Inventory Models (Stock Control)

Transcription

1 Inventor Models (Stock Control) Reference books: Anderson, Sweene and Williams An Introduction to Management Science, quantitative approaches to decision making 7 th edition Hamd A Taha, Operations Research, An Introduction, 5 th edition T A Burle, G O sullivan, Operational Research L Lapin, Quantitative Methods for business decisions with cases, 5 th edition Lecture 1 Inventor, sometimes known as stock, refers to idle goods or materials that are held b an organisation for use sometime in the future. Items carried in inventor can be: raw materials purchased parts components subassemblies work-in-process finished goods and supplies One main reason that organisations maintain inventor is that it is rarel possible to predict sale levels, production times, demand, usage needs exactl. Thus, inventor serves as a buffer against uncertain and fluctuating usage and keeps a suppl of items available in case the item are needed b the organisation or its customers. While inventor serves an important and essential role, the expense associated with financing and maintaining inventories is a substantial part of the cost of doing business. In large organisations, the cost associated with inventor can run into millions of dollars. To minimise the inventor cost and to guarantee a smooth operation of the organisation the same time, the manager needs to answer the following two important questions: 1) How much should be ordered when the inventor for the item is replenished? ) When should the inventor for a given item be replenished? The purpose of the discussion on inventor models is to show how quantitative models can assist in making the above decisions. We face three situations: Deterministic inventor models, where it is reasonable to assume that the rate of demand for the item is constant or nearl constant; 1

2 Probabilistic inventor models, where the demand for item fluctuates and can be described in probabilistic terms; Just-in-time (JIT), a philosoph of material management and control, of which the primar objective is to eliminate all sources of waste, including unnecessar inventor. 1. A Generalised Inventor Model It has been mentioned earlier that the ultimate objective of an inventor model is to answer the how much to order and when to order questions. The answer to the first question is the order quantit. It represents the optimum amount that should be ordered ever time an order is placed and ma var with time depending on the situation under consideration. The answer to the second question depends on the tpe of the inventor sstem: periodic review at equal time intervals - the time for acquiring a new order usuall coincides with the beginning of each time interval; continuous review - a review point is usuall specified b the inventor level at which a new order must be placed. The order quantit (how much) and reorder point (when) are normall determined b minimising the total inventor cost that can be expressed as a function of these two variables. The total inventor cost is generall composed of the following components: Total inventor cost = Purchasing cost + Setup cost + Holding cost + Shortage cost The purchasing cost becomes an important factor when the commodit unit price becomes dependent on the size of the order. This situation is normall expressed in terms of a quantit discount or a price break, where the unit price of the item decreases with the increase of the ordered quantit. The setup cost (also known as ordering cost) represents the fixed charge incurred when an order is placed. Thus, frequent smaller orders will result in a higher setup cost than less frequent larger orders. The holding cost, which represents the costs of carring inventor in stock (e. g., interest on invested capital, storage, handling, depreciation, and maintenance), normall increases with the level of inventor. The shortage cost, is a penalt incurred when we run out of stock of a needed commodit. It generall includes costs due to loss of customer s goodwill as well as potential loss in income.. Deterministic Models.1 Single-item static (EOQ) model This is the best known and most fundamental inventor model, which is applicable when the demand for an item has constant, or nearl constant, rate and when the

3 entire quantit ordered arrives in inventor at on point in time (instantaneous replenishment). It assumes no shortage in this inventor model. Different reference books ma use different notations for the inventor cost parameters. We will use the following: D = demand rate for an item (quantit/unit time) = the order quantit = maximum inventor level K = the setup cost ever time an order is placed h = the holding cost per item per unit time TC = the total cost per unit time From the definitions of D and, we know that the inventor level reaches zero from its maximum level in /D time units (this being also known as the ccle time), and that the number of orders to be placed per unit time is D/. Figure 1 below illustrates the inventor pattern for the EOQ inventor model. Inventor level Max. inventor level / Average inventor t 0 =/D Time Figure 1 Generall, the total inventor cost is as follows: Total inventor cost = Purchasing cost + Setup cost + Holding cost + Shortage cost However, in the current situation the shortage cost is non-existent because it assumes there is no shortage at all. Moreover, the purchasing cost will be a mere constant since in this model we assume that the commodit price is independent of the size of the order. The total cost function is to be used mainl for finding the optimal inventor level so that total cost is minimal. A constant in such a function will not alter the optimal solution. Therefore, in the EOQ model, we express the total inventor cost as the sum of the setup cost and the holding cost, as follows: Total inventor cost = Setup cost + Holding cost 3

