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An economic lot-size model with non-linear holding cost hinging on time and quantity
Int. J. Production Economics 145 (2013) 294–303
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
An economic lot-size model with non-linear holding cost hinging on time and quantity Valentín Pando a, Luis A. San-José b,n, Juan García-Laguna a, Joaquín Sicilia c a
Departamento de Estadística e Investigación Operativa, Universidad de Valladolid, Valladolid, Spain IMUVA, Instituto de Matemáticas, Universidad de Valladolid, Paseo de Belén 15, 47011 Valladolid, Spain c Departamento de Estadística, Investigación Operativa y Computación, Universidad de La Laguna, Tenerife, Spain b
art ic l e i nf o
a b s t r a c t
Article history: Received 25 May 2012 Accepted 26 April 2013 Available online 21 May 2013
This paper develops an economic lot size inventory model where the demand rate depends on the stock level and the cumulative holding cost is non-linear on both the quantity and the time they are stored. More concretely, it is supposed that the demand rate is a concave potential function of the inventory level and the holding cost is potential on both time and quantity. Moreover, shortages are not allowed. A general procedure to determine the optimal lot size and the maximum inventory profit is developed. Also, some results about the profitability of the inventory system are presented. This work extends several inventory models previously considered in the literature. Finally, numerical examples, which help us to understand the theoretical results, are also given. & 2013 Elsevier B.V. All rights reserved.
Keywords: Inventory management Lot-size models Stock-dependent and time-dependent holding cost Stock-dependent demand rate
1. Introduction Since Harris presented his EOQ model in 1913, many efforts have been made to adjust their assumptions to more realistic situations in inventory management. As is known, in order to obtain a simple mathematical model, Harris assumed that shortage was not allowed and both demand rate and holding costs per unit and per unit time were constant. Thus, the hypotheses of the classic model ensured that the incomes were independent of the stock level and, therefore, the optimal lot size was obtained by minimizing the sum of the carrying cost and the ordering cost per unit of time. However, as is also well-known, empirical evidence shows that these hypotheses are an excessive simplification of reality. So, many models developed later worked with the idea of relaxing these assumptions. In what follows, we will focus on inventory models which maintain the assumption of not allowing shortage, but the other hypotheses have been revised; that is, we do not consider that both demand rate and holding cost per unit and per unit of time are constant. Over time, some practitioners and researchers have found that an increase in exposed merchandise may bring about increased sales of some items. So, Wolfe (1968) presented empirical evidence of this relationship, showing that the sales of style merchandise, such as women's dresses or sports clothes, are proportional to the amount of displayed inventory. After this observation, mathematical
n
Corresponding author. E-mail addresses: [email protected] (V. Pando), [email protected] (L.A. San-José), [email protected] (J. García-Laguna), [email protected] (J. Sicilia). 0925-5273/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2013.04.050
models for inventory systems captured this idea by regarding the use of stock level dependent demand rate in their formulation. As a starting point, Baker and Urban (1988) defined a model in which the demand rate was supposed to be a concave potential function of the inventory level. This new approach to the problem brought about the loss of independence between revenues and total inventory costs, because a larger order size results not only in higher inventory costs but also in higher revenues. Therefore, in this situation, besides the ordering and holding costs, it becomes necessary to consider a third component for the inventory system: the gross profit from the sale of the item (the difference between the selling price and the purchasing cost). Moreover, in this case, the principal objective should be to maximize the profit of the inventory system per unit time instead of minimizing the total inventory cost per unit time. For this reason, Baker and Urban (1988) raised the issue from the perspective of maximizing the profit per unit time. Since then, many papers have appeared considering stock dependent demand rate, but some approach the problem from the perspective of minimizing the total inventory cost per unit time. Urban (2005) provides a detailed overview of the related literature published up to that date. Dye and Ouyang (2005) proposed an inventory model with a demand rate linearly dependent on the stock level and shortages with a time-proportional backlogging rate, which is an extension of the model developed by Padmanabhan and Vrat (1995). Teng and Chang (2005) considered an EPQ model with maximizing profits where the demand rate is simultaneously dependent on the stock level and the selling price but with a ceiling. Also, You (2005) and Roy (2008) analyzed EOQ models with price dependent demand.
