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Un metodo heuristico para el control de inventarios de productos de corto ciclo de vida.

A heuristic method for the inventory control of short life-cycle products

1. Introduction

1.1 Background

More demanding customer needs and requirements, increasing competitive pressures, and accelerated technological changes currently urge companies to adopt innovation as a key factor to ensure their sustainabiliry. Commonly, four or five product cycle stages are considered: introduction, growth, maturity, saturation, and decline. For many products, the maturity and saturation stages last for a long time. For other products, their full life cycle might last for less than a year or even for a shorter period of time. Kurawarwala & Matsuo (1996), Kapuscinski et al. (2004), and Nair & Closs (2006) state that, at present, it is common to find products whose demand only occurs during a short period of time, after which the products usually become obsolete even if their intrinsic characteristics or quality features do not change. Such products are known as shortlife-cycle products (SLCPs).

Given the long manufacturing and distribution lead times that SLCPs might have and their high transportation and fixed production costs, production orders and purchase decisions are usually made prior to the sales season when information may be very limited. When shortages occur during the sales season, it is possible that no inventory replenishments can be done. Therefore, the retailer and the manufacturer usually hold large inventories at the beginning of the season. On the other hand, holding large inventories may result in excessive costs of obsolescence because SLCPs cannot be regularly sold in future seasons. In some cases, when SLCPs are shipped to other locations where they can still be sold, companies must pay significant relocation costs. In some manufacturing sectors, the combined shortage and obsolescence costs exceed manufacturing costs (Fisher & Raman, 1999; Nair & Closs, 2006), and in many cases the products become obsolete only after several time periods (Silver et al., 1998). In these cases, it is possible to place more than one order during the sales season.

Companies that used to produce products with long life cycles, have nowadays to manage very short life cycles and a large variety ofproducts. Lee (2002) says that because SLCPs have transient and highly erratic demand, their design, manufacturing, physical distribution, and inventory management are difficult to execute. Some of the sectors that are most affected by the short life cycle of their products are: computer manufacturing, cell phones, garments, and the publishing industry.

According to Benjaafar et al. (2005), in a decentralized system with N identical and independent locations, the amount of inventory varies linearly with N, whereas, in a centralized system, the amount of inventory is approximately proportional to [square root of (N)]. Consequently, in a SLCP system, centralized decisions and collaborative strategies, inventory consolidation, and product postponement can generate important benefits by lessening the bullwhip effect and reducing shortages, excess inventory, safety stock, and the risk of obsolescence. Vendor Managed Inventory (VMI) is a very good alternative to establish such collaborative strategies because it attempts to solve coordination problems by giving the supplier the responsibility to manage the replenishment process and decide when and how much to ship to the customer (Kaipia et al., 2002). VMI can reduce inventory levels and simultaneously improve customer service. It is a way to improve total channel profit, particularly when there are disagreements in the inventory systems of the retailer and the supplier, which is usually the case in the publishing industry (Yao et al., 2007; Dong & Xu, 2002).

1.2 Relevant literature review and problem definition

As stated by Kurawarwala & Matsuo (1996), the literature on demand forecasting and inventory management does not adequately address topics about SLCPs. Nair & Closs (2006) state that traditional forecasting techniques may not be useful because most of them require at least one year of data. Even if these data were available, they would not generate satisfactory forecasts because some of the forecast technique assumptions do not hold. In addition, forecasting becomes more complex when price adjustments and advertising campaigns are implemented during the sales season.

Fisher et al. (2000) state that sales during the first part of the season are a good predictor of the total sales of a new product. Empirical results suggest that forecasts based on early sales are much more accurate than subjective predictions (Fisher & Raman, 1999). Rodriguez (2007) proposes a forecasting method based on the linear regression of cumulative demand through the sales season, using data from the demands of products with similar life cycle to estimate the regression parameters.

When the length of the season is much shorter than the replenishment time or when replenishment costs are high, a widely-studied approach has been to make a single shipment at the beginning of the sales season for a quantity that optimally balances the costs of understock and overstock. This approach is the well-known newsvendor or newsboy problem, which has had numerous extensions (e.g., Chung & Flynn, 2001; Matsuyama (2006); Chung et al., 2008).

Other researchers have considered multi-shipment scenarios. Kurawarwala & Matsuo (1996) involve a stochastic inventory model with finite horizon to determine the optimal replenishments along the life cycle. This model disregards fixed transportation and order processing costs. Zhu & Thonemann (2004) propose a heuristic to determine near-optimal inventory policies for SLCPs. Their model includes an ordering cost per unit and the holding and shortage costs, but does not consider fixed ordering costs either. Chen et al. (2007) formulate a nonlinear mixed-integer programming model to determine the optimal number of replenishments over a finite planning horizon. They assume a deterministic curve of demand and do not consider shortage costs.