4 The setup cost equals to the product of the number of orders per unit time and the setup cost per order. That is, setup cost = (D/)K = DK/ (cost/unit time) The holding cost can be calculated b multipling the average inventor level to the holding cost per item per unit time. The average inventor level in the EOQ model is ½. Therefore, holding cost = (½ ) h = ½ h (cost/unit time) Hence, the total cost function is TC = DK/ + ½ h (cost/unit time) (1) The how-much-to-order decision Equation (1) is a function of the order quantit. The optimal value of, the answer to the how-much-to-order question, can be obtained b minimising TC with respect to in the following procedure: 1) differentiate TC with respect to dtc d DK h = + ) let the above be zero DK h + = 0 3) solve the equation for *, the optimal order quantit KD * = () h The order quantit is the economic order quantit, sometimes also known as Wilson s economic lot size. The corresponding ccle time is: t 0 * = * /D and the minimal total inventor cost is: TC * = KDh The when-to-order decision 4

5 To answer this question we need first to introduce the concept of inventor position. The inventor position for an item is defined as the amount of the inventor on hand plus the amount of inventor on order. The when-to-order decision is expressed in terms of a reorder point - the inventor position at which a new order should be placed. In practice, an order takes time to be filled. The time lag from the point at which an order is placed until the order is delivered is called the lead time, denoted b L. Figure illustrates the situation where reordering occurs L time units before deliver is expected. Reorder points L L Figure In general, the reordering point (R) can be calculated b the following equation: R = L D (3) In equation (3), D is the demand per unit time, L is the lead time. Note that the lead time L ma be either longer or shorter than the ccle time t 0. If L < t 0, use L directl in equation (3); if, on the other hand, L > t 0, (L-nt 0 ) should be used in equation (3) instead of L where n is the largest integer not exceeding L/t 0 *. Example: The dail demand for a commodit is approximatel 100 units. Ever time an order is placed, a fixed cost K of 100 is incurred. The dail holding cost h per unit inventor is Pence. If the lead time is 1 das, determine the economic order quantit and the reorder point. Solve: We know that K = 100 per order D = 100 units/da h = 0.0 per unit per da L = 1 das Using equation (1), the EOQ is 5

6 * = KD = h 00. = 1000 units And the ccle time is t 0 * = * /D = 1000/100 = 10 Because in this case L > t 0, the reorder point is calculated as R = (L-t 0 ) D = (1-10) 100 = 00 units The above calculation shows that it is most economical to order 1000 units of the commodit when the inventor level for this commodit reaches 00 units. Lecture. Single item static (EOQ) model with price breaks In the EOQ model, the purchasing cost per unit time is neglected in the analsis because it is constant and hence should not affect the level of inventor. It often happens however, the purchasing price per unit ma depend on the size of the quantit purchased. This situation usuall occurs in the form of discrete price breaks or quantit discounts. In such cases, the purchasing price should be considered in the inventor model. Consider the inventor mode with instantaneous stock replenishment and no shortage. Assume the cost per unit is c 1 for < q and c for q, wherec 1 > c and q is the quantit above which a price break is granted. The total cost per unit time will now include the purchasing cost in addition to the setup cost and the holding cost. The total cost per unit time with price break is Cost Dc 1 + KD/ + ½ h for <q TC = (4) Dc + KD/ + ½ h for q I TC 1 m Figure 3 II q 1 III TC These two functions are shown graphicall in Figure 3. Disregarding the effect of price breaks for the moment, we let m be the quantit at which the minimum values of TC 1 and TC occur. 6