V. Pando et al. / Int. J. Production Economics 145 (2013) 294–303
Another issue presented by inventory managers was that, sometimes, the holding cost per unit of item is not proportional to the time held in stock, and/or the holding cost per unit time is also not proportional to the amount held in inventory. This question raised the need to consider other structures for the holding cost in the mathematical models of inventory systems. Under the hypothesis of a constant demand rate, Naddor (1982) proposed three inventory models based on the holding cost function per cycle for those situations. The first model considered that the holding cost per cycle was non-linear with time, but linear with the quantity of items (the perishable-goods system), the second one assumed that the holding cost per cycle was nonlinear with the quantity of items, but linear with time (the expensive-storage system) and the third model supposed that the holding cost per cycle was non-linear with both time and quantity of items (the general system). In this environment, Weiss (1982) introduced a deterministic model with constant demand rate, supposing that the holding cost per unit of product was a convex potential function of the time in stock. This situation can occur, for example, in the storage of perishable items such as food products. In this case, the longer the products are kept in storage, the more sophisticated the storage facilities and services are needed, and therefore, the higher the holding cost. In fact, this model proved to be equivalent to the perishable-goods system cited above, as noted by Ferguson et al. (2007). Later on, Goh (1994) generalized this situation to the case of stock-dependent demand rate, minimizing the total inventory cost per unit time. Alfares (2007) considered this same situation with two types of discontinuous step functions. Urban (2008) extended Alfares' work by using a profit maximization objective. Recently, Pando et al. (2012a) have studied the previous model of Goh (1994), but from the perspective of maximizing the profit. Inventory models with holding cost rate per unit time nonlinear in the stock level are scarcer. Nevertheless, this situation can be encountered in real inventories when the value of the item is very high and many precautionary steps are to be taken to ensure its safety and quality. This occurs, for example, when carrying luxury items like expensive jewelry and designer watches. From the perspective of minimizing the total inventory cost per unit of time and using a stock-dependent demand rate, Goh (1994) supposed that the holding cost rate per unit of time was a convex potential function of the quantity of items held in stock. Later on, Giri and Chaudhuri (1998) extended this model to the case of deteriorating items with a small constant fraction of deterioration. Berman and Perry (2006) also worked with stock-dependent holding cost and two types of demand rate functions. Scarpello and Ritelli (2008) gave some theoretical results for EOQ models when the holding cost grows with the stock level. Recently, Pando et al. (2012b) have analyzed one of the models of Goh (1994) from the perspective of maximizing the profit. To the best of our knowledge, the work of Naddor (1982) is the only paper dedicated to study inventory models for the situation in which the holding cost is non-linear with both time and quantity. This structure of the holding cost function is adequate for the representation of some real life situations. This occurs, for example, in the storage of very expensive products (then many precautionary steps are to be taken to ensure their safety) which deteriorates over time (so more sophisticated storage facilities and services are needed). However, in the work of Naddor (1982), the demand rate was considered constant. According to the previous comments, the objective of this paper is to study an inventory system with stock dependent demand rate and a structure of holding cost non-linear in both time and stock level from the perspective of maximizing profits per unit time. Thus, the main difference between our model and the one in Naddor (1982) is the assumption about the demand
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rate. So, instead of a constant demand rate, we consider the case where the demand rate is a concave potential function of the inventory level. Moreover, we develop the necessary theory to calculate the holding cost per cycle when it depends non-linearly on both time and stock level. The structure of this paper is as follows. Section 2 presents the assumptions and notation to be used throughout the paper. Section 3 deals with the mathematical formulation of the inventory model, the procedure to obtain the optimal policy and the main theoretical results. In Section 4, we provide some numerical examples to illustrate the theoretical results and show the difference between the optimal solutions for the problems of maximum profit and minimum cost. Finally, conclusions are drawn in Section 5.
