Inventory Analysis Deterministic - Shortages
Infinite Replenishment Rate and Backorders

In these models, we allow the inventory to go to 0 some time before replenishment. There are two assumptions regarding the response of a customer to the unavailable item. The first response is that the customer will accept later delivery. This is called a backorder. We provide three cost models for the backorder case:

• There is an expense for each cycle in which a backorder occurs. This cost is independent of the number of backorders. The expense is measured in \$.
• There is an expense for each backordered item. This cost is independent of the time the customer must wait for delivery. The expense is measured in \$/unit.
• The expense for each item is proportional to the time the customer must wait. For a particular cycle the total cost is proportional to the integral of the shortage amount during the cycle. The expense is measured in \$/unit-time.

The figure below illustrates on-hand inventory in blue and backordered quantities in red.

Since replenishments are instantaneous, backordered items are delivered at the time of replenishment and these items do not remain in inventory. We signify backorders as a negative inventory, so the minimum inventory is a negative number. The difference between the minimum and maximum inventory is the lot size.

Notation

When shortages are allowed, several new quantities must be defined and several reinterpreted.

• Backorder cost: When a customer seeks the product and finds the inventory empty, the demand can either go unfulfilled or be satisfied later when the product becomes available. The former case is called a lost sale, and the latter is called a backorder. We consider the lost sales cost later. The total backorder cost depends on the measured adopted for backorder cost.

• Backorder level: This is the average level of backorders over time. It is the average area of the red areas in the figure above.

• Fill Rate(v): This is the proportion of the customers served directly from the inventory. The proportion of the customers who must wait for a backordered item is 1 - v. We use this factor as a design variable when shortages are allowed. For the backordered case the service level is I_Max/q.

• Maximum inventory level (I_Max): Many books call this quantity the order level. When backorders are not allowed, this quantity is the same as q. When backorders are allowed, it is less than q. (units)

• Minimum inventory level (I_Min=I_Max-q): When backorders are not allowed, this quantity is 0. With backorders its absolute value is the level of maximum number of backordered units. (units)

• Mean Residence Time: This is the time a remains in inventory, given that it is placed in inventory. With shortages, not all units appear in inventory. Those units that are backordered, go to the customer as soon as they are available. For the infinite replenishment rate case, this is when an order arrives. We compute the mean residence time using Little's law by dividing the average inventory level by the rate of items entering inventory. This rate is D(1 - v).

• Mean Backorder Time: When an item is backordered, a customer must wait for a period of time before the backorder is satisfied. Again we use Little's law. For the mean backorder time, we divide the average backorder level by the rate of items being backordered.

Single Cycle

 The results of the analysis depend on the lot size and the service level. The figure shows that the maximum inventory is the same as the order level. In addition to the other results computed for the no-shortages case we now can compute: Minimum inventory level: This is vq - q. Maximum inventory level: This is vq.

The Inventory Model

 As before, we create a model by selecting Add Inventory from the menu. We select Allowed from the Shortages frame on the dialog. We select per unit-time as the Shortage Cost measure.
 The results for this case using a backorder cost of \$2 per unit-week is shown at the left. For the evaluation of an instance we choose a lot size of 400 and a service level of 90%. The optimum lot size is greater, 876, and the optimum fill rate is 86.96%. The optimum solution balances ordering, inventory and backorder costs.

Infinite Replenishment Rate and Lost Sales

A second response of the customer to an inventory stock out is for the customer to leave without purchasing the product. This is a Lost Sale. The cost of a lost sale is measured in \$ per unit. The cost should be at least the lost profit from the sale, but will probably be greater due to penalty costs and to loss of good will from the disappointed customer.

The figure below illustrates the on-hand inventory in blue. The inventory reaches 0 before the end of the cycle, but no shortage is shown since a shortage is never recovered. The lot size is the maximum inventory level.

Notation

The shortage cost has a different meaning with lost sales.

• Lost sales cost: When a customer seeks the product and finds the inventory empty, the demand goes unfulfilled in the lost sales case. In this case the shortage cost is the lost profit of the lost sale plus any other any costs associated with the lost sale. (\$/unit-time)

• Order level (q): The maximum level reached by the inventory is the order level. With lost sales, the order level is q. (units)

• Fill rate (v): This is the proportion of the customers served directly from the inventory. The proportion of the customers that are lost is 1 - v. We use this factor as a design variable.

Single Cycle

 The results of the analysis depend on the lot size and the service level. The figure shows that the maximum inventory is the same as the order level. In addition to the other results computed for the no-shortages case we now can compute: Maximum inventory level: This is q. Minimum inventory level: This is 0. Service Level: This is v. The satisfied demand in a cycle is Dt', where t' = v*(cycle time).

The Inventory Model

 As before, we create a model by selecting Add Inventory from the menu. We select per lost sale from the Shortage Cost options frame on the dialog. The lost sales case is interesting because the optimum solution is to allow no shortages or to satisfy none of the demand. The two cases are below for an example. On the left, Ex4 A, when the unit cost of a lost sale is \$3, the optimum policy is the same as the policy when no shortages are allowed. In the case at the right, Ex4 B, when the lost sales cost is \$2, the optimum policy has the lot size and service level both at 0. The optimum cost is the cost when all demand is lost. We have modified the example for this illustration by setting the product revenues and costs both to zero. The lost sales cost represents the net revenue loss on a sale plus any additional penalties due to the lost sale.

Finite Replenishment Rate and Backorders

The figure below illustrates this case. On-hand inventories are in blue and backordered quantities are in red.

No new notation is required for this case.

Single Cycle

 At the start of the cycle, the inventory grows at the rate P - D until the lot size is produced and then declines at the rate D. From time 0 to t1' the production is satisfying backorder demand and current demand. From t1' to t2' production adds to inventory. From t2' until the end of the cycle, the inventory decreases at the demand rate D.

The Inventory Model

 As before, we create a model by selecting Add Inventory from the menu. We select Finite from the Replenishment options and per unit-time from the Shortage Cost options on the dialog.
 The results depend on both the lot size and fill rate. The optimum solution balances order cost, inventory cost and backorder cost.

Finite Replenishment Rate and Lost Sales

The figure below illustrates this case. Inventories are blue. Since shortages are lost there is no backorder region.

No new notation is required for this case.

Single Cycle

 At the start of the cycle, the inventory grows at the rate P - D until the lot size is produced and then declines at the rate D. From time 0 to t1' the production is satisfying current demand and building inventory. From t1' to t2' inventory decreases until it reaches 0. From t2' until the end of the cycle, the inventory remains at 0 while sales are lost.

The Inventory Model

 As before, we create a model by selecting Add Inventory from the menu and select the appropriate options Again we see the result that the optimum either has no shortages or satisfies none of the demand. With a shortage cost of \$2 per unit, the optimum policy is to allow no shortages as shown on the left. With a cost of \$1 per unit, no inventory is maintained as shown on the right.

Operations Management / Industrial Engineering
Internet
by Paul A. Jensen