7 TC 1 TC TC 1 TC TC 1 TC Thus KD = m h The cost functions TC 1 and TC in Figure 4 reveal that the determination of the optimum order quantit * depends on where q, the price break point, falls with respect to the three zones I, II, and III shown in the figure. These zones are defined b determining q 1 (> m ) from the equation TC 1 ( m ) = TC (q 1 ) Since m is known, the solution will ield the value of q 1. The optimum order qualit * is, as illustrated in Figure 4, m if 0 q < m (Zone I) * = q if m q < q 1 (Zone II) m if q q 1 (Zone III) Cost Cost I q m II q 1 II I m II q q 1 II Cost I m II q 1 II q Figure 4 The procedure for determining * ma thus be summarised as follows: 1. Determine m = (KD/h). If q < m (zone I), then * = m, and the procedure ends. Otherwise, 7

8 . Determine q 1 from equation TC 1 ( m ) = TC (q 1 ) and decide whether q fallsin zone II or zone III. 1) If m < q < q 1 (zone II), the * = q; ) if q q 1 (zone III), the * = m. Example: Consider the inventor model with the following information. K = 10, h = 1, D = 5 units, c 1 =, c = 1, and q = 15 unit. First, we need to compute the m. m = (KD/h) = ( 10 5/1) = 10 units Since q > m, it is necessar to check whether q is in zone II or zone III. The value of q 1 is computed from TC 1 ( m ) = TC (q 1 ) namel, Dc 1 + KD/ m + ½ h m = Dc + KD/q 1 + ½ hq 1 Substitution ields q 1-30q = 0 This is obviousl of the tpe ax + bx + c = 0 whose solutions are b b 4a x1, = ± c a So, the solutions for q 1 are found to be either q 1 = 6.18 units or q 1 = 3.8 units. B definition, q 1 must be larger than m. Therefore, q 1 = 6.18 units. Since m < q < q 1, q is in zone II. Therefore, the optimum order quantit * = q = 15 units. The associated total cost per unit time is TC ( * ) = TC (15) = Dc + KD/ h/ = 15.83/da..3 EOQ model with planned shortages Lecture 3 A shortage is a demand that cannot be supplied. In man cases, shortages are undesirable and should be avoided if possible. However, in the cases where the value of a unit commodit is ver high, shortages are then desirable and used commonl in practice. An example of this tpe of situation is a new-car dealer s inventor. It is uncommon for a specific car ou want to be in stock. What the do is ask ou to specif our requirements for the car and place an order for ou after ou have showed our commitment (sizeable deposit for example). 8

9 In this model, we assume that a. the shortage cost is small; b. no demand is lost because of shortage as the customers will back order, i.e., accept delaed deliver (this tpe of shortage being known as backorders); and c. the replenishment is instantaneous. The EOQ inventor model with planned shortages is illustrated in Figure 5. t b t 1 T Figure 5 In Figure 5, = the total ordering quantit b = the inventor shortage, or the backorder t 1 = the time units in which the inventor reaches the zero level t 1 = (-b)/d t = the time period from stock-out to the arrival of a new order t = b/d T = the inventor ccle, T = /D In the above, as in the EOQ model D is the demand rate of the commodit. In this case, the total inventor cost can be expresses as the sum of the holding cost, setup cost, and the shortage cost, i.e., Holding cost TC = holding cost + setup cost + shortage cost In the EOQ model, it is assumed that the holding cost per item per unit time is h. The component of holding cost is the product of the average inventor level and h. The average inventor level is calculated as follows: [½ (-b) t t ]/T = (-b) /() 9

10 Therefore, the holding cost is h(-b) /() (cost/unit time) Setup cost It is known from the EOQ model that the cost per order is K. The setup cost equals to K multipling the number of orders in the given period of time. The number of orders can be calculated b dividing the demand b the ordering quantit, i.e., D/. The setup cost rate is K per order, hence the total setup cost is KD/ (cost/unit time) Shortage cost If we assume that the shortage penalt is p per item per unit time, then the shortage cost is p average backorder level Since the average backorder level is the shortage cost is (0 t 1 + ½ bt )/T = b /() b p/( ) (cost /unit time) Therefore, the total cost is TC h ( b ) KD b p = + + (cost/unit time) (5) Minimise TC with respect to the order quantit and to the planned backorders ields the optimal values of order quantit * and backorder b * : DK h+ p * = h p h b * = h p * + (6) (7) Proof That gives ( TC) h ( ) ( b) = bp p b + b h h = + b = 0 10