2. Assumptions and notation The inventory system studied in this paper is developed on the basis of the following assumptions. The inventory is continuously reviewed and the planning horizon is infinite. The item is a single product and no shortage is allowed. The replenishment is instantaneous when the stock is depleted and the lead time is zero. The demand rate is a known function of the inventory level. The order cost is fixed, regardless of the lot size. The unit purchasing cost and the selling price are known and constant. The cumulative holding cost for x units of product that have been stored during t units of time is a potential function of t and x. Thus, this function may not be linear in any of these two variables. Table 1 summarizes the notation used in this paper. We suppose that the demand rate is described by the potential function of the inventory level DðtÞ ¼ λ½IðtÞβ
ð1Þ
with λ 4 0 and 0≤β o 1. With this functional form, as the inventory level decreases, so does the demand rate. Thus, at the beginning of a cycle, the inventory level decreases rapidly because the demand is bigger at a high level of stock. At time t¼ 0 the inventory level and the demand rate are at their highest level. As more inventory is depleted, the rate of decrease of the stock level slows down. The elasticity of the demand rate with respect to the stock level β represents the relative change in demand rate with respect to the corresponding relative change in the stock level (that is, β ¼ ð∂DðtÞ=∂IðtÞÞ=ðDðtÞ=IðtÞÞ). Note that, if the demand is inelastic (that is, β ¼ 0), we have DðtÞ ¼ λ and the model reverts to the wellknown inventory model with constant demand rate. Furthermore, it is supposed that the cumulative holding cost for x items stored during t units of time is given by the following Table 1 List of notation. q T t I(t) K p s Hðt; xÞ h γ1 γ2 D(t) λ β α ξ
Order quantity or lot size per cycle ( 40, decision variable) Length of the inventory cycle ð4 0Þ Time spent in inventory ð≤TÞ Inventory level at time t ð≤qÞ Ordering cost per order ð 4 0Þ Unit purchasing cost ð4 0Þ Unit selling price ð≥pÞ Cumulative holding cost for x items stored during t units of time ð 4 0Þ Scaling constant for holding cost ð4 0Þ Elasticity of the holding cost with respect to time ð≥1Þ Elasticity of the holding cost with respect to the stock level ð≥1Þ Demand rate at time t Scaling constant of demand rate ð 4 0Þ Elasticity of demand rate with respect to the stock level ð0≤β o 1Þ Auxiliary parameter, α ¼ 1−β ð0 o α≤1Þ Auxiliary parameter, ξ ¼ αγ 1 þ γ 2 ð 41Þ
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equation:
potential function of t and x: γ1 γ2
Hðt; xÞ ¼ ht x
ð2Þ
where h 40, γ 1 ≥1 and γ 2 ≥1. These three parameters can be estimated via a non-linear regression or a multiple linear regression between the logarithm of the cumulative holding cost and the logarithms of time and quantity. Appendix A provides a numerical example where this function is shown to be useful for evaluating the holding cost in an inventory model. Note that h ¼ Hð1; 1Þ, that is, the holding cost of one unit held in stock during one unit of time is the scaling parameter for the holding cost. The elasticity of the holding cost with respect to time γ 1 represents the relative change in this cost with respect to the relative change in time (that is, γ 1 ¼ ð∂H=∂tÞ=ðH=tÞ). Similarly γ 2 represents the relative change in this cost with respect to the relative change in the quantity (that is, γ 2 ¼ ð∂H=∂xÞ=ðH=xÞ). Also, if γ 1 ¼ 1 in formula (2), we obtain the same cumulative holding cost function considered by, among others, Naddor (1982, expensive-storage system), Berman and Perry (2006) and Pando et al. (2012b). Similarly, if γ 2 ¼ 1 in (2), we have the cumulative holding cost of the model by Naddor (1982, perishable-goods system), which was later used by Ferguson et al. (2007) and Pando et al. (2012a). We assume the general condition ðs≥pÞ, instead of s 4 p, because this assumption allows several inventory models presented by other authors to be included in our study. We refer to some models that have been developed from the perspective of minimizing the average cost, without considering the revenues and purchasing costs. Formally, they can be obtained from the model developed here by taking s ¼p and, in this article, we refer to them as inventory models of minimum cost. However, in the model studied here, our objective is to maximize the average profit and, therefore, the revenues and purchasing costs have to be included. Hence, we will name this class of models as inventory models of maximum profit. Consequently, those inventory models of minimum cost which do not include revenues or purchasing costs analyze only a part of the systems associated to the models of maximum profit; that is, those systems are subsystems of the system considered here. In a few inventory models (for example, in the EOQ models studied by Harris, 1913 and Naddor, 1982), resolving the first situation is equivalent to solving the second because the demand rate is constant, but this does not always occur. In fact, this last case is what occurs in the model studied here. More specifically, if we solve a model of minimum cost, as in Naddor (1982), and the revenues and the purchasing costs are added later on, then the profit obtained is smaller than or, eventually, equal to the profit obtained when the revenues and the purchasing costs are included in the formulation of the model, that is, if we consider the model of maximum profit. In this paper, as in the model of Naddor (1982, general system), a lot-size inventory model with non-linear holding cost in both time and stock level is studied, but here we are considering the more general situation with a stock-level dependent demand rate and the problem is handled from the proper perspective of maximizing profit per unit time. Therefore, taking into account the previous comments, we present a model which generalizes the models studied by Goh (1994). The following section includes the mathematical formulation of the model and the solution procedure to obtain the optimal policy.
