inventory and waiting line unit 12 inventory...

Inventory and Waiting Line Models

UNIT 12 INVENTORY CONTROL :

PROBABILISTIC MODELS Objectives

After studying this unit, you should be able to

• discuss various probabilistic models.

• explain various approaches to the probabilistic problems,

• set various inventory levels under various probabilistic conditions.

• describe the role of simulation study for inventory problems.

• clarify the problems in terms of variability in demand and lead time,

Structure

12.1 Introduction

12.2 Inventory Models with Probabilistic Demand

12.3 Single Period Probabilistic Models

12.4 Multi-Period Probabilistic Models

12.5 Inventory Control Systems

12.6 Fixed Order Quantity System

12.7 Periodic Review System

12.8 Other Variants of Probabilistic Models

12.9 Summary

12.10 Key Words

12.11 Self-assessment Exercises

12.12 Further Readings

12.1 INTRODUCTION In previous unit, we have discussed Simple deterministic inventory models where each and every influencing factor is known completely. But, in actual business life complete certainty never occurs. Therefore, we will discuss here some practical situations of inventory problems by relaxing the condition of certainty for some of the factors. The major influencing factors for the inventory problems are price, demand and lead time. Other factors such as carrying cost, ordering and stock-out costs are also affecting the inventory problems, but their nature is not so much disturbing. This is because their estimation provides almost, on the average, as known values. Even price can also be averaged out to reflect the condition of certainty. But sometimes, price fluctuations are too much in the market and hence they influence the inventory decisions. Similarly, variability in demand or consumption of an item as well as the variability in lead time influences the overall inventory policy. 12.2 INVENTORY MODELS WITH PROBABILISTIC

DEMAND Inventory models where only demand is probabilistic or random will be discussed here. Demand pattern may be having discrete probability distribution or continuous probability distribution as explained in Figure 12.1.

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Inventory Control – Probabilistic Models

12.3 SINGLE PERIOD PROBABILISTIC MODELS These models deal with the inventory situation of the items-such as perishable goods, spare parts and seasonal goods requiring one time purchase only. The demand for such items may be discrete or continuous. Since purchases are made only once, the lead time factor is least important in these models. In single period models, the problem is studied using marginal (or incremental) analysis and the decision procedure consists of a sequence of steps. In such cases, there are two types of costs involved, namely (a) Over-stocking cost, and (b) Under-stocking cost. These two costs represent opportunity losses incurred when the number of units stocked is not exactly equal to the number of units actually demanded. The following symbolic notations are to be used : D = demand of an item in units (a random variable) Q = the number of units stocked (or to be purchased) C1 = Over-stocking cost (also known as over-ordering cost). This is an opportunity

loss associated with each unit left unsold. = C + Ch - V C2 = Under-stocking cost (also known as under-ordering cost). This is an opportunity

loss due to not meeting the demand. = S-C-Ch/2+Cs

where C is the unit cost price; Ch, the unit carrying cost for the entire period; Cs, the shortage cost; S, the unit selling price and, V, the salvage value. 1) Single Period Discrete Probabilistic Demand Model (Incremental Analysis

Method) The cost equation for this type of problem may be developed as follows. For any quantity in stock Q, only D units are consumed. Then for specified period of time, the cost associated with Q units in stock is either: a) (Q-D)C1, where D, the number of units used or demanded is less than or equal to

the number of units Q, in stock, i.e., D Q≤ b) (D-Q)C2, where the number of units required is greater than the number of units

in stock, i.e. D>Q. Since, the demand D is random variable, its probability distribution of demand is known. p(D) denotes the probability that the demand is D units, such that total probability is one, i.e.,

If Q* is the optimal quantity stocked, then the total expected cost f(Q*) will be minimum. Thus, if we stock one unit more or less than the optimal quantity, the total expected cost will be higher than the optimal. Thus,

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Therefore, the optimal stock level Q* satisfies the relationship (6). For practical application of (6), the three step procedure is as follows: Step 1:From the data, prepare a table showing p(D), the probability and the

cumulative probability P(D≤Q) for each reasonable value of D.

Step 2 :Compute the ratio 2

1 2

CC C+

which is known as service level.

Step 3 :Find the value of Q which satisfies the inequality (6). Example 1 A trader stocks a particular seasonal product at the beginning of the season and cannot re-order. The item costs him Rs. 25 each and he sells at Rs. 50 each. For any item that cannot be met on demand, the trader has estimated a goodwill cost of Rs. 15. Any item unsold will have a salvage value of Rs. 10. Holding cost during the period is estimated to be 10 per cent of the price. The probability distribution of demand is as follows:

Determine the optimal number of items to be stocked. Solution As per step 1, we put the data regarding demand distribution in the Table 1 below:

Table 1 : Probability Distribution of Demand

Looking at Table 1, this ratio lies between cumulative probabilities of 0.60 and 0.80 which in turn reflect `the values of Q as 3 and 4. That is,

P(D≤ 3)=0.60<0.69<0.80 = P(D≤ 4).

