36 Inventory Control Systems: Periodic Review System

Vikas Singla

36.1 INTRODUCTION

Previous chapter discussed one of the inventory control systems where stock level was tracked after

every withdrawal of items. Though this method helps in mitigating the effects of uncertainty in demand patterns but becomes time consuming and sometimes irrelevant. For instance, checking inventory level less important items and done very frequently such as nuts & bolts would be time consuming and unnecessary. Also when there is certainty of supply from the vendors after a particular time period then there is no need to check stock level after every usage. In such cases, periodic review system is applicable. This chapter focuses its discussion on this type of inventory control system.

36.2 PERIODIC REVIEW (P) SYSTEM

In contrast to the continuous-review inventory system considered, we now assume that the system is only being monitored periodically. At the end of each period, when the current inventory level is determined, a decision is made on how much to order (if any) to replenish inventory for the next period. Each of these decisions takes into account the planning for multiple periods into the future. We begin with the simplest case where the planning is only being done for the next two periods and no setup cost is incurred when placing an order to replenish inventory.

One option with a stochastic periodic-review inventory system is to plan ahead only one period at a time, using the stochastic single-period model from the preceding section to make the ordering decision each time. However, this approach would only provide a relatively crude approximation. If the probability distribution of demand in each period can be forecasted multiple periods into the future, better decisions can be made by coordinating the plans for all these periods than by planning ahead just one period at a time. This can be quite difficult for many periods, but is considerably less difficult when considering only two periods at a time.

36.2.1 Assumptions:

Except for having two periods, the assumptions for this model are basically the same as for the one- period model presented in the preceding section, as summarized below.

Each application involves a single stable product.
Planning is being done for two periods, where unsatisfied demand in period 1 is backlogged to be met in period 2, but there is no backlogging of unsatisfied demand in period 2.
The demands D1 and D2 for periods 1 and 2 are independent and identically distributed random variables.
The initial inventory level (before replenishing) at the beginning of period 1 is x1 >=0.
The decisions to be made are y1 and y2, the inventory levels to reach by replenishing (if needed) at the beginning of period 1 and period 2, respectively.
The objective is to minimize the expected total cost for both periods, where the cost components for each period are

C = unit cost for purchasing or producing each unit,

H = holding cost per unit remaining at end of each period,

P = shortage cost per unit of unsatisfied demand at end of each period.

For simplicity, we are assuming that the demand distributions for the two periods are the same and that the values of the above cost components also are the same for the two periods. In many applications, there will be differences between the periods that should be incorporated into the analysis. For example, because of assumption 2, the value of p may well be different for the two periods. Such extensions of the model can be incorporated into the dynamic programming analysis presented below, but we will not delve into these extensions.

36.2.2 Two scenarios of P model

Under this system inventory level or remaining number of units in the stock is checked after certain and fixed time period. This system is also termed as fixed interval reorder or periodic reorder system. As the title suggests next order is placed after a certain fixed time period. For instance, a soft drink vendor makes weekly round to every retailer to check the inventory level. This helps the retailer as well because retailer would be sure of the time period when the vendor would visit for supplies.

But this system has its limitations which have been discussed in following two sections:

(i) If demand is uncertain before reorder point: As P system checks inventory level only after fixed time period so if the demand is strong during the time period before decided time of checking stock level then next order of the batch would be higher to compensate for high Similarly if demand is slow during such time period then number of items left in the stock when it is checked would be higher and so retailer would place a smaller order.
(ii) If demand is uncertain after reorder point: After placing next order at reorder point stock is not checked till order is received. Variation in demand whether high or low would impact size of new batch.

36.2.3 Periodic review system with uncertain demand before reorder point:

Case 1: Suppose batch size i.e. Q = 100 units, daily demand d = 10 units and lead time L = 5 days. Lead time is the time taken by the supplier to replenish stock by Q after order has been placed. If daily demand would have been fixed and certain of 10 units then 100 units would be consumed in 10 days (Table 1) making Time Between Order (TBO) to be of 10 days as shown by Fig.1(a). This TBO and quantity would remain same in case of certain daily demand.

Case 2: Suppose from a batch size of 100 units daily demand before reorder point varies in such a manner that demand during day 1 is of 20 units, during day 2 is of 10 units, during day 3 is of 5 units and during day 4 is of 15 units. For remaining days daily demand is constant at 10 units daily (Table 1). With lead time of 5 days and review being done only after fixed time period of 5 days the variation in demand would lead to situation of stock out. Under the given scenario as shown in Fig. 1(b) first 50 units are consumed in first four days and as next 50 units would be consumed in next five days making Time Between Order (TBO) as 9 days so entire batch would be consumed in 9 days leading to stock out on 10th day. Under such situation next order should have been placed after 4th day. But because of P system order would be placed only after 5th day of Q=110 units.

