with $\,r = D$.)

In honor of the two variables, a policy of this kind is called a reorder-point/order-quantity or $\,(r,q)$ policy.

Figure 2 illustrates the behavior of $\,IN(t)$ and $\,IP(t)$ under such a policy.

As in the EOQ model the graph retains the same sawtooth pattern, but now the pattern can shift vertically, depending on the choice of $\,r$.

## Performance Criteria

The relevant criteria include average inventory $\,\overline{I}$ and average frequency $\,\overline{OF}$.

The primary backorder-related performance measure is the following:

$\,\overline{B}$ - long-term average outstanding backorders,

$\,\overline{B}=\mathop{\lim }\limits_{T\to \infty } \frac{1}{T} \int _{0}^{T}B(t)dt$

(The limit here is analogous to the one defining $\,\overline{I}$).

$\,v$ -- safety stock,

$\,v=r-D,$

$\,y$ -- safety time,

$\,y=\frac{v}{\lambda } .$

Like $\,v$, $\,y$ can be negative.

For any given $\,q$, only certain values of $v$ make sense: Equation (1) implies that the net inventory $\,IN(t^{-}$ ) just at the end of a cycle is precisely $\,v$, so $\,IN(t)=v+q$ at the beginning of a cycle. Therefore,

If $\,v>0$, then for all $\,t$

$\,I(t)>v>0$ and $\,B(t)=0$

If $\,v<-q$, then for all $\,t$

$\,I(t)=0$ and $\,B(t)>-(v+q)>0$

Neither conclusion is appealing: In case 1, we have more inventory than we actually need, and in case 2 we never fill all backorders. So, we can and do restrict attention to the range $\,-q\le v\le 0$. So, $\,v$ is negative (more precisely, nonpositive), as is $\,y$, and each arriving order fills all current backorders.

Thus, a cycle consists of two parts, one of length $\,u+y=(q+v)/\lambda$, during which inventory is held, and a second part of length $\,-y=-v/\lambda$, when backorders accumulate (see fig. 3). These intervals correspond to the fractions $\,(q+v)/q$ and $\,-v/q$, respectively, of the full cycle.

The average inventory is simply $\,\frac{1}{2} (q+v)$ during the first part and zero during the second. The average over a full cycle is a weighted average of these quantities:

$\,\overline{I}=\left(\frac{q+v}{q} \right)\left(\frac{1}{2} (q+v)\right)+\left(\frac{-v}{q} \right)(0)=\frac{1}{2} \frac{(q+v)^{2} }{q} .$

Likewise, the average backorders in the first part of the cycle is zero, and $\,\frac{1}{2} (-v)$ in the second, so

$\,\overline{B}=\left(\frac{q+v}{q} \right)(0)+(-v/q)\left(\frac{1}{2} (-v)\right)=\frac{1}{2} \frac{v^{2} }{q} .$

Finally, the cycle length is $\,u=\frac{q}{\lambda }$, so

$\,\overline{OF}=\frac{\lambda }{q} .$

Clearly these criteria are in direct conflict. Let us translate them into monetary terms. We continue to use the cost factors $\,k$, $\,c$ and $\,h$ defined earlier. Suppose we can also estimate a factor for backorders analogous to $\,h$:

$\,b$ - penalty cost for one unit backordered during one time-unit (moneys/[quantity-unit time-unit])

This parameter summarizes all the drawbacks of backorders mentioned above. The total average cost then becomes

$\,C(v,q)=(k+cq)\overline{OF}+h\overline{I}+b\overline{B}=$

$\,=c\lambda +\frac{k\lambda }{q} +\frac{1}{2} \frac{h(q+v)^{2} }{q} +\frac{1}{2} \frac{bv^{2} }{q} .$

## The Optimal Policy and Sensitivity Analysis

The cost $\,C(v,q)$ is now a function of two variables. To minimize it, we equate its partial derivatives to zero. ($\,C$ is continuously differentiable and strictly convex on its domain, so this approach works.) That is,

$\,\frac{\partial C}{\partial v} =\frac{h(q+v)}{q} +\frac{bv}{q} =0$

$\,\frac{\partial C}{\partial q} =\frac{-k\lambda }{q^{2} } +\frac{1}{2} \frac{h(q^{2} -v^{2} )}{q^{2} } -\frac{1}{2} \frac{bv^{2} }{q^{2} } =0$

Defining the cost ratio

$\,\omega =\frac{b}{b+h}$

gives the unique solution to these equations as

$\, q^{*} =\sqrt{\frac{2k\lambda }{h} } \sqrt{\frac{1}{\omega } }$

$\, v^{*} =-(1-\omega )q^{*}$

$\, C^{*} =C(v^{*} ,q^{*} )=c\lambda +\sqrt{2k\lambda h\omega }$

## References

1. Zipkin P. (2000) Foundations of inventory management; The McGraw-Hill Companies, Inc.