Few real SLCP cases have been reported in the literature. For instance, Kapuscinski et al. (2004) present a forecasting technique and a system for controlling the inventory of components. The system has been implemented at Dell Computer Inc. and has helped to save at least 43 million dollars of inventory cost of one component employed in computer manufacturing. As Wagner (2002) states, analysts have made many efforts to develop effective inventory control methods that are easy to apply. However, customers continue to deal with empty shelves and excessive shortages appear to be unavoidable. Consequently, and given the complexity of a SLCP inventory control system, we believe that it is necessary to develop heuristic methods that can be easily applied to real cases and that yield satisfactory results. We have found that these methods outperform empirical inventory control strategies typically found in practice. Accordingly, in the present work, we develop an easy-to-implement heuristic method to control the retailer inventory of SLCPs under a VMI environment within a supply chain with a manufacturing plant, one warehouse, and Nretailers. In contrast to the majority ofprevious works, our model includes fixed ordering costs in the decision process and is applied in a real case. The heuristic has been divided into two parts. First, based on the newsvendor model, we consider a single shipment at the beginning of the sales season. Second, we allow the system to have two or more shipments during the sales season, based on the forecast of the remnant demand. The heuristic then selects the policy that most likely produce the least total cost.

We focus on a real case of distributing textbooks from one warehouse to 34 different retailers. Currently, the organization does not have enough capacity to produce during the sales season and therefore, it must produce ahead of time. Our results confirm that the developed heuristic significantly improves current practices. The heuristic provides the shipment plan through the season and minimizes the amount shipped, the shortages and excess inventory, and the total relevant cost.

The rest of this paper is organized as follows. Section 2 describes the inventory control heuristic. In Section 3, we evaluate the behavior of the heuristic based on real data from a book publishing company, and discuss these results in Section 4. Lastly, in Section 5 wepresent some conclusions.

2. Development of the heuristic method

2.1 Model assumptions

The development of the heuristic method is based on the following assumptions:

* We consider a supply chain with one warehouse, and Nretailers that share demand information and make centralized decisions, so facilitating the application of VMI concepts.

* Product demand is transient, stochastic, nonstationary, and non-correlated with the level of inventory at each retailer.

* The product cost, sale price, and replenishment costs are known and stable along the sales season.

* Any unsatisfied demand is considered as a lost sale.

* During the sales season, there are no product shipments among retailers.

* After the sales season, each unsold unit has a salvage value that is equal to a fraction of the unit sale price to the retailer. The salvage value includes the inventory holding cost during the season at the point of sale. This cost is assumed to be proportional to the number of units kept in inventory but not to the time that the products are maintained in stock. It is based on the assumption that sales season lasts for a short time.

2.2 The single-shipment submodel

We model the total demand during the sales season as atriangularprobability distribution because it is a representation easily compatible with aggregate forecasting systems for short life cycle products. It also constitutes a simple representation of an unknown distribution that lies within certain bounds (hair & Closs, 2006). It is therefore necessary, from pre-season information, to estimate the minimum expected or pessimistic demand a (the lower bound), the optimistic demand c (the upper bound), and the most likely demand b (the mode).

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (1)

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (2)

By using the known critical fractile of the newsvendor model (see, for instance, Chopra & Meindl, 2004) and the inverse cumulative distribution function of the demand [[PSI].sup.-1] (*), we determine the optimal size of the shipment [Q.sup.*] by means ofEgs. (1) and (2).

In Eq. (1), the unit cost of understock [C.sub.F] is equal to the difference between the product cost v and the unit salvage value s. Since Eq. (1) is valid for any continuous nonnegative random variable, we can apply it to the inverse cumulative function of the triangular distribution, and obtain the optimal shipment size [Q.sup.*] that maximizes the total expected profit (see Appendix A).

Given that we are looking for the best solution for the entire supply chain under a VMI environment, the parameters of the model must be defined from a global perspective, in order to avoid 'double marginalization'. This expression means that each of the supply chain parties perceives a different portion of the supply chain profit margin and thus that party makes its own decisions, reducing the total profit of the supply chain as a whole. To avoid double marginalization, therefore, one should consider p as the unit sale price to the final customer, v as the total unit variable cost paid by the supply chain to place the product in the point of sale (including the variable transportation cost), and s as the net unit salvage value of the product after discounting the inventory holding cost during the season as well as the return and any product recovery costs. [Q.sup.*] will then be the optimal shipment size for the entire supply chain.

