**Inventory control for point-of-use locations in hospitals**

This is a simplified version of a paper published as *Journal of the Operational Research Society* (2012) **63,** 497–510, authored by M Bijvank and I F A Vis

**1. Introduction**

The main objective of a hospital is to provide high-quality health care. Sufficient medical items need to be on hand to enable hospital staff to perform their daily work. Typically, medical supplies are stored at many locations in a hospital and in large quantities to prevent stock outs as much as possible. However, hospitals lack available storage space and millions of dollars are tied up in inventories that consume on average 20% of net patients revenues and represent the second largest expense after labour. Therefore the available storage capacity should be used efficiently, and it is important to find a balance of the service quality and the desired inventory levels between the different items. The goal of this paper is to develop inventory models which consider the service level as well as the capacity limitations at hospitals. Our main contribution is the development of new solution techniques to support hospital management decisions with regard to inventory control.

Two types of models are proposed to deal with the capacity limitations and the service requirements: a capacity model and a service model. In a *capacity model*, the objective is to maximise the service level when the storage capacity is limited. This model can be used in a multi-item inventory control system with a storage capacity constraint. In a *service model*, the required capacity is minimised while satisfying a service constraint instead of having a cost objective. The service model results in the lowest inventory levels within a service level restriction. Consequently, the model will minimise inventory holding costs.

**2. POU inventory systems in hospitals**

We roughly distinguish three types of inventories in hospitals, namely perishable items including medicines and blood, non-disposables (eg, instruments) and disposables (eg, gloves, needles, sutures). Our main focus in this research is on disposable items since this type of products is stored in almost all locations in hospitals and, therefore, more difficult to control.

Traditionally, health-care supply chains are characterised by a multitude of different suppliers, products and patient care units that arbitrarily order multiple items. A hospital storage room receives these items and distributes them to the right lower-level *point-of-use* (POU) locations, such as nursing units and operating rooms (see Figure). Another option is to outsource the replenishment activities and make suppliers deliver directly to the POU locations (see Figure).

A traditional inventory system *versus* a Just In Time (JIT) stockless system: (a) traditional inventory system; (b) JIT stockless system.

In the literature two concepts are proposed to improve the performance of health-care supply chains: product standardisation and selecting prime vendors. The former concept reduces the number of different items that have to be stocked. Consequently, it reduces the volatility of the demand. The latter concept can reduce lead times and, therefore, the safety stock as well. Besides these tactical and strategical decisions, it is also essential to decide on the time instant to place a new replenishment order (i.e. the reorder level) and the corresponding order size for each item on stock at a single POU location. Such operational decisions are the focus in this paper.

Inventory management of disposable items at each POU location can be described as a system where all items are stored in bins. Each bin has a total storage capacity of *C*(*i*) units for item *i* that is used to fulfil demand of medical staff whenever required. If *s* or less units of an item are available in the bin a signal is given (eg, a nurse puts a bar code of the item on an ordering board). This level *s* is called the *reorder level*. These signals are scanned at prespecified time intervals of length *R* (i.e. *periodic reviews*) that may range from days to weeks. Items are usually ordered in *fixed quantities* of size *Q* to provide a transparent and easy-to-understand inventory policy for hospital staff. After *L* time units the ordered items are resupplied from the higher-level stock point to the specific bins (see Figure). In a hospital setting the lead time *L* is known and relatively short due to the high product availability at nearby higher-level stock points. Therefore, another characteristic for the inventory system at hospitals is that the lead time *L* is shorter than the length of the review period *R*. This is referred to as *fractional lead times*.

If a required item is not available in the right quantity at a specific POU location to provide the required health-care service, a substitute product is used or an emergency delivery is performed (eg, from another POU location). The original demand for the item is considered to be *lost*. Such situations are time costly and should be avoided as much as possible. Therefore, we define the service level as the fraction of demand to be satisfied directly from stock on hand (ie, *item fill rate*). Note that this definition does not include the fraction of demand that is satisfied due to a substitution or an emergency delivery in case of a stock out.