Therefore, the optimal number of units to stock is 4 units.

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Estimating Cost of Under-stocking Suppose that in example 1, the cost of under-stocking is not known, but the decision maker's policy is to maintain a stock level of 5 units. We can find for what values. of C2, the under-stocking cost, does Q*=5? We have the following inequality:

Inventory for Perishable Products Many organisations handle merchandise which has negligible utility if it is not sold almost immediately. Products in this category include newspapers, printed programmes for special events, fresh produce and other perishable commodities. Such items commonly have high mark-up. The large difference between the wholesale cost and the retail price is due to the risk vendor faces in stocking the item. He faces obsolescence costs on the one hand and opportunity costs on the other. All such problems can be very easily solved with the help of the model discussed above. Example 2 A newspaper boy buys papers for Rs. 0.35 each and sells them for Rs. 0.60 each. He can not return unsold newspapers. Daily demand has the following distribution:

If each day's demand is independent of the previous day's demand, how many papers should he order each day? Solution Transferring the given data into the following table, we have

gives Q*=280. Thus, newspaper boy should buy 280 papers each day.

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2) Single Period Discrete Probabilistic Demand Model (Payoff Matrix Method) Example 3 Consider again Example No. 1 here. The trader has five reasonable courses of action, or strategies. He can stock the items from 2 to 6 units. There is no possible reason to stock more than 6 since he can never sell more than 6 and there is no possible reason for ordering less than 2. Since there are five alternative courses of action for stocking and five levels of demand, it follows that there are 25 combinations of one strategy and one level of demand. For these 25 combinations, we find the trader's payoffs in the form of payoff matrix. As per the cost information given, the payoffs are obtained for the two situations (a) when demand is not more than the stock level, and, (b) when demand is more than the stock quantity. That is,

The payoff matrix will be five by five. Each element of the matrix can be calculated by above total payoffs for demand less than, equal to, or greater than the order size. When demand is less than or equal to the order size, we have the following contributions to the payoff. The trader buys the items for Rs. 25Q, he sells D of them for Rs. 50D, he earns salvage of Rs. 10(Q-D) for items not sold, and he incurs the carrying cost of Rs. (.10)(25) (Q-D) on unsold items and average carrying cost of (.10) (25) D/2 on the items sold during the period. Thus, total payoff comes out to be -17.50 Q+41.25D for demand less or equal to order size. If demand is more than the order size, the contributory payoff will consist of (i) the cost of purchase Rs. 25Q, (ii) selling revenue of Rs. 50Q, (iii) goodwill loss of Rs. 15(D-Q), and (iv) carrying cost of (.10) (25) Q/2. Thus, the total payoff for demand more than order size is 38.75Q-15D. The payoff matrix is given as follows:

We now find the expected payoff for each order size (or strategy). The procedure for calculating the expected values is simple: for any given strategy multiply each possible payoff for that strategy by the corresponding probability of the given level of demand and add all of these products up. Thus, for first strategy of order size 2 units, the expected value of payoffs is (47 50)(0.35) +(32.50)(0.25) + (17.50)(0.20)(2.50)(0.15) + (-12.50)(0.05) = Rs. 28. Proceeding similarly, we calculate all of the expected values: Order size Q 2 3 4 5 6 Expected value Rs. 28 Rs. 47.0625 Rs. 52.0625 Rs. 45.8125 Rs. 31.125 Objective is to select that strategy which provides the highest payoff. Thus, the trader should order for 4 units for the highest expected payoff of Rs. 52.0625. If we compare the two methods of finding the solution we see that incremental analysis provides only the optimum level of purchase quantity and does not indicate about the level of expected profit. The payoff matrix method provides both the answers i.e. optimum purchase quantity as well as, the optimum expected return.