dIðtÞ ¼ −λ½IðtÞβ ; dt
ð3Þ
0≤t≤T
with the initial condition Ið0Þ ¼ q. The solution of (3) is IðtÞ ¼ ðqα −αλtÞ1=α
ð4Þ
where α ¼ 1−β. Taking into account that the replenishment is done when the stock is depleted, we have IðTÞ ¼ 0 and, therefore, the corresponding length of the inventory cycle T is given by T¼
qα αλ
ð5Þ
Therefore, (4) can be rewritten as 1=α t IðtÞ ¼ q 1− T
ð6Þ
Once the mathematical expression for the inventory level is obtained, the β parameter is better understood. Specifically, the cumulative demand in the second half of the inventory cycle is calculated as 1=ð1−βÞ Z T Z T 1 λ½IðtÞβ dt ¼ − dIðtÞ ¼ IðT=2Þ ¼ q 2 T=2 T=2 and the proportion of sales in this second half of the cycle with respect to the sales over the whole cycle is given by ρ ¼ ð12Þ1=ð1−βÞ . Evidently, as 0≤β o 1, it is always 0 o ρ≤ 12 which makes sense for this type of demand with higher sales in the first half of the cycle. Then, the β parameter can be estimated as a function of ρ as β ¼ 1 þ ðln 2Þ=ðln ρÞ. The realization of the inventory level is depicted in Fig. 1. With the assumptions given in the previous section, the total profit per cycle is calculated as the difference between the revenue per cycle and the sum of the ordering cost, the purchasing cost and the inventory holding cost per cycle. Obviously, the ordering cost is K, the revenue per cycle is sq and the purchase cost per cycle is pq. The holding cost per cycle can be calculated as (please see Appendix B) Z T HCðqÞ ¼ hγ 1 t γ1 −1 ½IðtÞγ 2 dt 0
Using (5) and (6), we have Z T γ1 −1 t t γ 2 =α dt HCðqÞ ¼ hγ 1 qγ 2 T γ 1 1− T T T 0 γ ¼ hγ 1 qγ 2 T γ 1 B γ 1 ; 2 þ 1 α hγ 1 B γ 1 ; γα2 þ 1 αγ 1 þγ 2 qξ ð7Þ ¼ q ¼ γ1 ðαλÞ Δ R1 where Bða; bÞ ¼ 0 za−1 ð1−zÞb−1 dz is the well-known beta function, ξ ¼ αγ 1 þ γ 2 and Δ ¼ ðαλÞγ 1 =hγ 1 Bðγ 1 ; ðγ 2 =αÞ þ 1Þ. Obviously, ξ≥1 þ α 4 1 and Δ 4 0. Therefore, the total profit per cycle is given by ðs−pÞq−K−ðqξ =ΔÞ and, by (5), the profit per unit time, or gain, is GðqÞ ¼ ½ðs−pÞq− K−ðqξ =ΔÞ=T, which can be expressed as GðqÞ ¼ αλðs−pÞq1−α −αλKq−α −
αλ ξ−α q Δ
ð8Þ
Thus, the problem consists in determining the decision variable q 4 0 such that the function G(q) given by (8) is maximized. Since GðqÞ ¼ −ðαλ=ΔÞgðqÞ where 3. The model
gðqÞ ¼ qξ−α −ðs−pÞΔq1−α þ KΔq−α
Considering the expression for the demand rate given in (1), it is clear that the stock level I(t) verifies the following differential
the optimal solution that minimizes g(q) is the same as the optimal solution that maximizes G(q).
ð9Þ
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length of the cycle time using (5). Moreover, from (8), (9) and (11) it follows that αλ n −α n ξ ðq Þ ½ðq Þ −ðs−pÞΔqn þ KΔ Δ αλ ¼ − ðqn Þ−α ½uqn þ v−ðs−pÞΔqn þ KΔ Δ
Gðqn Þ ¼ −
Substituting the values of u and v given in (12) and (13) respectively into the previous expression, the maximum average profit is given by ð1−αÞðs−pÞ αK −ðs−pÞ qn þ þK Gðqn Þ ¼ −αλðqn Þ−α ξ−α ξ−α αλ ðξ−1Þðs−pÞqn −Kξ ¼ ξ−α ðqn Þα
Fig. 1. Behavior of the inventory level over time.