Case 3: Suppose from a batch size of 100 units daily demand before reorder point varies in such a manner that demand during day 1 is of 5 units, day 2 it is of 10 units, during day 3 and day 4 it is of 5 units and during day 5 it is of 15 units. For remaining five days daily demand remains constant at 10 units. Under this scenario in first five days only 40 units are consumed and as review is done after 5 days an order of Q=100 units is placed. For next five days a total of 50 units are consumed making it a total of 90 units at the end of one ordering cycle as shown in Fig. 1 (c). When new order is made available the stock still had 10 units thus increasing the stock to 110 units. So for next order manager would order only 90 units to keep the stock level at optimum. This indicates that though TBO remains same of 10 days but quantity ordered changes.

These three cases clearly indicates that in P system

The TBO remains same of 10 days.
Batch size varies because of varying demand. Q was 100 units in case 1, 110 units in case 2 and 90 units in case 3.
Reorder point i.e. time when order is placed is same i.e. next order is placed only after 5 days.
In P system as seen from above illustrations estimation of reorder point does depend on varying demand before lead time as stock level varies after every ordering cycle.

36.2.4 Periodic review system with uncertain demand after reorder point:

As shown in previous section estimation of reorder point in P system does depend on variation in demand before lead point. But what happens if demand is varying after lead time? This has been discussed by illustrating following cases.

Suppose a batch size of 100 units is in the stock and for first five days 10 units daily are consumed. Thus after 5 days a stock of 50 units is left. Now as lead time i.e. time between placing and receiving an order is of 5 days so order is placed after 5 days assuming that by the time next batch of 100 units is obtained remaining stock of 50 units would be consumed. Also stock is reviewed after every 5 days so making reorder point as when 50 units are left.

Case 1: Suppose demand after 5th day varies in such a fashion that 5 units are consumed on 6th day, 10 on 7th day, 5 on 8th day, 15 on 9th day and 5 on 10th day as shown in Fig. 2(a). Thus a total of only 40 units out of available 50 units are consumed. This results in formation of safety stock of 10 units (Table 2). Also after a periodic review at 5th day if an order of 100 units is ordered then total inventory position would be of 110 units increasing holding cost. So estimation of reorder point depends on variation in demand after during lead time period.

These extra 10 units are called as safety stock. Thus,

Stock at ROP level (R = 50) – demand during Lead time (L = 40) = Safety stocky (SS = 10) So, ROP = L + SS

Case 2: Suppose demand during 6th and 7th day is of 20 units and during 8th day is 10 units so all remaining 50 units are consumed in 3 days as shown in Fig. 2(b). Whereas order would be replenished only on 10th day making a situation of stock out during 9th and 10th day (Table 2).

These two cases clearly indicate that:

If demand is weak during lead time it would result in creation of safety stock.
If demand is strong during lead time it would result in out of stock situation and might result in loss of customers.
This result in uncertainty in estimation of reorder point. Above cases clearly shows that in P system estimation of reorder point is dependent on variation in demand both before and during lead time. Resolution of this uncertainty requires understanding the concept of service level.

36.2.5 Estimation of reorder point: How much to order? As seen in above discussed sections inventory position in Q system is effected by variation in demand both before and during lead time. So batch size to be ordered should have protection level for both time periods. In case of P system protection level was determined only for the demand variation during lead time. Suppose time period before lead time is denoted as ‘P’ after which periodic review of stock is done and during lead time is denoted as ‘L’ then total time per ordering cycle would be ‘P + L’. So quantity that has to be ordered to maintain a specific inventory level would be made for this time period.

This is done in similar fashion as was in case of Q system. Demand variation data would be collected for both time periods rather than only for during lead time period. The quantity to be ordered or desired inventory level to be maintained would be sum of average variation in demand during time period P+L and safety stock to buffer against variation in demand during same time interval. Thus,

Desired inventory level = (average demand during P+L) + (safety stock during P+L)

= d * (P+L) + z*[standard deviation*square root of (P+L)] Where d is average demand

P is time period after which stock would be reviewed

L is lead time

z is number of standard deviations calculated from desired service level standard deviation is the deviation in demand d.

Example: It is given that average daily demand of a product is 6 units with a standard deviation of 1.2 units. The variation shows normal distribution and stock is reviewed after every 60 days. Lead time is 5 days and company intends to maintain a service level of 95%. Estimate number of units to be ordered per batch.

Solution:

d = 6 units

standard deviation = 1.2 units

P = 60 days

L = 5 days

At 95% service level z = 1.65.

Putting these values in the formula

Desired inventory level = 6 * (60 + 5) + 1.65 * [1.2 * square root (60 + 5)]

= 390 + 16

= 406 units.

36.3 COMPARISON BETWEEN P AND Q SYSTEMS OF INVENTORY CONTROL

The application of one type of inventory control systems depends on number of factors such as type of demand, type of product etc. Selection of a particular system depending on advantages and disadvantages is difficult as one’s advantages are disadvantages of other and vice versa. However, because of increasing usage of technology in inventory control Q system has more application than P system. Technology automatically indicates number of units left in the stock and how much is withdrawn in case of every withdrawal process as desired in Continuous review system. Following is the comparison of two systems

without getting in to advantages and disadvantages.