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The payoff matrix can be converted to opportunity cost matrix, where the opportunity cost is, in short, a cost sustained because the decision taken is not the best in terms of the level of demand which actually occurs. It is easy to calculate the opportunity cost matrix directly from the payoff matrix. Take any column of the payoff matrix corresponding to a specific level of demand and select the largest payoff if the payoffs are profits, the smallest payoff if the payoffs are costs. Then subtract each payoff in the same column from the largest payoff to get the corresponding opportunity costs in the case of profits. For costs, subtract the smallest payoff from each payoff in the same column to get the opportunity costs. In our example, we get the opportunity cost matrix as follows:

We now, find the expected opportunity costs for each alternative. The objective is to select that strategy which provides minimum expected opportunity cost. The expected opportunity cost for the first alternative is : (0) x (0.35)+(38.75) (0.25)+(77.50) (0.2)+(116.25) (0.15)+(155) (0.05) = 50.375 Proceeding similarly, we calculate all the expected values: Order size, Q 2 3 4 5 6 Expected cost Rs. 50.375 31.3125 26.3125* 32.5625 47.25 Thus, the decision is to select the minimum expected cost i.e. the trader should store 4 units for the lowest cost of Rs. 26.3125. We find a relationship between the expected opportunity costs and expected payoffs as follows:

E.O.C. = K-E.P. where E.O.C. = expected opportunity cost

E.P. = expected payoff (profit) K = constant or K = 78.375

= (47.50)(0.35) +(71.25)(0.25) + (95)(0.2) +(118.75)(0.15) +(142.5) (0.05)

= sum of the expected value of the largest entries in each column of the payoff matrix

= the expected value of the payoffs for all the best strategies. or, the expected opportunity cost for a given strategy. = K (78.375)-Expected payofffor each strategy (8) Thus, it is obvious from the equation (8) that the maximum value of E.P. will simultaneously produce, the minimum of E.O.C. The two analyses namely, the payoff matrix method and the opportunity cost matrix method produce the came result. If the original payoff matrix is in terms of costs it can be similar reasoning, be shown that the above relationship (8) will be of the form:

EOC=E.p.-K, where E.p. is now in terms of costs. 3) Single Period Continuous Probability Demand Model This model is similar to that discussed through (1) to (6) with the difference that the demand is not discrete. The demand, though probabilistic, is treated as continuous.

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The probability distribution described by equation (1) is now written as

where f(D) is a continuous probability density function indicating the probability for small interval of demand D. dD is known as the differential of the demand D. Equation (9) is read as the integral of probability function f(d) dD over the interval from demand D=0 to D=∞ is one. This is the total probability. The cumulative probability function is obtained as follows:

representing that F(Q) is the probability that demand lies between 0 and Q. For optimal results, we use the incremental analysis in the same way as has been used in (1) to (6). The optimal quantity to stock Q* is the point where

Example 4 An item sells for Rs. 25 per unit and costs Rs. 10. Unsold items can be sold for Rs. 4 each. It is assumed that there is no shortage penalty cost besides the lost revenue. The demand is known to be any value between 600 and 1000 items. Determine the optimal number of units of the item to be stocked. Solution It is assumed that the demand distribution is a uniform distribution between 600 and 1000 units. The uniform distribution is represented by the relation

1f(D)=b-a

where b is the upper limit of the demand and a is the lower limit. Thus, for our problem,

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Activity 1

List out the various factors influencing the decision for producing ice-creams during the summer season.

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Activity 2

How will you formulate the payoff matrix and the opportunity cost matrix for the purchase of spare parts along with the main heavy equipment for an industrial organisation?

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Activity 3

Two products are stocked by a company. The company has limited space and cannot store more than 40 units. The demand distributions for the products are as follows:

The inventory carrying costs are Rs. 5 and Rs. 10 per unit of the ending inventories for the first and second product, respectively. The shortage costs are Rs. 20 and Rs. 50 per unit of the ending shortage for the first and second product respectively. Find the economic order quantities for both the products.

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Activity 4

A television dealer finds that cost of holding a television stock for a week is Rs. 30 and the cost of unit shortage is Rs. 70. For one particular model of television the probability distribution of weekly sales is as follows:

Weekly Sales 0 1 2 3 4 5 6

Probability 0.05 0.10 0.20 0.25 0.20 0.15 0.05

How many Units per week should the dealer order?

…………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………

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12.4 MULTI-PERIOD PROBABILISTIC MODELS In single period models, only demand is the major variable factor and lead time does not play any role in the decision process. But, in multi-period models, both demand and lead time play major role in the decision process. These factors may be changing according to certain laws of probability. The variation in demand and/or in lead time imposes risks. We cushion the effects of demand and lead time variation by absorbing risks in carrying larger inventories, called bufferstocks or safety stocks. The larger we make these safety stocks, the greater our risk, in terms of the funds tied up in inventories, the possibility of obsolescence and so on. However, we minimize the risk of running out of stock. While minimizing the risk of out of stock we can minimize the risk of inventories by reducing the buffer inventories which in turn lead to increase in the risk of poor inventory service. Therefore, our objective is to find a rational decision model for balancing these risks.