To summarise, the inventory system in most hospitals is characterised by periodic reviews, an (*R*,?*s*,?*Q*) replenishment policy, short lead times, lost sales, capacity restrictions and a service level objective. Besides this replenishment policy, order-up-to policies are also commonly observed at hospitals. In this policy the inventory position (inventory on hand plus inventory on order minus backorders) is raised to an order-up-to level *S* at each review instant. This policy is denoted as an (*R*,?*S*) policy. More generally, in an (*R*,?*s*,?*S*) policy the inventory position is increasing to level *S* at a review instant when the inventory position is at or below reorder level *s*. When *s *= *S*-1, this policy corresponds to the order-up-to policy. Both policies are considered in this paper as alternative replenishment policy next to the (*R*,?*s*,?*Q*) policy. The advantage of this latter policy is its simplicity to understand and implement in practice since order sizes are fixed. However, it does not use the available capacity optimally. The available capacity is utilised more efficiently when the replenishment policy incorporates an order-up-to level that equals the available capacity. Although such policies require more effort to process the replenishments, we show the benefits in terms of service levels and capacity utilisation compared to fixed order size policies in our case study and our numerical experiments.

**3. Models **

In this section we show how we developed a service model and a capacity model for the inventory control system at a POU location as described in Section 2. Therefore, we decompose the inventory system in single-item models and we embed these models in a multi-item inventory system with a storage capacity constraint. This multi-item model can be used to determine the available capacity for each item in the capacity model.

**3.1. Single-item inventory system**

We started by deriving a single-item capacity model and a service model for the (*R*,?*s*,?*Q*) policy with lost sales, fractional lead times and a service level constraint.

The demand during *t* time units is modelled as a discrete random variable *Dt*, which is assumed to be independent for non-overlapping time intervals.

**3.2. Multi-item inventory system**

The inventory control problem in hospitals is more complex than the single-item system considered so far. The capacity limitation for each item is part of a larger inventory system with multiple items, where the limited capacity is shared by all items stored at a POU location.

The allocation of the limited storage capacity available and the determination of the values for the inventory control variables can be solved simultaneously. We propose a knapsack kind of approach in which a trade-off has to be made between the increase of the service level for an item and a decrease of the remaining capacity available for the other items. The ratio of this service level increment divided by the extra assigned capacity to this item is computed for each item in every iteration. We assign an extra bin to the item with the highest ratio until all capacity is assigned. To compute the increase in the service level, we determine the optimal control values of the replenishment policy and the corresponding service.

**4. Hospital sample data**

In this section we describe the performance of the (*R*,?*s*,?*Q*) policy and the (*R*,?*s*,?*S*) policy based on the models developed. The main goal of this section is to apply the capacity model and the service model in practice and to demonstrate how the models can be used to increase the service level.

We observed the (*R*,?*s*,?*Q*) replenishment policy at the VU University Medical Centre (VUmc) in Amsterdam and at Hospital Amstelland in Amstelveen. In particular, we consider many items and report in detail on a representative example about infusion liquids at three POU locations (paediatrics, intensive care and obstetrics).

Based on the results of our experiments we conclude that the (*R*,?*s*,?*Q*) replenishment policy results in service levels of about 70% up to 98% (without emergency replenishments). The capacity *C* is insufficient and results in stock outs. There are several solutions to minimise stock-out occurrences:

- use the alternative (
*R*,?*s*,?*S*) replenishment policy with the same order frequency, - shorten the length of the review period,
- increase the available capacity.

**5. Approximation procedure**

In this section we propose an approximation procedure in which closed-form expressions are derived for performance measures of interest. Such expressions are much easier to implement in practical applications compared to our Markov models, and large computation times are avoided to set the inventory control variables. Moreover, the expressions can be used for any lost-sales inventory system (even when there is a cost objective). No assumptions are imposed on the lead time or the available capacity. However, we make the assumption that at most one order is outstanding at any time. Such assumptions are common in the literature for lost-sales inventory models. This assumption is always satisfied when the lead time is fractional, otherwise we impose the restriction *Q *> *s* or *S*-*s *> *s* on the (*R*,?*s*,?*Q*) or (*R*,?*s*,?*S*) policy, respectively. Furthermore, our approximation procedure can be used for different types of replenishment policies.