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (3)

Although the newsvendor model attempts to optimize order size, it does not explicitly determine whether it is convenient to actually ship the products. To overcome this shortcoming, before shipping the products, it is reasonable to verify whether the net profit remains positive after considering the fixed replenishment cost [A.sub.0], given in $ / shipment. To be precise, the system will only ship the products if, for any shipment size Q, the expected value of the profit is greater than zero, that is:

E(U|Q) > 0 (4)

We calculate this expectedprofit by using Eq. (3) (see Appendix B).

2.3 The multiple-shipment submodel

We now develop a submodel that operates under a VMI approach where the supplier knows the stock level at each retailer during the sales season. At the beginning of the sales season, we have a forecast of the minimum expected demand a by point of sale. As in the single-shipment submodel, it is necessary to check whether the expected profit of the first shipment compensates the fixed shipping cost. If the first shipment size is equal to the expected minimum demand a, then the conditionthatmustbe satisfied is:

a(p-v) [greater than or equal to] [A.sub.0] (5)

If this constraint does not hold, then the shipment size should not be equal to a ; in that case, we should apply the singleshipment submodel described in the previous section.

After making the first shipment, we should determine at each time period whether it is necessary to replenish. Under the multipleshipment scenario, it is not easy to determine the expected profit of a specific shipment because, except for the last shipment of the sales season, any overstock at the end of a period cannot be considered as a product return. Consequently, instead of determining the economic feasibility of making each shipment, its size is calculated by using the Economic Order Quantity (EOQ) model. At each time period, the shipment size Q is defmedby the interval:

Max([Q.sub.A],[Q.sub.B]) [less than or equal to] Q [less than or equal to] [Q.sub.c] (6)

where Q is the amount (in units) to ship if shipping is feasible, [Q.sub.A] is the forecast of the amount (in units) required to satisfy the expected demand of the period plus the corresponding safety stock, [Q.sub.B] is the economic order quantity (in units), and [Q.sub.C] is the forecast of the amount (in units) required to satisfy the remaining expected demand of the sales season plus the corresponding safety stock.

We will consider the possibility for replenishment whenever [Q.sub.A] > 0. We calculate [Q.sub.A] based on the equation:

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (7)

where [I.sub.j]. is the inventory level (in units) at the retailer at the end of period j , k is the safety inventory factor for the period, L is the replenishment lead time (in time units), R is the review interval (in units), R {??] 1, [x.sub.j+R+L] is the forecast demand (in units) from period j + 1 to period j + R + L, and [[sigma].sub.j+R+J] is the estimation of the standard deviation (in units) of the forecast error demand from period j + 1 to period j + R + L, to calculate the corresponding safety stock.

As stated by Kapuscinski et al. (2004), the safety factor k can be determined by using the critical fractile of the newsvendor model:

k = [[PHI].sup.-1] ([C.sub.F]/[C.sub.F]+[C.sub.E.sup.']) (8)

where [[PHI].sup.-1] (o) is the inverse cumulative standard normal distribution and [C.sub.E.sup.'] is the holding cost of an unsold unit between consecutive periods, given in $ / (unit * period). Since any unsold unit in a period may be sold in the next period, we assume that

N[C.sub.E.sup.'] = [C.sub.E] (9)

where N is the total number of periods into which the sales season has been divided. Since N[C.sub.E.sup.'] stands for the holding cost of a unit for the rest of the sales season, this expression is equivalent to the holding cost that the standard EOQ model contains. So, from Eq. (9) and the known EOQ expression, we obtain:

[Q.sub.B] = [square root of (2X[A.sub.0]/[C.sub.E])] (10)

where X is the forecast (in units) of the total demand of the season.

In addition, it is necessary to verify that [Q.sub.B] plus the stock at the point of sale do not exceed the estimated amount to satisfy the demand of the remainder of the season and the corresponding safety stock. To accomplish this, we calculate the following quantity:

[Q.sub.C] = [x.sub.N] + [k.sub.N][[sigma].sub.N] - [I.sub.j] (11)

where [k.sub.N] is the safety inventory factor for the remainder of the season, determined by means of Eq.(8) by using [C.sub.E] instead of [C.sub.E.sup.'], [x.sub.N] is the forecast demand (in units) over the rest of the season, and [[sigma].sub.N] is the standard deviation (in units) of the forecast errors of the remaining demand over the rest of the season, used to calculate the corresponding safety stock.

It is important to note that the economic order quantity [Q.sub.B] may be slightly less than the amount necessary to satisfy the remnant demand of the season, [Q.sub.c]. For instance, suppose that the required amount to satisfy next week's demand forecast plus the safety stock is equal to 20 units, that the EOQ is 91 units, and that the expected total demand including safety stock for the rest of the season is 98 units. If we decided to ship an EOQ of 91 units, then it would be likely that a possible upcoming shipment to satisfy the remaining demand would not be profitable.