12.5 INVENTORY CONTROL SYSTEMS When, either demand or lead time, or both vary randomly, the concept of operating doctrine is introduced to take into account the possibility of stock-outs. These operating doctrines are based on inventory control system adopted for the purpose of control of inventory. While following an inventory control system, the various levels of operating the system are established. There are various kinds of stock-levels, but the following are fundamentals to the control systems:

- Minimum Level

- Ordering Level

- Hastening Level

-Maximum Level

Minimum Stock Level : This is also known as safety or buffer stock. This is the level below which stock is not allowed to fall. When this level is reached, it triggers of urgent action to bring forward delivery of the next order, it is sometimes called the danger level. In fixing their level, the main factor to be taken into account is the effect which a run-out of stock would have upon the flow of work or operations.

Ordering Stock Level : This is known as Re-order level. This is the level at which ordering action is taken for the material to be delivered before stock falls below the minimum. Two main factors are involved in deciding the re-order level, (i) the anticipated rate of consumption, and (ii) the estimated time which will elapse between the raising of a provision demand and the actual availability of goods in store after receipt and inspection, i.e. the lead time.

The Hastening Stock Level : This is the level at which it is estimated that hastening action is necessary to request suppliers to make early delivery. It is fixed between the minimum level and the re-order level. This level is subjectively based on experience.

The Maximum Stock Level : This is the level of stock above which the stock should not be allowed to rise. The purpose of this level is to curb excess investment. In fixing the maximum, the main consideration is usually financial, and the figure is arranged so that the value of stock will not become excessive at any time. Other points affecting this level are the possibility of items becoming obsolete, and the danger of deterioration in perishable commodities.

These levels are fixed for implementing various levels of inventory control systems.

12.6 FIXED ORDER QUANTITY SYSTEM This is also known as Perpetual Inventory System; Re-order Inventory System or Q-system. In this system, the count of the number of units in inventory is continuously maintained. With lead time is less than the re-order cycle, an order for a fixed quantity (mostly, it is EOK)) is placed when the inventory level drops to a pre-determined re-order R.

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Figure 12.2: Fixed Order Quantity System

Setting of various levels for Q system For fixed order quantity system, the various levels can be fixed for the following conditions: 1) 2) 3)

Variable Demand and Constant Lead Time Constant Demand and Variable Lead Time Variable Demand and Variable Lead Time

1) Control Levels for Variable Demand and Constant Lead Time In this case we want to find an operating doctrine that takes into account the possibility of a stockout. We define the few additional variables: U = random variable representing demand during lead time σ = standard deviation of demand during lead time

U = Expected lead time demand tL = lead time

d = average daily demand

dσ = standard deviation of daily demand

D = expected annual demand B = buffer stock or safety stock Z = number of standard deviations needed for a specified confidence level R = Re-order level Thus, re-order level is given as:

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Example 5

The daily demand of an item is normally distributed with a mean of 50 units and standard deviation of 5 units. Lead time is 6 days. The cost of placing' an order is Rs. 8, and annual holding costs are 20% of the unit price of Rs. 1.20. A 95% service level is desired. Back-orders are allowed but there is no stock-out cost. Find the various levels.

Solution

Operating doctrine is to order 2700 units when inventory level reaches to the re-order point of 2662 units.

Maximum Level = 2700+862 = 3562 units

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2) Control Levels for Constant Demand and Variable Lead Time For this model, the various levels are determined as follows:

Re-order level = R* = U + B

where U = Ldt = (Daily demand x expected lead time) and Buffer stock, B = z Uσ

= Z (demand) (Standard deviation of lead time) Another, simple way of determining the safety stock is : B = (Maximum Lead Time - Normal Lead Time) (Demand during Lead Time) Further more, the levels of safety stock depend upon what extent an organisation is prepared to accept stock-out risk (SOR). Since it is difficult to obtain an accurate estimate for the shortage cost, the management must specify reasonable service level (SL) so as to determine safety stock necessary to keep the stock-out risk within the prescribed limits. The service level is the probability of not running out of stock on any stock cycle, i.e. per cent of order cycle in which all the demand can be supplied from the stock: Service level (SL) = 100% - Stock-out Risk (SOR) ... (18)

Optimal stock-out risk (SOR) = 1

2

CC

... (19)

where (19) is obtained by marginal analysis of evaluating the cost of shortage for one unit i.e. (1 unit short) (average stock-out per year) (Cost per unit short) = C1 or (SOR/yr) C2 = C1 ... (20) Example 7 A company is ordering an item 4 times a year and has specified a service level of one stock-out per 3 years. The history of re-order lead times is shown below. Daily demand of the item is 40 units. Find the re-order level.