Consequently, the maximum-profit problem is equivalent for solving the non-linear program MinimizefgðqÞ : q 4 0g
ð10Þ
3.1. Solution procedure The first two derivatives of the function g(q) are g′ðqÞ ¼ ðξ−αÞqξ−α−1 −ð1−αÞðs−pÞΔq−α −αKΔq−α−1 ξ−α ð1−αÞðs−pÞΔ αKΔ q− ¼ αþ1 qξ − ξ−α ξ−α q and g″ðqÞ ¼ ðξ−αÞðξ−α−1Þqξ−α−2 þ αð1−αÞðs−pÞΔq−α−1 þ αð1 þ αÞKΔq−α−2 Since g″ðqÞ 40 for q 40, it follows that g(q) is a strictly convex function; moreover, limq↓0 gðqÞ ¼ limq-∞ gðqÞ ¼ ∞. Therefore, the function g′ðqÞ has only one positive root which is the value qn where g(q) attains the global minimum, that is, the solution for the problem. Obviously, the equation g′ðqÞ ¼ 0 is equivalent to this other PðqÞ ¼ qξ −uq−v ¼ 0
ð11Þ
where u¼
ð1−αÞðs−pÞΔ ≥0 ξ−α
ð12Þ
v¼
αKΔ 40 ξ−α
ð13Þ
It is clear that if u ¼0 (that is, α ¼ 1 or s¼p) the solution is αKΔ 1=ξ qn ¼ v1=ξ ¼ ð14Þ ξ−α Therefore, it is only necessary to solve Eq. (11) when u 4 0 (that is, 0 o α o1 and s 4p). The following result answers this question Theorem 1. Let the function PðqÞ ¼ qξ −uq−v, with ξ 41, u 4 0, v 4 0 and q 40. Write qo ¼ maxfðu þ vÞ1=ξ ; ðu þ vÞ1=ðξ−1Þ g. For i ¼ 1; 2; 3; …, "consider the sequence qi ¼ qi−1 −ðPðqi−1 Þ=P′ðqi−1 ÞÞ ¼ ððξ−1Þqξi−1 þ vÞ=ðξqξ−1 i−1 −uÞ. Then P(q) has a unique" positive root qn which is the limit of the sequence fqi g∞i ¼ 0 . Proof. Please see Appendix C.
□
Using this theorem, we can numerically calculate the optimal lot-size qn with the desired precision and evaluate the optimal
ð15Þ
Although calculating the optimal lot size qn (resolving the equation PðqÞ ¼ 0) can be easily done with any solver software, we include below a simple algorithm to evaluate the solution using the Newton–Fourier method. It can be applied directly by the inventory manager without specific software. Solution algorithm. Obtaining the optimal policy and the optimal profit in the model. Step 1. Calculate Δ ¼ ðαλÞγ 1 =hγ 1 Bðγ 1 ; ðγ 2 =αÞ þ 1Þ, ξ ¼ αγ 1 þ γ 2 , u ¼ ð1−αÞðs−pÞΔ=ðξ−αÞ and v ¼ αKΔ=ðξ−αÞ. Step 2. If u ¼0 then the optimal lot-size is qn ¼ v1=ξ . Go to Step5. Step 3. If u þ v ¼ 1 then the optimal lot size is qn ¼ 1. Go to Step5. Step 4. Calculate qo ¼ maxfðu þ vÞ1=ξ ; ðu þ vÞ1=ðξ−1Þ g and r ¼ ðu=ξÞ1=ðξ−1Þ . (a) Select a positive tolerance value, say TOL 4 0, with 0 o TOL o r. (b) Calculate qi ¼ ððξ−1Þqξi−1 þ vÞ=ðξqξ−1 for i−1 −uÞ i ¼ 1; 2; 3; …; until Pðqi −TOLÞ o 0. (c) Take as optimal lot-size the value qn ¼ qi . Step 5. Compute the optimal inventory cycle T n ¼ ðqn Þα =αλ and the maximum profit Gðqn Þ ¼ ðαλ=ðξ−αÞÞ½ððξ−1Þðs−pÞqn −KξÞ=ðqn Þα Remark. In order to understand the previous algorithm and its criterion of stop, it is convenient to see the proof of Theorem 1 in Appendix C. Thus we have: (i) the function P(q) attains its minimum at point r; (ii) PðqÞ o 0 if and only if q∈ð0; qn Þ; (iii) the succession qi is strictly decreasing and r o qn o qi for all i ¼ 0; 1; 2; …; and (iv) as the tolerance TOL belongs to the interval ð0; rÞ, it follows that qi −TOL 40 for all i ¼ 0; 1; 2; …. Then, when the condition Pðqi −TOLÞ o 0 is satisfied, it follows qi −TOL o qn , that is, qi −qn o TOL. This allows us to conclude that Pðqi −TOLÞ o 0 is a suitable stopping rule. 3.2. Meaning and discussion of optimal policy An interesting question