However, since we know that the EOQ model is not very sensitive to lot size, we may define a range, i.e., from 90% to 110% of the optimum [Q.sub.B], to adjust the shipment size in order to reduce the risk of shortages in the last periods of the season. Specifically, in our heuristic, whenever 1.1 [Q.sub.B] > [Q.sub.C], we set the shipment size equal to [Q.sub.C]. If we ship this quantity, we can assume that it will not be necessary to make an additional shipment, unless an unexpected increase in demand occurs by the end of the season. Therefore, the expected understock and overstock of the shipment will then be the expected understock and overstock of the rest of the season. Accordingly, we can compare the expected profit of the shipment with the expectedprofitwhennothingis shipped.

Similarly, the expected profit of the last shipment of the season should be calculated to determine whether such a shipment is profitable. We assume a normal distribution of the forecast errors of the demand through the end of the season. So, to calculate the expected profit U as a function of the necessary inventory T to fulfill the demand of the remainder of the season, we used Eq. (12) (see Appendix C).

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (12)

where [G.sub.Z](*) is the unitnormal loss function, and:

T = {[I.sub.j when the shipment is not sent [I.sub.j] + [Q.sub.j] when the shipment is sent, (13)

where [Q.sub.j] is the amount (in units) to ship in period j, if shipping is feasible.

To develop the heuristic for the multiple-shipment case, we need to define some additional variables: Fal is the variable to accumulate shortages (in units), [I.sub.0] is the initial inventory (in units) at the point of sale, j is the period counter, M is the shipment counter, [Q.sub.1] is the size (in units) of the first shipment, X is the variable to accumulate demand (in units), and [x.sub.j] is the observed demand (in units) in period j .

Figure 1 shows the complete heuristic method. It is important to note that, according to our experiments, we have found that by defining the first shipment [Q.sub.1] equal to:

[Q.sub.0] = a + [square root of (2[A.sub.0][Q.sup.*]/[C.sub.E])] (14)

we obtain a total cost lower than the cost corresponding to [Q.sub.1] = a. The rationale for this expression is the following. Since a is the minimum expected demand, it should be the minimum feasible initial shipment. However, this value should be adjusted by considering the optimal shipment [Q.sup.*] and the associated fixed cost [A.sub.0], incorporating the triangular probability distribution of demand and the relationship between shortages and overstock. As shown in Eq. (14), we do this by adding a quantity derived from an adaptation of the EOQ model.

Another interesting result that we found is that, for all the cases we tested, the ratio [Q.sub.0] / [Q.sup.*] is an indicator of whether the single-shipment submodel or the multiple-shipment submodel will produce the least total costs. This has been implemented in the heuristic by defining a threshold P that can be specified based on empirical results. In any particular case, however, the heuristic can be run for both submodels with preliminary data and then the user can estimate the critical value P that produces the best performance ofthe heuristic.

It is well known that the EOQ model is not highly sensitive to variations in the lot size. Therefore, we can establish the following empirical rule. Whenever [Q.sub.0](1 + [alpha]) is greater or equal than [Q.sup.*], then it is better to send a single shipment of optimal size [Q.sup.*]. Therefore, the critical value P can be calculated as

P = [Q.sub.0]/[Q.sup.*] = [Q.sub.0]/[[Q.sub.0]/(1-[alpha])] = 1/(1-[alpha]) (15)

3. Application of the heuristic method

3.1 Data

The current policy of the company is to make a single shipment at the beginning of the season, based on a simple empirical rule, i.e., the shipment amount is 125% of the total expected demand.

To compare the behavior of the heuristic method with the inventory policies of the company, we used the data shown in Table 1. These data correspond to the sales of a SLCP in 34 retailers during the 2006 season, which lasted for 12 weeks approximately. Figure 2 illustrates the behavior of the sales at six randomly selected retailers. These data are representative because they reflect different life cycle patterns of products with different sales volume over the season. We used sales instead of demand data because the latter are extremely difficult to obtain from self-service retailers.

[FIGURA 1 OMITIR]

[FIGURA 2 OMITIR]

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (16)

3.2 Model parameters

A common practice of system and market analysts and demand planners is the direct parameter estimation, commonly based on information from other SLCPs because these products are almost always new and their demand information is very limited. In order to emulate the forecasting system of the company, we applied the STATFIT[R] software to obtain estimations of the pessimistic, optimistic, and most likely demands from the data in Table 1. Our results yielded a = 143 units, b = 189 units, and c = 311 units. In addition, we used the following parameters: p = 45$ / unit, s = 10$ / unit, Ao = 20$ / shipment, L = 0, and R = 1 week. Implementing changes in these values is straightforward.