Solution

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Example 8

Consider Example No. 7 with the following change in demand. The average daily demand of the item is 40 units per day and variance of daily demand is (30 units/day)2

Solution

Activity 5

Mention below the products for which fixed order system is applicable. What are the benefits of this system?

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12.7 PERIODIC REVIEW SYSTEM This system involves the reviewing of stock levels at a fixed interval of time known as review period and placing replenishment orders at the end of each period. The replenishment quantity is variable and corresponds to the amount of stock required to bring the stock ordered and the stock on hand up to a target level. Thus, the key variable for system are T, the review period (fixed time between reviews of

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inventory records) and TI, the target inventory level on which orders are based. This system is also known as Fixed Period System or Replenishment Inventory System or P-system.

Figure 12.3 summarises the operation of a forced ordering periodic review system, as an order must be placed at the end of each review period. Each review period is exactly T days long; at the end of this period, an order is placed for a quantity sufficient to replenish inventory to the target level T1. After the re-order lead time tL, the shipment arrives and goes into inventory. The order quantities are different each period, being calculated as Q=Tl minus inventory on hand minus previous orders not yet received. The formula for target inventory for a fixed interval T is as follows :

1) Control Levels for Variable Demand and Constant Lead Time

The average demand during the lead time and review period is d (T+tL). Buffer stock, to be hold during this period is, B = Z ... (20) d (T+tL)σ

Where Z is the number of standard deviations to give the required service level. The target inventory level is

TI = dd(T+tL) Z (T+tL)σ+ …..(21)

2) Control Levels for Constant Demand and, Variable Lead Time

Average review period demand = Td

Average lead time demand = Lt d

Buffer stock = B = Lt

Zdσ

Thus,

where tL = standard deviation of lead time 3) Control Levels for Variable Demand and Variable Lead Time

Average review period demand = Td

Average lead time demand = Lt d

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Example 9 Consider example 5. This is a case of constant lead time and variable demand. Solution Review period is fixed equal to

2 8T= 0.06 22 days

(.2)(1.2)(50)(365)×

= =

Lead time = 6 days Average demand during review period T = 22x50 = 1100 units Average demand during lead time, tL = 6x50 = 300 units Safety stock = dZ T+tσ L = 1.645 x5 x = 43.5 or 44 units

Therefore, target inventory level TI = 1100 + 300 + 44 = 1444 units. Example 10 Consider Example 7 - a case of constant demand and variable lead time. Solution For this example, review period is fixed as

T = 14

years = 90 days

Then, average demand during review period = 90x40 = 3600 units Average lead time, Lt = 14.83 days. Therefore, average demand during average lead time = 40x 14.83 = 593.3 units Variance of lead time = 34.97 (days)2 Safety stock for the service level of 91.7% confidence = 1.39 40 34.97 = 351.4 units Therefore, the target inventory level TI = 3600+593.3+351.4 = 4544.7 or 4545 units. Example 11 Consider example 8 where demand and lead time both are varying. Solution

Review period T = 14

yrs = 90 days

Thus average consumption during review period = 90x40=3600 units Average consumption during average lead time = 40x14.83 = 593.3 units

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Therefore, the Target Inventory level = 3600+593.3+414.9 = 4608.2 units

12.8 OTHER VARIANTS OF PROBABILISTIC MODELS The other variants of probabilistic models are: 1) Variable Lead Times This situation is illustrated with the help of an example. Example 12 An item is used uniformly at a rate of 1150 units per year. Each unit costs Rs. 40. Ordering and Inspection cost is Rs. 55 per order. Holding costs are 25% of the value of the average inventory in storage. The distribution of lead time is as follows: Lead Time Days 6 7 8 9 10 11 12 13 14 Relative frequency 0.00 0.04 0.08 0.38 0.24 0.12 0.09 0.03 0.02 The opportunity cost of running out of stock is estimated to be Rs. 40 per unit. The number of working days in a year are 230 days. Find the optimum stocking policy. Procedure for solving such problems is illustrated below: i)

ii)

iii)

Determine the order size, orders per year, order interval and the daily usage rate under the assumption of certainty disregarding lead times. Find the average lead time, and the lead time needed for 95% or 99% confidence level. Determine the EOQ, for the lead time obtained above by the formula

Where (EOC)i = expected opportunity cost for the ith lead time provided

iv) Determine the total cost for the value of Qi, the ith order size, by the formula:

where (EHC)i is the expected holding cost for the ith lead time. Solution The following information is known Co = Rs.55, Ch = Rs. 10, D = 1150 units Therefore,