The approximation procedure is based on the average performance during a replenishment cycle, whereas our exact models are based on the average performance during a review period. A *replenishment cycle* is the time between two consecutive orders (either order placement or order delivery). The time period from when the inventory position reaches the reorder level to the actual order delivery is called the *risk period*. It consists of the waiting time until the next review plus the lead time. Furthermore, the *undershoot* is defined as the satisfied demand during this waiting time (ie, it represents the difference between the reorder level and the inventory position at the order placement).

**6. Inventory rule**

The goal of this section is to develop a heuristic inventory rule for the capacity model with an (*R*,?*s*,?*Q*) replenishment policy that can easily be understood by hospital staff to decide upon the reorder level and the order size. This inventory rule can also be used in the multi-item model of Section 4.2.

The inventory rule consists of several tests. First, we check if the capacity *C* is sufficient to satisfy the demand. If the capacity is restrictive, we need to check whether it is likely that this restriction results in out-of-stock occurrences. Therefore, we examine if the reorder level *s* could be sufficient to be used as safety stock in order to fulfil demand until the next delivery. If this seems to be sufficient, we can determine the value of *s* such that stock outs are minimised. Otherwise, we need to find a balance between the reorder level and the order quantity.

The capacity is not restrictive when the order quantity is at least the average amount that is asked for during a review period, that is *Q* = *µR*. Another characteristic for this situation is that when no order is placed (ie, inventory level larger than *s*) the remaining inventory is sufficient to fulfil the demand until the next possible order delivery (ie, the demand until the next review and order delivery), or *s*+1 = *µR*+*µL*. Since *s*+*Q *= *C*, the capacity is not restrictive if *C*+1 = 2*µR*+*µL*. Therefore, we can set *s *? [*µR*+*µL*-1;*C*-*µR*] to obtain high service levels. We have chosen to set the value of *s* equal to the middle of this interval.

When there is a shortage of capacity, we want to order at least the average number of units that are asked for during a review period, that is *Q *= *µR*. This order quantity is on average sufficient to satisfy demand between two order deliveries when orders are placed every review period. Owing to the stochastic nature of the demand, we cannot guarantee that an order is placed at each review. Therefore, we introduce an approximation for the probability that orders are placed every two succeeding reviews. This is only likely when *Q* = *µR* (ie, we assume the inventory level to be zero when an order arrives). A new order is placed when the delivered quantity minus the demand between order delivery and the next review is equal to or less than *s*.

**7. Conclusion**

The inventory replenishment system at POU locations in hospitals can be classified as a lost-sales inventory system where the lead time is shorter than the length of a review period and the focus is on service levels. Another characteristic of hospital inventory management is the lack of available storage capacity. We developed capacity and service models for such inventory systems and compared their performances for (*R*,?*s*,?*Q*) policies and (*R*,?*s*,?*S*) policies. Both types of replenishment policies are common in hospitals. The fixed order size policy results in a more insightful replenishment process for hospitals with the use of bar codes. However, the (*R*,?*s*,?*S*) policy uses the capacity more efficiently. If inventory levels are monitored automatically (for instance, when RFID-chips are used) such policies are recommended. However, both replenishment policies perform equally well when the service level is rather high.

We also derived closed-form expressions that approximate performance measures (like the service level) in order to set the inventory control variables. We demonstrated that this approximation procedure can also be used in more general settings other than a hospital inventory system, including a cost objective. The procedure performs very good when the average order size is larger than 1.5 times the average demand in a review period. Furthermore, we developed a simple inventory rule that finds near-optimal values for the reorder levels and order quantities for the capacity model. This inventory rule can easily be embedded in multi-item algorithms that assign items to the available capacity at different POU locations. It can also be used to determine the required capacity.

One possible aspect for future research is the influence that substitution products have on the service level in case of a stock out. Another interesting aspect to investigate would be the interaction of the inventory control between the POU locations and the higher-level stock points like the central storage room. Especially how to set the lead time and the review period length, since they influence the performance of the POU location but they are typically determined by the supplier (ie, the higher-level stock point).