for inventory managers is to establish conditions to ensure that the inventory system will generate profits. If this happens using the optimal lot size qn , then Gðqn Þ 4 0 and it is called a profitable inventory system. However, if Gðqn Þ o 0, we say that it is a non-profitable inventory system, because, even using the optimal lot size, it generates losses. Finally, if Gðqn Þ ¼ 0 then, for the optimal situation, it matches costs and revenues and it is called a break-even inventory system. With these definitions, the inventory managers are interested in characterizing the system as profitable, non-profitable or breakeven without calculating the optimal lot size qn and the maximum profit Gðqn Þ. Obviously, if s ¼p then Gðqn Þ o0 and the system is non-profitable. The following result gives a characterization for the type of inventory system as a function of the input parameters when s 4p.
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Theorem 2. Assuming that s 4p, we have 1. The inventory is profitable if and only if s−p 4 ðξ=ðξ−1ÞÞððξ−1ÞK ξ−1 =ΔÞ1=ξ . 2. The inventory is a break-even system if and only if s−p ¼ ðξ=ðξ−1ÞÞððξ−1ÞK ξ−1 =ΔÞ1=ξ . 3. The inventory is non-profitable if and only if s−p o ðξ=ðξ−1ÞÞððξ−1ÞK ξ−1 =ΔÞ1=ξ . Proof. From (15) it is clear that Gðqn Þ 40 if and only if qn 4Kξ=ððξ−1Þðs−pÞÞ. Thus, by the properties of the function P(q) (please see Appendix C), this last inequality is equivalent to P½Kξ=ððξ−1Þðs−pÞÞ o 0. Now, as ξ Kξ Kξ KΔ ξð1−αÞ ¼ þα P − ðξ−1Þðs−pÞ ðξ−1Þðs−pÞ ξ−α ξ−1 ξ Kξ KΔ ¼ − ðξ−1Þðs−pÞ ξ−1 Fig. 2. Optimal policy for profit maximizing.
we deduce that the inventory is profitable if and only if ½Kξ=ððξ−1Þðs−pÞÞξ o KΔ=ðξ−1Þ, that is, s−p 4 ðξ=ðξ−1ÞÞððξ−1ÞK ξ−1 =ΔÞ1=ξ , which is assertion 1. The same reasoning applied to break-even and non-profitable systems proves assertions 2 and 3. □ As a consequence of this theorem, it is clear that s ¼ p þ ðξ=ðξ−1ÞÞððξ−1ÞK ξ−1 =ΔÞ1=ξ is the minimum selling price that the inventory manager should overcome to obtain a profitable system. Another interesting issue for practitioners is to have a rule to know if a replenishment policy is the optimal one. For this purpose, the following theorem allows us to characterize the optimal policy for the model under consideration. Theorem 3. Let the functions Ψ ðqÞ ¼ ðs−pÞq (gross profit per cycle with lot size q) and HCðqÞ ¼ qξ =Δ (holding cost per cycle with lot size q). Then, the lot size q is optimal if and only if ðξ−αÞHCðqÞ ¼ αK þ ð1−αÞΨ ðqÞ
ð16Þ
Proof. If q is optimal (that is, q ¼ qn ), from (7), (11), (12) and (13), we have HCðqn Þ ¼ ðqn Þξ =Δ ¼ ðð1−αÞðs−pÞ=ðξ−αÞÞqn þ αK=ðξ−αÞ ¼ ðð1−αÞ =ðξ−αÞÞΨ ðqn Þ þ ðα=ðξ−αÞÞK. That is, ðξ−αÞHCðqn Þ ¼ αK þ ð1−αÞΨ ðqn Þ. On the other hand, if q verifies ðξ−αÞHCðqÞ ¼ αK þ ð1−αÞΨ ðqÞ, subtracting this equation from the other one with qn , we have ðξ−αÞ½ðqn Þξ =ΔÞ−ðqξ =ΔÞ ¼ ð1−αÞðs−pÞðqn −qÞ; or, equivalently, from (12), ½ðqn Þξ −qξ ¼ uðqn −qÞ. Now, as qn is the solution of Eq. (11), we have ðqn Þξ ¼ uqn þ v and thus qξ −uq−v ¼ 0. Hence q ¼ qn because P (q) has a unique positive root. □ It is possible to obtain an interpretation of the above result if we consider the gross profit per unit time instead of the gross profit per cycle. Thus, dividing both sides of the equality (16) by the optimal cycle time T n , we have ðξ−αÞ