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We now calculate the expected costs of lead time alternatives of 12, 13 and 14 days respectively. Since, the opportunity cost is Rs. 40 per unit, the one can sustain a loss of Rs. 40x5= Rs. 200 per day for late deliveries based on the daily usage of 5 units. Multiplying the stock deficiencies by Rs. 40 per cent and the excess stock by holding cost of Rs. 10 per unit, we develop an expected value table below:

Table 4 : Expected costs of Lead time alternatives*

Figures to the left of zeros are holding costs and that to the right of zeros are opportunity costs. The two costs on the right-hand side of the table, the expected holding costs and expected opportunity costs affect the ordering policy in two ways. Opportunity costs occur only when an order takes longer to arrive than the lead time then was allowed. This can be treated as ordering expenses or set up expenses. Holding costs are annual expenses and reflect the extra inventory or safety stock that is held in storage. These are calculated as follows: For the lead time alternative of 12 days, we have (EHC)12 = 250x0.04+200x0.08+150x0.38+100x0.24+50x0.12 = Rs. 113. = The expected holding cost of safety stock for 12 days lead time. Similarly (EOC)12 = 200x0.03+400x0.02 = Rs. 14.

= Expected opportunity costs for 12 days' lead time. and EOQ for 12 days lead time is given as

It becomes obvious from Table 5 that most economic policy is to allow 13 days lead time with EOQ=117 units with least total cost of Rs. 1326. An order should be placed whenever the stock level declines to 13x5=65 units. By following this policy one can expect that run out of stock twice in 100 order periods. There should not be too much concern if opportunity costs cannot be estimated with the desired degree of accuracy because a little change in the opportunity costs does not affect the overall result significantly. One can get the simpler result just after calculating the lead time for over 95% of time 13 days which indicates that the re-order level should be kept for 13x5=65 units and EOQ=117 units.

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2) Variable Demand Rates Variations in demand or usage rates produce situations similar to variable lead times. If the lead time is fixed, the risk of stock-out is dependent on the actual demand exceeding the anticipated demand. Holding the safety stock reduces the risk. The problem then becomes one of the selecting a safety stock against the cost of stock shortage. Example 13 Annual holding cost for an item are Rs. 2 per unit and the stock is replenished about times a year. Opportunity costs (or stock out costs) are estimated to be Rs. 1/- per week. The item has the following usage rate during lead time of one week period. Usage rate per week 100 150 200 250 300 Relative frequency 0.10 0.20 0.40 0.10 0.20 A series of stock-outs has indicated that the policy of keeping 200 units as re-order level be reviewed. It was decided to retain the same lot size but a safety stock could be carried for the item. Find the optimal policy. Solution In order to solve this problem, we will follow the same general principle as in case I above. A zero safety stock is to be associated with a re-order point of 200 units. From the demand pattern we see that for 200 units of re-order level, stock outs may be up to 30% of the time. Holding an extra 50 units would limit running out of stock to 20% Of the time and 100 units cushion would completely eliminate the chances of stock-out for the same demand pattern. The opportunity costs when we require 50 units in safety stock, but actually we don't have a safety stock, would be 50x 1x6=300. All these are illustrated below:

Table 6 : Expected value of opportunity costs

The total cost of each of the alternatives is the sum of EOC and the holding costs. For each safety stock increment of 50 parts the annual holding cost is Rs. 50x2=Rs. 100. The total costs are calculated in Table 7.

Table 7 : Total cost table

From Table 7 it is clear that the original ordering policy was sound and re-order point should remain at 200 units. 3) Variable Demand and Variable Lead Time When both the lead time and demand rate vary significantly from average values, the task of determining a satisfactory safety stock is considerably more complicated. No direct mathematical techniques are generally available to handle these situations. But we can apply the technique, often called, the Monte Carlo simulation. One procedure for generating simulated experience is utilization of random numbers. An abbreviated tabulation of random numbers is given in Table 8.

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Table 8 : Random Numbers

32867 53017 39106 31111 28325 65501 38947 60207 27102 71684 74859 18107 15606 14543 22345 82244 67549 78910 55847 56155 81211 94095 95970 00401 11751 08995 62845 69902 69469 92810

These groups of numbers follow no pattern or order; they are randomly distributed. The main concern a user should have is imposing a pattern by repeatedly using the same set in a consistent order. The figures can be read in any manner desired by rows or columns, diagonally up or down etc. For an example bf simulation, the pie-charts in Figure 12.4 may be assumed to represent the relative frequency of lead times and usage rates for a commodity. The relative size of the slices in the "Pies" corresponds to the chance occurrence of each increment of lead time or demand. Thus, possible lead times are 8, 9, 10 and 11 days occurring with respective relative frequencies of 0.20, 0.20, 0.30 and 0.30.