11 h b * = h p + ( TC) h ( b) ( b) = ( ) = DK h + p b h h = 0 DK b p h Substituting b * = h p * in the above equation gives + DK h+ p * = h p From equation (6), The interval between orders can be calculates as follows: T * = * /D (8) Example: Suppose the North-west Electronics Compan has a product for which the assumptions of the EOQ inventor model with backorders are valid. Information obtained b the compan is as follows: The annual demand D = 000 units/ear The cost per order K = 5 The holding cost rate h = 10 per unit per ear The backorder cost p = 30 per unit per ear Work out the optimal order quantit *, the optimal backorder level b *, and the ccle time T. Solve: The optimal order quantit * = [ 000 5/10 (10+30)/30] = units 115 units The optimal backorder level b * = 10/(10+30) 115 = 8.75 units 9 units The ccle time for place orders T = * /D = 115/000 = ear 11

12 Since there are 50 working das in a ear, the ccle time is = 14.4 working das. The total annual cost for operating the inventor is TC = holding cost + setup cost + shortage cost = 10 (115-9) /( 115) / /( 115) = = 867/ear If the compan had chosen to prohibit backorders and had adopted the regular EOQ model, the recommended inventor decision would have been * = ( 000 5/10) = 100 units.4 Economic production-quantit model Lecture 4 In the models we discussed so far, we assumed that the orders are filled instantaneousl. However, in some situations, that is not the case. Consider the follow situation. When the inventor reaches its zero level, a production sstem starts its operation and add new items to the inventor while the inventor keeps on its normal operation. Of course, the production rate of items must be higher than that of the demand rate. This section discusses the model for this situation. This inventor model is represented in Figure 6. Max inv. level Production rate Demand rate T 1 T T As in the EOQ model, we are now dealing Figure with 6 two costs, the holding cost and the setup cost. Namel 1

13 TC = holding cost + setup cost Holding cost We know that the holding cost equals to the product of the average inventor level and the holding cost per item per unit time. Let D = the demand rate per unit time P = the production rate per unit time h = the holding cost per item per unit time Then, according to Figure 6, T = /D, T 1 = /P, and T = T - T 1 = (P-D) / DP The maximum inventor level in this situation is the build-up speed T 1 = (P-D) /P = (P-D)/P From the EOQ model, we know that the average inventor level is half of its maximum. In this case, it is ½ (P-D)/P. Therefore, Setup cost the holding cost = ½ (P-D)/P h = ½ h(p-d)/p The setup cost is based on the number of production runs per unit time and the setup cost per run. The number of production runs is D/. If the setup cost per run is K, which includes the labour, material, and other relevant cost during the production, then the setup cost is Total cost the setup cost = D/ K = DK/ The total cost for this model is TC = ½ h(p-d)/p + DK/ (9) The minimum-cost production quantit, often referred to as the economic production quantit, can be found b minimise TC with respect to, leading to = DK h P P D (10) Example 13

14 Beaut Bar Soap is produced on a production line that has an annual capacit of 60,000 cases. The annual demand is estimated 6,000 cases, with the demand rate essentiall constant through out the ear. The cleaning, preparation, and setup of the production line cost approximatel $135. The manufacturing cost per case is $4.50, and the annual holding cost is figured at a 4% rate. Thus the holding cost per case per ear is h = 0.4 $4.50 = $1.08. What is the recommended production quantit? Solve: We know that P = 60,000 cases/ear D = 6,000 cases/ear K = $135 per run h = $1.08 per case per ear So, = DK P = h P D = 3387 cases The total annual cost according to * is TC * = DKh P D = = $073 P The ccle time between production runs is T = * /D = 3387/6000 = ear Considering that there are 50 working das in a ear, T = working das Thus, we should plan a production run of 3387 units about ever 33 working das. Lecture 5 3. Probabilistic Models In practice, a large number of inventor situations cannot be described b the deterministic inventor models we discussed so far. In these cases, the demand is no longer constant and deterministic, but probabilistic instead. This tpe of demand is best described b the probabilit distribution. 3.1 Single-period inventor model with probabilistic demand 14

