Contents

Base stock model

Assumptions[edit]

1. Products can be analyzed individually
2. Demands occur one at a time (no batch orders)
3. Unfilled demand is back-ordered (no lost sales)
4. Replenishment lead times are fixed and known
5. Replenishments are ordered one at a time
6. Demand is modeled by a continuous probability distribution

Variables[edit]

• ${\displaystyle L}$ = Replenishment lead time
• ${\displaystyle X}$ = Demand during replenishment lead time
• ${\displaystyle g($x)} = probability density function of demand during lead time
• ${\displaystyle G($x)} = cumulative distribution function of demand during lead time
• ${\displaystyle \theta }$ = mean demand during lead time
• ${\displaystyle h}$ = cost to carry one unit of inventory for 1 year
• ${\displaystyle b}$ = cost to carry one unit of back-order for 1 year
• ${\displaystyle r}$ = reorder point
• ${\displaystyle SS=r-\theta }$, safety stock level
• ${\displaystyle S($r)} = fill rate
• ${\displaystyle B($r)} = average number of outstanding back-orders
• ${\displaystyle I($r)} = average on-hand inventory level

Fill rate, back-order level and inventory level[edit]

In a base-stock system inventory position is given by on-hand inventory-backorders+orders and since inventory never goes negative, inventory position=r+1. Once an order is placed the base stock level is r+1 and if X≤r+1 there won't be a backorder. The probability that an order does not result in back-order is therefore:

${\displaystyle P(X\leq r+1)=G(r+1)}$

Since this holds for all orders, the fill rate is:

${\displaystyle S($r)=G(r+1)}

If demand is normally distributed ${\displaystyle {\mathcal {N}}(\theta ,\,\sigma ^{2})}$, the fill rate is given by:

${\displaystyle S($r)=\phi \left({\frac {r+1-\theta }{\sigma }}\right)}

Where ${\displaystyle \phi ()}$iscumulative distribution function for the standard normal. At any point in time, there are orders placed that are equal to the demand X that has occurred, therefore on-hand inventory-backorders=inventory position-orders=r+1-X. In expectation this means:

${\displaystyle I($r)=r+1-\theta +B(r)}

In general the number of outstanding orders is X=x and the number of back-orders is:

${\displaystyle Backorders={\begin{cases}0,&x<r+1\\x-r-1,&x\geq r+1\end{cases}}}$

The expected back order level is therefore given by:

${\displaystyle B($r)=\int _{r}^{+\infty }\left(x-r-1\right)g(x)dx=\int _{r+1}^{+\infty }\left(x-r\right)g(x)dx}

Again, if demand is normally distributed:^[2]

${\displaystyle B($r)=(\theta -r)[1-\phi (z)]+\sigma \phi (z)}

Where ${\displaystyle z}$ is the inverse distribution function of a standard normal distribution.

Total cost function and optimal reorder point[edit]

The total cost is given by the sum of holdings costs and backorders costs:

${\displaystyle TC=hI($r)+bB(r)}

It can be proven that:^[1]

Where r* is the optimal reorder point.

Proof
${\displaystyle {\frac {dTC}{dr}}=h+(b+h){\frac {dB}{dr}}}$ ${\displaystyle {\frac {dB}{dr}}={\frac {d}{dr}}\int _{r+1}^{+\infty }(x-r-1)g($x)dx=-\int _{r+1}^{+\infty }g(x)dx=-[1-G(r+1)]} To minimize TC set the first derivative equal to zero: ${\displaystyle {\frac {dTC}{dr}}=h-(b+h)[1-G(r+1)]=0}$ And solve for G(r+1).

If demand is normal then r* can be obtained by:

${\displaystyle r^{*}+1=\theta +z\sigma }$

See also[edit]

• 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
• Continuous replenishment with backorders: (Q,r) model
• Demand varies deterministically over time: Dynamic lot size model
• Several products produced on the same machine: Economic lot scheduling problem

References[edit]