To forecast weekly demands and the total demand of the season at each retailer, as required by the multiple-shipment model, we applied a doublelinear regression model of cumulative demands of 50 products with similar life cycle for the 2005 season. Eq. (16) was applied for that purpose.

where N is the number of periods of the season, [X.sub.j], is the cumulative sales up to period j, [X.sub.m] is the forecast of cumulative demand up to period m [[beta].sub.0] is the estimator of the independent term in the double-linear regression model, [[beta].sub.1] is the estimator of the coefficient of cumulative demand up to period j in the double-linear regression model, [[beta].sub.2] is the estimator of the coefficient of cumulative demand up to period j-1 in the double-linear regression model.

Eq. (16) indicates that the regression model must be applied for m =N and for m =j + 1, because it is necessary to estimate the cumulative demand from period j to the end of the sales season and the demand of the next period.

This process is to be carried out from period j = 2 to period j = N-1. In Table 2, we show the value of the estimators and the typical regression errors that we obtained. The typical regression errors were used in the multiple-shipment model as estimators ofthe typical forecast errors.

3.3 Computational results

We applied the heuristic method to the data of the 34 retailers shown in Table 1, and randomly selected six retailers to present our results.

Figure 3 illustrates the behavior of the inventory level at each selected retailer and Figure 4 shows the inventory levels when we forced the selected retailers to have a single shipment of optimal size during the season. From the two figures, it is clear that, when we applied the heuristic, the inventory level at the six chosen retailers, for most periods, was significantly less than that for the single-shipment approach. This fact has a positive effect on the total relevant cost, the level of returns, and the level ofshortages, as will be noted below.

In addition, Figure 5 exhibits the distribution of shipments through the sales season for the six selected retailers after applying the heuristic. It is important to note that the shipments were distributedthrough the season in all cases. We did not observe a constant pattern of shipments for different retailers; this behavior suggests that the management of the model and its specific application to each retailer depend on the retailer, according to its particular demand through the season.

Table 3 shows the expected total relevant cost and the expected returns and shortages estimated for the current policy of the company, and contrasts them with those obtained from the application of the single-shipment submodel and of the heuristic to the shipment planning at the 34 retailers. The heuristic achieved a total relevant cost that is 52.7% less than that of the current system and 36.1% less than the cost obtained by applying the single-shipment submodel. These cost decreases are mainly caused by the decrease in returns yielded by the heuristic (63.3% with respect to the current situation and 26.5 regardingthe single-shipment model).

[FIGURA 3 OMITIR]

It is important to note that the current policy reaches the least percentage of shortages with respect to the total amount shipped (0.51%), but at the highest total relevant cost, total amount shipped, and returns. Under the single-shipment scenario, the shortage percentage is 2.6%, whereas the percentage reached by the heuristic is 0.59 %,which is a satisfactory result.

4. Discussion

In Table 3, we illustrated the advantages of applying the designed heuristic for the inventory control of SLCPs. Now, we analyze the sensitivity of these results caused by different fluctuations in product price, salvage value, and fixed shipping costs.

[FIGURA 4 OMITIR]

[FIGURA 5 OMITIR]

The results are shown in Table 4, where the highlighted cells represent the least total cost for each combination ofparameters. It is important to note that, in this case, by defining a threshold value P = 0.87 (representing an a value of 0.15), the heuristics determined whether the single-shipment submodel or the multiple-shipment submodel would produce the best results. This means that even if the company were to merely apply the newsvendor model to optimize a single shipment at the beginning of the season, the heuristic would produce at least equal or, for most cases, better results. Additionally, from Table 4 it is clear that the fixed shipping cost [A.sub.0] may have a significant impact on the best shipment approach and on the minimum total relevant cost.

Table 5 summarizes the response of the heuristic to extreme changes in the costs of overstock and understock. In this analysis, the sale price p has been kept constant. The changes in the costs of overstock and understock were obtained by varying the salvage value s or the unit variable cost v. The blank spaces in the table correspond to infeasible combinations of p, v, and s, since the latter would take on negative values.

As expected, when the costs of overstock or understock are negligible, the best is to apply the single-shipment approach. In this case, the critical value P = 0.87 is not a good predictor of the best submodel. Therefore, although it is not easy to fmd these cases in practice, if they occur, the single-shipment approach is highly recommended.

5. Conclusions

In this paper, we develop a heuristic method for the inventory control of products with a short life cycle in a one-warehouse N-retailer supply chain under a VMI environment. The heuristic combines single and multiple shipment approaches for the sales season. The developed models search for the best profit for the entire supply chain rather than for the particular interest ofsome ofthe links ofthe chain.