15 It is necessar clarif the term single period. This term refers to the situation where the inventor will onl be demanded in one time duration, and cannot be transferred to the next time duration. Newspaper selling is such an example. The newspaper ordered for toda will not be sold tomorrow. Fashion selling is another example. Spring-summer designs will not sell during the autumn-winter season. Increment analsis is a method that can be used to determine the optimal order quantit for a single-period inventor model. The increment analsis addresses the how-much-to-order question b comparing the cost or loss of ordering one additional unit with the cost or loss of not ordering one additional unit. Let c o = cost per unit of overestimating demand This cost represents the loss of ordering one additional unit which will not sell. c u = cost per unit of underestimating demand This cost represents the loss of not ordering one additional unit which could have been sold. Suppose that the probabilit of the demand of the inventor items being more than a certain level is P(D>), and that the probabilit of the demand of the inventor items being less than or equal to this level is P(D ). Then, the expected loss (EL) will be either of the following: For overestimation: EL(+1) = c o P(D ) For underestimation: EL() = c u P(D>) The optimal value of the demand level, *, being the optimal ordering quantit as well, can be found when This means EL(*+1) = EL(*) (11) c o P(D * ) = c u P(D> * ) (1) From the stud of Probabilit, it is known that P(D> * ) = 1 - P(D * ) (13) Substituting (13) into (1), we have c o P(D * ) = c u [1 - P(D * )] Solving for P(D * ), we have P(D * ) = c u /(c u +c o ) (14) 15

16 The above expression provides the general condition for the optimal order quantit * in the single-period inventor model. The determination of * depends on the probabilit distribution. Example1: (uniform probabilit distribution) Emma s Shoe Shop is to order some new design men s shoes for the next spring-summer season. The shoes cost 40 per pair and retail 60 per pair. If there are still shoes not sold b the end of Jul, the will be put on clearance sale in August at the price of 30 per pair. It is expected that all the remaining shoes can be sold during the sale. For the size 10D shoes, it is found that the demand can be described b the uniform probabilit distribution, shown in Figure 7. The demand range is between 350 and 650 pairs, with average, or expected, demand of 500 pairs of shoes. Expected demand Figure Solve: We are asked to help Emma s Shoe Shop to determine the order quantit. It is essential to work out the overestimating cost c o and the underestimating cost c u. The cost per pair of overestimating demand is equal to the purchase cost minus the sale price per pair; that is c o = = 10 The cost per pair of underestimating demand is the difference between the regular selling price and the and the purchase cost; that is c u = = 0 Putting c o and c u into equation (14), we have P(D * ) = c u /(c u +c o ) = 0/(0+10) = /3 We can find the optimal order quantit * b referring to the assumed probabilit distribution shown in Figure 7 and finding the value of that will provide P(D * ) = /3. To do this, we note that in the uniform distribution the 16

17 probabilit is evenl distributed over the entire range of Thus, we can satisf the expression for * b moving two-thirds of the wa from 350 to 650. That gives /3 ( ) = 550 pairs namel, the optimal order quantit of size 10D shoes is 550 pairs. Example : (Normal probabilit distribution) Kremer Chemical Compan has a contract with one of its customers to suppl a unique liquid chemical product. Historicall, the customer places order approximatel ever 6 months. Since the chemical needs months ageing time, Kremer will have to make its production quantit decision before the customer places an order. Kremer s inventor problem is to determine the number of kilograms of the chemical to produce in anticipation of the customer s order. Detailed information: Kremer s manufacturing cost: 15/kg Fixed selling price: 0/kg Underestimation (Kremer to bu substitute + transportation): 4/kg Overestimation (Kremer to reprocess and sell the surplus) 5/kg Normal probabilit distribution (Figure 8) best describes the possible demand. Expected demand μ Standard deviation σ = kg Figure 8 Solve: Underestimating cost: c u = 4-0 = 4/kg Overestimating cost: c o = 15-5 = 10/kg Therefore, P(D * ) = 4/(4+1) = 0.9 In the table of area of for the standard normal distribution, all the area under the curve is 1, representing Probabilit =1. However, onl half of the area under the curve (probabilit) is concerned in this table because of the smmetr of the standard normal distribution curve. The area used to find z is 17