Figure 12.4 : Relative frequencies of Lead Times and Usage Rates

The simulation of stock movement is conducted by randomly selecting both a lead time and a usage rate. This could be done by mounting spinners on the two pies. At the pin on the lead time pie may show a day, and the associated usage pie could point to 30 units. The two spins are combined to indicate one level of total demand that could occur after an order has been placed. A random number table could commonly be used in place of spinners. In a group of 10 numbers, any digit from 0 to 9 is equally likely to occur. By letting each digit equal to 10%, we can assign certain digits to each increment of lead time and usage according to its likelihood of occurrence. An arbitrary assignment might be: Days Lead Time

Digits Units Usage Rate per day

Digits

8 0,1 20 0 9 2,3 25 1,2 10 4„5,6 30 3,4 11 7,8,9 35 5,6,7,8 40

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Then each pair of numbers in the random number table could represent a lead time duration and the number of units used per day during that duration.

Using the first two numbers in the first column of Table 8 we see that 3 is the first number of 32 and 2 is the second, then the digit 3 corresponds to a lead time of 9 days and usage rate of 30 units per day. Then the inventory pattern is as follows:

Table 9 : Simulation of Ten Re-ordering Periods

The re-order level is based on an average lead time of 10 days and an average usage rate of 30 units per day. A typical order is placed when the stock hand falls to 10x 30=300 units. The right-hand column of Table 9 shows that we do note require safety stock or buffer stock 60% of the time while it is required for 40% of their time.

If the simulation procedure started in Table 9 is continued, a distribution can be determined for expected stock levels at the time of delivery. From this distribution an inventory policy can be delivered which establishes a minimum cost balance between holding and opportunity costs. The computation would be similar to those followed for a variable lead time inventory policy. A more direct tactic would be to set a tolerable limit for stock-outs per year and hold a safety stock which confirms to this limit. Most of the time, the buffer stock is equal to K times the standard deviation of the demand distribution where K=1, 2 or 3. It is logical that the greater the number of trails, the more closely the simulation will correspond to the actual inventory pattern.

Owing to the large number of repeated calculations required to give a reliable estimate of probable stock-outs, it is desirable to use a computer for such Monte-Carlo simulations.

Activity 6

For any hypothetical demand distribution and lead time distribution simulate to find out average demand and safety stock requirements for 95% confidence.

…………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………

12.9 SUMMARY This unit has highlighted the role of inventory control under the actual business conditions of uncertainty desirable by any probability distribution. Various inventory models under the conditions of probabilistic demand and/or probabilistic lead time have been illustrated with the help of various examples. Two major Inventory Control systems have been described by the various situations of probabilistic pattern of demand and lead time. Monte-Carlo simulation method has been applied for illustrating the inventory model when both demand and lead time vary.

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12.10 KEY WORDS

Buffer Stock : Extra-inventory held against the possibility of stock-out.

Continuous Probability Distribution : A probability distribution in which the variable is allowed to take on any value within a given range.

Deterministic Model : A model where every influencing factor is known completely.

Discrete Probability Distribution : A probability distribution in which the variable is allowed to take only limited number of values.

Expected Value : The average value or mean. This is obtained by summing the multiplication of the variable values with their respective probabilities.

Expected Payoff : Expected value of the variable indicating payoffs.

Expected Opportunity Cost : Expected value of the variable indicating opportunity costs.

Monte-Carlo Simulation : Quantitative procedure which conducts a series of organised trial and error experiments on a model of a process to predict the behaviour of that process over time.

Multi Period Models : Models dealing with multiple ordering in a plan period.

Opportunity Cost : The cost of the opportunities that are sacrificed in order to take a certain action.

Opportunity Cost Matrix : Matrix (or values put in a table form) of opportcosts.

Over-stocking cost : This is the cost of keeping more units thap demanded.

Payoff The benefit which accrues from a given combination of decision alternand state of nature.

Payoff Matrix : Arrangement of payoffs in proper order or in table form.

Perishable Products : The goods that deteriorate with time.

Periodic Review System : An inventory which reviews the inventory status of item under system control at stipulated intervals review periods.

Probability Distribution : Values of a variable known as outcomes of an experiwith the associated probabilities is known as probability distribution.

Probabilistic Inventory Models : Models dealing with inventory having demalead time with probability diitributions.

Random Numbers : The numbers that simulates the output that one would getsampling a uniform random variable.