In the literature, we could not identify an inventory management model for SLCPs that considers fixed shipping costs under a multipleshipment scenario. However, taking the fixed shipping cost into account is important as is shown in this work. The model exhibits a good behavior when the cost of shortages and returns are not negligible, which is the case in most real cases.

The heuristic provides the shipment plan through the season, automating the decisions of ' when to ship' and' in what quantities'. Moreover, it allows a better inventory allocation through the supply chain, thus reducing total system inventory and the risk of obsolescence. By applying the heuristic in a textbook publishing company with 34 retailers, we could significantly reduce the total relevant cost, the amount shipped, the shortages and the excess inventory, which is precisely one of the most complex problems of the inventory control of SLCPs. Implementing the heuristic on electronic sheets is relatively simple, which can be very useful for the inventory management of many SLCPs.

6. Appendices

6.1 Appendix A

The probability density function of the triangular distribution (a < b < c) is:

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (A1)

Since

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (A2)

the cumulative distribution function is given by:

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (A3)

When solving the integrals in Eq. (A3), including the intervals where the density function is not defined, we obtain:

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (A4)

This result may be used to get the value ofx from a given cumulative probability R such that R = P(X = x). Finding x, we nnally get

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (A5)

6.2 Appendix B

Consider the triangular probability density function described in Appendix A above. To determine the expected returns and the expected shortages, two cases must be taken into account:

6.2.1 Case Q < b

Since Q is less than the most likely value of demand, the expected returns are given by:

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (B1)

By substituting the function on this interval, we get:

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (B2)

which is equivalent to:

E(Q - x) = [(Q - a).sup.3]/3(c - a)(b -a) (B3)

To determine the expected value of shortages, we may consider:

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (B4)

Now,

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (B5)

For the triangular distribution

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (B6)

and from Eqs. (B1) and (B3), we get to the final result:

E(x - Q) = (a + b +c)/3 - Q + [(Q -a).sup.3]/3(c - a)(b- a) (B7)

6.2.2 Case Q [??] b

Here, we follow a similar procedure as in the previous case, that is

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (B8)

By substitution, we get:

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (B9)

That is, the expected shortages are:

E(x - Q) = [(c - Q).sup.3]/3(c - b)(c - a) (B10)

Now, following a similar procedure, the expected returns wil l then be:

E(Q - x) = [(c - Q).sup.3]/3(c - b)(c - a) + Q - (a + b + c)/3 (B11)

The expectedprofit can be calculated as the sum of the revenues from the units we expect to sell, plus the revenue from the units returned at the end of the season, minus the cost of the purchased units, minus the fixed shipping cost. This is equivalent to:

E(U) = [E(x) - E(x -Q)]p + E(Q - x) s - vQ - [A.sub.0] (B12)

By substituting the above expressions for E(x - Q) and E (Q-x) in Eqs. (B7) and (B11), respectively, into Eq. (B 12), the expected profit (given the shipment size Q) is given by the following equation, which corresponds to Eq. (4):

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (B13)

6.3 Appendix C

Given a random demand x with normal probability distribution, with standard deviation 6 and expected value x for a given level of available inventory I, the expected number of shortages is given by (Thomopoulos,1980):

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (C1)

where f (*) is the normal probability density function, [G.sub.z] (*) is the unit normal loss function, and kis the standard normal variable, such that

k = 1 - [bar.x]/[sigma] (C2)

Similarly, the expected number of returns is given by:

E(1 - ) = [sigma][G.sub.z](-k) (C3)

The expected profit, given an available inventory I can be calculated as follows:

E(U|I) = [E(x) - E(x - 1)]p + E(1 - x)s - vl - [A.sub.0] (C4)

where p, s, v, and [A.sub.0] have been already defined in Section 2.3.

By substituting Eqs. (C1)-(C3) into Eq. (C4) be obtain Eq.(CS) which matches Eq.(12) with its corresponding variables:

[EXPRESION MATEMATICA IRREPRODUCIBLE EN ASCII.] (C5)

(Recibido: Septiembre 15 de 2008--Aceptado: Mayo 13 de 2009)

7. Bibliographic references

Benjaafar, S., Cooper, W. L., & Kim, J.-S. (2005). On the benefits of pooling in productioninventory systems. Management Science 51 (4), 548-565.

Chen, C.-K., Hung, T.-W., & Weng, T.-C. (2007). A net present value approach in developing optimal replenishment policies for a product life cycle. Applied Mathematics and Computation 184 (2), 360-373.

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Jesus A. Rodriguez *, Carlos J. Vidal ** [seccion]

* Universidad del Tlalle, Buga, Colombia

** Escuela de Ingenieria Industrial y Estadistica, Universidad del Valle, Cali, Colombia [seccion] e-mail: cjvidal@univalle.edu.co
Table 1. Weekly sales(in units) in 34 retailers during the 2006 season.