18 that between the vertical line through z and the vertical line in the middle. For example, if this area is , then the corresponding z value is 1.5. Now, for this problem, P(D * ) = 0.9. This is corresponding to about 1/3 of the area covered b the curve on the left. The area which is to be used to find z is = 0.1. This gives us z = from the table. To go back to the de-standardised form, we have z = ( * -μ)/σ. Thus * = μ + σz = = 945 kg With the assumed normal probabilit distribution of demand, the Kremer Chemical Compan should produce 945 kg of the chemical in anticipation of the customer s order. Area = 0.9 σ = 100 z 1000 kg Figure 9 Note that in this case the cost of underestimation is less that that of overestimation. Thus, the Kremer Chemical Compan is willing to risk a higher probabilit of under estimation and hence a higher probabilit of stockout. In fact, Kremer s optimal order quantit has a 0.9 probabilit of having a surplus and a = 0.71 probabilit of a stockout. 3. An order-quantit, reorder-point inventor model with probabilistic demand In the previous section, we considered a single-period inventor model with probabilistic demand. In this section, we will extend our discussion to a multi-period order-quantit, reorder-point model with probabilistic demand. This multi-period model has the following characteristics: 1) the inventor sstem operates continuousl with man repeating periods or ccles; ) inventor can be carried from one period to the next; 3) an order is placed whenever the inventor position reaches the reorder point; 4) since demand is probabilistic, the following cannot be determined in advance: 18

19 the time the reorder point will be reached; the time between orders; the lead time. The inventor pattern can be described b the Figure 10. Figure 10 The how-much-to-order decision Although we are in a probabilistic demand situation, we can appl the EOQ model as an approximation of the best order quantit. That is KD * = h where, in this case, D is the expected annual demand. The when-to-order decision If the expected (or mean, average) demand is μ per unit time, and the standard deviation is σ, then the reorder point r can be expressed as r = μ + zσ (15) where z is the number of standard deviation necessar to obtain the stockout probabilit, and it can be find from the standard normal probabilit distribution table according to the tolerance of stockout probabilit. Example: Dabco lighting distributors purchases a special high intensit light bulb for industrial lighting sstems. Dabco would like a recommendation on how much to order and when to order so that a low-cost inventor polic can maintained. 19

20 The data: ordering cost is K = 1 per order; one bulb costs 6, and Dabco uses a 0% annual holding cost for its inventor (h = = 1.0) demand during 1 week lead time is best represented b a normal probabilit distribution with a mean of 154 light bulbs per week and a standard deviation of 5 light bulbs; Dabco is willing to tolerate an average of 1 stockout per ear. Solve: How-much-to-order Since the mean or expected demand is 154 bulbs, the annual demand is weeks = 8008 bulbs/ear. KD * = = ( /1.0) 400 bulbs h Using * = 400, Dabco can anticipate placing approximatel D/ * = 8008/400 0 orders per ear with an average of approximatel 50/0 = 1.5 working das between orders. When-to-order It is known that Dabco allows an average of 1 stockout per ear. It has been calculated that there will be approximatel 0 orders a ear. So, the probabilit of a stockout in a ear is 5% or 0.05). In other word, the probabilit for stockout not to happen is = This corresponds to z = Since μ = 154 and σ = 5 r = μ + zσ = bulbs Thus, the recommended inventor decision is to order 400 bulbs whenever the inventor level reaches the reorder point of 195. Since the mean or expected demand during the lead time is 195 units, the = 41 units serve as a safet stock, which absorbs higher-than-usual demand during the lead time. Roughl 95% of the time, the 195 units will be able to satisf demand during the lead time. The anticipated annual cost for this sstem is as follows: Ordering cost DK/ * = /400 = 40 Holding cost, normal inventor ½ * h = = 40 Holding cost, safet stock (41)h = = 49. Total