Review Period : A time interval fixed for reviewing the inventory position.

Re-order Level : The stock level which is sufficient for the lead time consumpti

Salvage Value : Revenue received by selling the item at comparative lower seprices.

Single Period Models : Inventory models where only single order is initiated.

Standard Deviation A measure of the spread of the date around the mean.

Service Level : The probability of not being out of stock or the percentage ofthat the demand during the re-order period can be satisfied.

Under Stocking Cost : Cost relating to the out-of stock situation undeprobabilistic situation.

Variance : This is the square of the standard deviation

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12.11 SELF-ASSESSMENT EXERCISES 1) Actual daily demand and lead time distributions are given below., What is the

expected demand during the lead time? What is the minimum or maximum demand during lead time will ever be?

Find the operation doctrine for 95% service level for P and Q systems.

2)

3)

4)

5)

6)

7)

Daily demand of an item is normally distributed with a daily mean of 60 units and standard deviation of 10 units. Supply is virtually certain for a lead time of 3 days. The cost of placing an order is Rs. 18, and annual holding costs are 20 of the unit price of Rs. 1.20. For 360 days a year find the operating doctrine for 90% and 95% service level. A Christmas tree supplier has evaluated weekly demand for November-December over the last 7 years. Demand appears normally distributed with a mean of 350 trees demanded weekly and standard deviation of 200. A Christmas tree sells for an average price of Rs. 60.00. If not sold, it can be salvaged for Rs. 20. The cost to raise the tree is Rs. 37.50 what should be the weekly, ordering quantity for the up-coming season? Consider an item for which the following data are available: Annual average demand=10000 units. Standard deviation of demand per week =50 units. Unit cost Rs. 6. Ordering cost = Rs. 100 per order. Inventory carrying charge is 30% per year. Average lead time = 4 weeks. Maximum delay in lead time = 2 weeks. Probability of delay = 0.25. Service level is 95%. Design an appropriate system for this item. From the following information determine the decision variables for proper inventory management, Annual demand = 1200 units, Annual carrying cost = Rs. 16/unit Ordering cost = Rs. 24/order, stock-out cost = Rs. 40 Lead time = 10 days, Working days/yr=300 Demand during lead time 38 39 40 41 42 Probability 0.10 0.20 0.38 0.24 0.08 A television dealer finds that cost of holding a television stock for a week is Rs. 30 and the cost of unit shortage is Rs. 70. For one particular model of television the probability distribution of weekly sales is as follows: Weekly Sales 0 1 2 3 4 5 6 Probability 0.05 0.10 0.20 0.25 0.20 0.15 0.05 How many units per week should dealer order? Also, find his optimal expected payoff. Find the optimal ordering policy for the following inventory problem: Demand/yr = 12 units, Price/Unit = Rs. 20 Ordering cost = Rs. 60, Carrying charge = 20% Unit shortage cost = Rs. 50. The lead time distribution is as follows: Lead time (days) 20 21 22 23 24 25 26 Probability 0.04 0.06 0.20 0.30 0.25 0.10 0.05

12.12 FURTHER READINGS Adam, E.E, Jr and K.J. Ebert (1986) : Production and Operations Management;

Prentice Hall International.

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Buchan, J. and E. Koenigsberg (1963): Scientific Inven'tory Management; Prentice Hall, Unc.

Buffa, Diwood S. (1990) : Modern Production/Operations Management, led. Wiley Eastern Limited.

Dobler, D.W., Leel, Jr., and Burt, D.N. (1984): Purchasing and Materials Management Text and Cases; Tata McGraw Hill.

Gupta, M.P. and J.K. Sharma (1987): Operations Research for Management; National Publishing House.

Hadley, G and T.N. Whitin (1983): Analysis of Inventory Systems; Prentice Hall. Levin, R.I. and C.A. Kirkpatrick (1978): Quantitative Approaches to Management;.

McGraw Hill Kogkusha International Student. IVIhstafi, C.K., (1988) : Operations Research : Methods and Practice; Wiley Eastern

Ltd. McClain, J.O. and L.J. Thomas (1987) : Operations Management; Prentice Hall of

India. Peterson, R and E.A. Silver (1979) : Decision Systems for Inventory Management

and Production Planning, Wiley, New York. Shenoy, C.V., V.K. Shrivastava and S.C. Sharma, (1986) ; Operations Research for

Management; Wiley Eastern Limited. Starr, M.K. and D.W. Miller (1977) : Inventory Control Theory and Practice,

Prentice Hall of India. Taha, H.A. (1982): Operations Research: Introduction; MacMillan Publishing Co.

inventory and waiting line unit 12 inventory...

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