                              WEEK

Retailer         1     2       3       4       5     6     7

            1    3    21      51      70      76    28    33
            2    3    15      54      61      67    37    21
            3    1    24      45      63      44    33    23
            4    1    22      52      54      61    17    23
            5    4    20      37      59      37    23    29
            6    0    26      66      43      46    23    35
            7    4    21      27      73      59    24    18
            8    2    28      41      72      59    24     3
            9    7    45      18      19      68    31    14
           10    7    52      62      47      31    22     5
           11    1    19      33      38      40    29    34
           12    3    22      40      55      42    20    18
           13    2    24      38      71      52    15     5
           14    2    15      43      59      59    12    16
           15    2    27      34      38      51    26    21
           16    2    37      29      27      44    25    29
           17    0    10      17      31      19     4    28
           18    5    30      63      77      23     3     1
           19    1    18      44      53      57     7    13
           20    3    27      38      50      38    19    16
           21    7    22      35      42      53    18    14
           22    0    23      37      60      39    17    17
           23    5    25      47      69      10     7    18
           24    1    18      38      41      47    20    15
           25    1    17      29      50      41    18    16
           26    0     0       0       0      64    62    25
           27    3    30      15      11      45    31    23
           28    2    19      41      51      37    13    12
           29    3    20      27      46      37    14    15
           30    2    24      48      42      23    11     9
           31    0     6      12      13      23     5    26
           32    0    16      26      27      41    18    15
           33    3    10      23      11      33    13    26
           34    0     3       6      18      10    10    11
TOTAL           80   736   1,216   1,541   1,476   679   627

                              WEEK

Retailer           8     9    10   11   12   TOTAL

            1      9     7     4    0    0   302
            2     13     1     1    1    2   276
            3     14     3     3    2    1   256
            4     12     2     7    3    0   254
            5     18    13     8    4    1   253
            6      8     3     0    0    0   250
            7     13     4     2    2    2   249
            8      5     2     1    0    1   238
            9     15     3     8    3    0   231
           10      3     3     0    0    0   232
           11     22     6     1    2    1   226
           12     11     6     5    2    0   224
           13     11     1     1    2    0   222
           14      7     5     1    0    2   221
           15      3     6     2    5    3   218
           16      6    11     7    1    1   219
           17     64    35     8    0    2   218
           18      6     5     0    1    0   214
           19      3     1     4    2    1   204
           20      5     3     0    0    2   201
           21      6     4     0    0    0   201
           22      3     2     0    0    1   199
           23      9     2     1    2    1   196
           24      5     4     2    0    0   191
           25      8     3     3    2    1   189
           26     21    11     4    2    0   189
           27     16     5     7    3    0   189
           28      6     3     4    1    0   189
           29     10     3     2    4    0   181
           30      4     3     2    1    1   170
           31     49    22     4    1    0   161
           32      8     8     2    1    0   162
           33     16     9     6    7    1   158
           34     49    40     7    0    0   154
TOTAL           8458   239   107   54   24

Table 2. Parameters of the regression model based on sales of
products with similar life cycle.

       Regression of sales for the whole season

       Typical                Parameter
       regression
       error        [[beta].   [[beta].   [[beta].
Week   (units)      sub.0]     sub.1]     sub.2]

1      124.57       193.13       6.15
2       83.45        61.98     -10.51      6.61
3       80.33        62.81       2.79      1.59
4       52.62        -4.45      -5.80      5.68
5       20.95         8.13      -2.08      2.76
6        8.52        -1.68      -0.89      1.82
7        4.77        -0.78      -0.69      1.70
8        2.33         1.40      -0.27      1.27
9        1.55         0.64      -0.28      1.28
10       1.09        -0.04      -0.26      1.26
11       0.46         0.19      -0.07      1.07

       Regression of sales for the next period

       Typical               Parameters
       regression
       error        [[beta].   [[beta].   [[beta].
Week   (units)      sub.0]     sub.0]     sub.0]

1      14.11        19.84       2.52
2      15.36        -3.95       0.28       2.34
3      10.47        11.26       0.61       1.26
4      16.73        -5.36      -2.21       2.88
5      11.65         5.42      -1.10       1.98
6       5.74        -0.54      -0.46       1.44
7       3.64        -1.76      -0.51       1.52
8       1.49         0.51      -0.19       1.19
9       0.93         0.53      -0.12       1.11
10      0.93        -0.2       -0.23       1.23
11      0.46         0.19      -0.07       1.07

Table 3. Comparison between the heuristic method application, the
current situation, and the single-shipment scenario.