21 Note that if Dabco had a constant demand, the annual cost would be 480 onl. The uncertaint in the situation costs an extra A periodic-review model with probabilistic demand The inventor model discussed in 3.1 is a continuous-review model sstem, where the inventor position is monitored continuousl so that an order can be placed whenever the reorder point is reached. With computerised inventor sstems, such continuousreview on the inventor can be easil achieved. However, if a compan dandles multiple products, continuous-review on each of the products ma mean heav work-load and probabl low efficienc. In such cases, an alternative inventor model, the periodic review model, is preferred, because this model enables the orders for several items to be placed at the same preset periodicreview time. In this model, we assume that for an single product, the lead time is less than the length of the review period. Then the how-much-to-order decision at an review period is determined using the following: = M - H (16) where, = the order quantit; M = the replenishment (or the maximum) level; H = the inventor on hand at the review period. Since the demand is probabilistic, the inventor on hand, H, will var. Thus, the order quantit that must be sufficient to bring the inventor position back to its maximum or replenishment level M can be expected to var each period. Under the periodicreview model, enough units are ordered each review period to bring the inventor position back to the replenishment level. A tpical inventor pattern for a periodic-review sstem with probabilistic demand is illustrated in Figure 11. Note that the time between periodic reviews is predetermined and fixed. 1

22 Figure 11 The decision variable in the periodic-review model is the replenishment level M. To determine M, we could begin b developing a total-cost model, including holding, ordering, and stockout (shortage) costs. Instead, we will describe an approach that is often used in practice. In this approach, the objective is to determine a replenishment level that will meet a desired performance level, such as a reasonabl low probabilit of stockout or a reasonabl low number of stockouts per ear. In the periodic-review model, the order quantit at each review period must be sufficient to cover demand for the review period plus the demand for the following lead time. If during the review period plus the lead-time period the demand can be expressed b the normal probabilit distribution, then the general expression for M is M = μ + zσ (17) where μ = the mean demand during the review period plus the lead-time period; σ = the standard deviation of demand; z = the number of standard deviations necessar to obtain the acceptable stockout probabilit. Example: Pound Discounts is a store selling a wide variet of products for household use. The compan operates its inventor sstem with a two-week periodic review. From past experience, Pound discount knows that the demand for an kind of its goods obes the normal probabilit distribution. For product, the mean demand during the -week review period and the lead-time period is 50 units with the standard deviation of demand being 45. If there is 1% chance that demand will exceed the replenishment level, work out the replenishment (or maximum) inventor level the store should keep for this product. Solve: Known that μ = 50, σ = 45, and there is a 1% chance of stockout From the normal probabilit distribution table and using 1% stockout probabilit, we know that

23 z =.33 Therefore, M = μ + zσ = units 3

24 Exercise 4 Inventor Models (Stock Control) 1. In each of the following cases, inventor is replenished instantaneousl and no shortage is allowed. Find the economic lot size, the associated total cost, and the length of time between two orders. a. K = 100 per order, h = 0.05 per unit per da, D = 30 units per da. b. K = 50 per order, h = 0.05 per unit per da, D = 30 units per da. c. K = 100 per order, h = 0.01 per unit per da, D = 40 units per da. d. K = 100 per order, h = 0.04 per unit per da, D = 0 units per da. The smbols used: K - the set-up cost per order h - the holding cost per item per unit time D - the demand rate in items per unit time. In each case in Problem 1, determine the reorder point assuming that the lead time is a. 14 das. b. 40 das. 3. A compan currentl purchases its stock of a certain item b ordering enough suppl to cover a 1-month demand. The annual demand of the item is 1500 units. It is estimated that it costs 0 ever time an order is placed. The holding cost per unit inventor per month is and no shortage is allowed. a. Determine the optimal order quantit and the time between orders. b. Determine the difference in annual inventor costs between the optimal polic and the current polic of ordering 1-month suppl 1 times a ear. 4. An item is consumed at the rate of 30 items a da. The holding cost per item per unit time is 0.05 and the set-up cost is 100. Suppose that no shortage is allowed and the purchasing cost is 10 for an quantit less than or equal to q = 300 and 8 otherwise. Find the economic lot size. What is the answer if q = 500 instead? 5. Consider the inventor situation with the following features: Single item Constant demand rate Instantaneous replenishment No shortage allowed Notations to be used: C o = the cost of placing one order, i.e. the set-up cost; C h = the holding cost per item per unit time; Y = the order quantit; and 4