36.4 P MODEL: SPECIAL CASE

EOQ model and some of its variations were dependent upon the assumption of a constant demand rate. When this assumption is relaxed, i.e., when the amounts that need to be withdrawn from inventory are allowed to vary from period to period, the EOQ formula no longer ensures a minimum-cost solution. Consider the following periodic-review model.

Planning is to be done for the next n periods regarding how much (if any) to produce or order to replenish inventory at the beginning of each of the periods. The order to replenish inventory can involve eitherpurchasing the units or producing them, but the latter case is far more common with applications of this model, so we mainly will use the terminology of producing the units. The demands for the respective periods are known (but not the same in every period) and are denoted by

ri demand in period i, for i 1, 2, . . . , n.

These demands must be met on time. There is no stock on hand initially, but there is still time for a delivery at the beginning of period 1.

The costs included in this model are similar to those for the basic EOQ model:

K = setup cost for producing or purchasing any units to replenish inventory at beginning of period, c = unit cost for producing or purchasing each unit,

h = holding cost for each unit left in inventory at end of period.

Note that this holding cost h is assessed only on inventory left at the end of a period. There also are holding costs for units that are in inventory for a portion of the period before being withdrawn to satisfy demand. However, these are fixed costs that are independent of the inventory policy and so are not relevant to the analysis. Only the variable costs that are affected by which inventory policy is chosen, such as the extra holding costs that are incurred by carrying inventory over from one period to the next, are relevant for selecting the inventory policy. By the same reasoning, the unit cost c is an irrelevant fixed cost because, over all the time periods, all inventory policies produce the same number of units at the same cost. Therefore, c will be dropped from the analysis hereafter.

The objective is to minimize the total cost over the n periods. This is accomplished by ignoring the fixed costs and minimizing the total variable cost over the n periods, as illustrated by the following example.

Example:

An airplane manufacturer specializes in producing small airplanes. It has just received an order from a major corporation for 10 customized executive jet airplanes for the use of the corporation’s upper management. The order calls for three of the airplanes to be delivered (and paid for) during the upcoming winter months (period 1), two more to be delivered during the spring (period 2), three more during the summer (period 3), and the final two during the fall (period 4). Setting up the production facilities to meet the corporation’s specifications for these airplanes requires a setup cost of $2 million. The manufacturer has the capacity to produce all 10 airplanes within a couple of months, when the winter season will be under way. However, this would necessitate holding seven of the airplanes in inventory, at a cost of $200,000 per airplane per period, until their scheduled delivery times. To reduce or eliminate these substantial holding costs, it may be worthwhile to produce a smaller number of these airplanes now and then to repeat the setup (again incurring the cost of $2 million) in some or all of the subsequent periods to produce additional small numbers. Management would like to determine the least costly production schedule for filling this order. Thus, using the notation of the model, the demands for this particular airplane during the four upcoming periods (seasons) are

r1 = 3, r2 = 2, r3 = 3, r4 = 2.

Using units of millions of dollars, the relevant costs are K = 2, h = 0.2.

The problem is to determine how many airplanes to produce (if any) during the beginning of each of the four periods in order to minimize the total variable cost. The high setup cost K gives a strong incentive not to produce airplanes every period and preferably just once. However, the significant holding cost h makes it undesirable to carry a large inventory by producing the entire demand for all four periods (10 airplanes) at the beginning. Perhaps the best approach would be an intermediate strategy where airplanes are produced more

than once but less than four times.

36.5 SUMMARY

Periodic Review system (P) of inventory control is used where orders are placed after every fixed and regular time interval resulting in certainty in order making. But such system gets affected adversely by fluctuation in demand both during and before lead time. Also as orders are made after specific interval thus time between orders for each cycle remains same whereas because of variation in demand quantity ordered could be different. For instance, if demand is high before lead time making consumption of inventory quicker so manager tends to order for higher quantity. Whereas if demand is slow then manager tends to place a

lower order to maintain desired inventory level.

36.6 GLOSSARY

Fixed time period model: is the model that specifics quantity to be ordered after every fixed time interval.
Safety stock: is the amount of units carried in addition to the expected demand.
Stock out: is the situation when the entire inventory gets used up due to high demand and no units are left to serve the customer.

36.7 REFERENCES/ SUGGESTED READINGS

Chase, B.R., Shankar, R., Jacobs, F.R. and Aquilano, N.J., Operations & Supply Chain Management, 12thEdition, McGraw Hill.
Stevenson, W.J., Operations Management, 9thEdition, Tata McGraw Hill.
Lee J. Krajewski, Operations Management, Prentice-Hall of India, New Delhi, 8th Edition.