                       Total           Total
Scenario               relevant cost   amount    Returns   Shortages
                       ($)             shipped   (units)   (units)
                                       (units)

Current situation      20,360          9,044     1,853        46

A single shipment of
optimal size           15,080          7,956       925       206

Applying the            9,630          7,841       680        46
heuristic method

Table 4. Response of total relevant cost ($) for the optimal single
shipment and for the heuristic method to variations in price p,
salvage value s, and fixed shipping cost Ao.

                                             Fixed shipping cost
  p       s      Q *   Scenario                 [A.sub.0] $)
($/u)   ($/u)   (u)                        0        30       60

                       Single-shipment     11,335   12,355   13,375
                       Heuristic            7,985    8,975   12,445
        15      266    [Q.sub.0] (units)      143      199      223
65                     [Q.sub.0]/Q *        0.538    0.748    0.838
                       Single-shipment     23,355   24,375   25,395
         5      239    Heuristic           14,115   14,970   21,030
                       [Q.sub.0] (units)      143      174      187
                       [Q.sub.0]/Q *        0.598    0.728    0.782
                       Single-shipment      9,105   10,125   11,145
        15      253    Heuristic            5,965    8,165   11,145
                       [Q.sub.0] (units)      143      198      221
45                     [Q.sub.0]/Q *        0.565    0.783    0.874
                       Singl-shipment      17,775   18,795   19,815
         5      223    Heuristic           11,060   13,510   18,175
                       [Q.sub.0] (units)      143      173      185
                       [Q.sub.0]/Q *        0.641    0.776    0.830
                       Single-shipment      4,745    5,765    6,785
        15      210    Heuristic            2,920    5,765    6,785
                       [Q.sub.0] (units)      143      193      214
25                     [Q.sub.0]/Q*         0.681    0.919    1.019
                       Single-shipment      7,115    8,135    9,155
         5      187    Heuristic            5,615    8,135    9,155
                       [Q.sub.0] (units)      143      170      182
                       [Q.sub.0]/Q *        0.765    0.909    0.973

                                              Fixed shipping cost
  p       s      Q *   Scenario                  [A.sub.0] ($)
($/u)   ($/u)   (u)                        90      120      150

                       Single-shipment     14,395   15,415   16,435
                       Heuristic           14,395   15,415   16,435
        15      266    [Q.sub.0] (units)      241      256      269
65                     [Q.sub.0]/Q *        0.906    0.962    1,011
                       Single-shipment     26,415   27,435   28,455
         5      239    Heuristic           23,145   27,060   28,455
                       [Q.sub.0] (units)      197      205      212
                       [Q.sub.0]/Q *        0.824    0.858    0.887
                       Single-shipment     12,165   13,185   14,205
        15      253    Heuristic           12,165   13,185   14,205
                       [Q.sub.0] (units)      238      253      266
45                     [Q.sub.0]/Q *        0.941    1.000    1.051
                       Singl-shipment      20,835   21,855   22,875
         5      223    Heuristic           20,835   21,855   22,875
                       [Q.sub.0] (units)      195      203      210
                       [Q.sub.0]/Q *        0.874    0.910    0.942
                       Single-shipment      7,805    8,825    9,845
        15      210    Heuristic            7,805    8,825    9,845
                       [Q.sub.0] (units)      230      243      255
25                     [Q.sub.0]/Q*         1.095    1.157    1.214
                       Single-shipment     10,175   11,195   12,215
         5      187    Heuristic           10,175   11,195   12,215
                       [Q.sub.0] (units)      190      198      204
                       [Q.sub.0]/Q *        1.016    1.059    1.091

Table 5. Response of total relevant cost ($) and [Q.sub.0]/Q *
ratio for a single shipment and for multiple shipments to variations
in the costs of overstock ([C.sub.E) and understock ([C.sub.F]).

Cost of                          Cost of understock [C.sub.F] ($)
overstock   Scenario
CE ($)                          1       15       30       44

            Single-shipment     1,629    3,241    3,660   3,843
 1          Multiple-shipment   2,384    3,527    3,632   3,814
            [Q.sub.0]/Q *       0.829    0.633    0.611   0.600
            Single-shipment     2,707   14,915   20,150
15          Multiple-shipment   4,592   12,895   12,690
            [Q.sub.0]/Q *       1.055    0.829    0.763
            Single-shipment     2,697   19,340
30          Multiple-shipment   6,826   17,245
            [Q.sub.0]/Q *       1.094    0.897
            Single-shipment     2,703
44          Multiple-shipment   8,644
            [Q.sub.0]/Q *       1.115
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Title Annotation:INGENIERIA INDUSTRIAL
Author:Rodriguez, Jesus A.; Vidal, Carlos J.
Publication:Ingenieria y Competividad
Article Type:Report
Date:Jun 1, 2009
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