25 D = demand rate for an item. a. Draw at least 4 ccles of the inventor variations against time, indicating the maximum inventor level and the ccle time. b. Decide the expression for the total inventor cost. c. Determine how much to order ever time an order is placed. d. Determine when to order in terms of inventor position R, assuming the lead time is L. 6. An item sells for 4 a unit but a 10% discount is offered for lots of size 150 units or more. A compan that consumes this item at the rate of 0 items per da wants to decide whether or not to take advantage of the discount. The set-up cost for ordering a lot is 50 and the holding cost per unit per da is Should the compan take advantage of the discount? 7. The ABC Compan purchases a component used in the manufacture of automobile engines directl from the supplier. ABC s engine production requires 1,000 item s of this component annuall. The cost for placing an order is $5, and the holding of the component costs $0.50 per item per ear. The Compan decided to operate with a backorder (shortage) inventor polic. Backorder costs are estimated to be $5 per unit per ear. Identif the following: a. The minimum-cost order quantit b. The maximum number of backorders c. The maximum inventor level d. The ccle time e. Total annual cost f. Assuming 50 das of operation per ear and a lead time of 5 das, what is the reorder point for the ABC Compan? 8. A manufacturing compan operates an inventor with the economic production lot size model, where the production rate is 8000 units per ear, the demand rate is 000 units per ear, the ordering cost is $300 for placing an order, and the holding cost is $1.60 per unit per ear. Find the following: a. the economic production lot size for this situation; b. the minimum cost of keeping the inventor; and c. the production frequenc. 9. Wilson Publish Compan produces books for the retail market. Demand for a current book is expected to occur at a constant annual rate of 700 copies. The cost of one cop of the book is The holding cost is based on an 18% annual rate, and the production set-up costs are 150 per set-up. The equipment on which the book is produced has an annual production volume of 5,000 copies. There are 50 working das per ear and the lead time for a production run is 15 das. Use the production lot size model to compute the following values: a. Minimum-cost production lot size b. Number of production runs per ear 5

26 c. Ccle time d. Length of production run e. Maximum inventor level f. Total annual cost g. Reorder point 10. A retail outlet sells a seasonal product for $10 per unit. The cost of the product is $8 per unit. All units not sold during the regular season for half the retail price in an end-of-season clearance sale. Assume that the demand for the product is uniforml distributed between 00 and 800. a. What is the recommended ordering quantit? b. What is the probabilit of a stockout using our order quantit in (a)? c. To keep customers happ and returning to the store later, the owner feels that stockouts should be avoided if at all possible. What is our recommended order quantit if the owner is willing to tolerate a 0.15 probabilit of stockout? d. Using our answer to (c), what is the goodwill cost ou are assigning to a stockout? 11. Flod Distributors, Inc., provide a variet of auto parts to small local garages. Flod purchases parts from manufacturers according to the EOQ model and ships the parts from a regional warehouse directl to its customers. For a particular tpe of muffler, Flod s EOQ analsis recommends orders with * = 5 to satisf an annual demand of 00 mufflers. There are 50 working das per ear and the lead time averages 15 das. a. What is the reorder point if Flod assumes a constant demand rate? b. Suppose that an analsis of Flod s muffler demand shows that the demand follows a normal probabilit distribution with μ = 1 and σ =.5. If Flod management can tolerate one stockout per ear, what is the revised reorder point? c. What is the safet stock for (b)? If the holding cost is $5 per unit per ear, what is the extra cost due to the uncertaint of demand? 6