HCðqn Þ K Ψ ðqn Þ ¼ α n þ ð1−αÞ n T T Tn
ð17Þ
where HCðqn Þ=T n , K=T n and Ψ ðqn Þ=T n are the optimal holding cost, ordering cost and gross profit per unit time, respectively. Therefore, the function G(q) attains its maximum at the cut point of the functions ðξ−αÞðHCðqÞ=TÞ and αðK=TÞ þ ð1−αÞðΨ ðqÞ=TÞ. Fig. 2 illustrates this idea by plotting the three functions. Note that this interpretation is an extension of a well-known result associated to the classical lot size model: “for the optimal order quantity, the holding cost happens to be exactly equal to the ordering cost” (see, for instance, Axsäter, 2000, p. 32), but adapted here to the model under consideration. Specifically, if α ¼ γ 1 ¼ γ 2 ¼ 1, Eq. (17) leads to HCðqn Þ=T n ¼ K=T n which is the result previously cited.
3.3. Particular models Next, let us deduce that several lot size models studied by other authors turn out to be particular cases of the model developed in this paper. 1. When β ¼ 0 (that is, α ¼ 1) and choosing the notation h ¼ c1 , γ 1 ¼ n, γ 2 ¼ m, K ¼ c3 and λ ¼ r, we obtain the general system developed by Naddor (1982). Thus, putting a ¼ γ 1 Bðγ 1 ; γ 2 þ 1Þ, we have u¼0 and v ¼ c3 r n =ac1 ðm þ n−1Þ. Applying Solution Algorithm, we obtain qn ¼ ½c3 r n =ðac1 ðm þ n−1ÞÞ1=ðmþnÞ which coincides with the optimal policy shown by Naddor. Substituting this value into (15), we see that the maximum average profit is Gðqn Þ ¼ rðs−pÞ−ðm þ nÞ½ac1 r m ðc3 =ðm þ n−1ÞÞmþn−1 1=ðmþnÞ . Of course, the last term in Gðqn Þ is the minimum cost given by Naddor (1982, p. 108). Note that this model admits the following ones as particular cases: (i) If γ 1 ¼ 1 (that is, the holding cost per cycle is non-linear in quantity of items but linear in time), we obtain the model studied by Naddor (1982, expensive-storage system) and (ii) if γ 2 ¼ 1 (that is, the holding cost per cycle is non-linear in time but linear in quantity of items), then we revert to the model analyzed by Naddor (1982, perishable-goods system) which has also been studied by Weiss (1982). 2. If we suppose that γ 2 ¼ 1, we revert to the model developed by Pando et al. (2012a). Let us remember that, in this case, various particular models can be deduced: (i) If we set s¼ p, we obtain the model developed by Goh (1994, model A) and (ii) if we suppose that s 4 p and γ 1 ¼ γ 2 ¼ 1, we revert to the model proposed by Baker and Urban (1988) when the order point is zero. 3. If γ 1 ¼ 1, then the problem is equivalent to the one studied by Pando et al. (2012b). Thus, our optimal policy qn is equal to the one shown by them. Also, if we further assume that s ¼p, we obtain the model analyzed by Goh (1994, model B).
4. Computational results To illustrate the proposed model and its associated optimal policies, in this section, some numerical examples are presented. We consider the same numerical example proposed by Pando et al. (2012a), but adding values for the parameter which were not considered in their model. Therefore, we assume the following input parameters for the inventory system: λ ¼ 1, K ¼10, h¼0.5,
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s¼ 62, p¼ 50 and β∈f0; 0:1; 0:3; 0:5; 0:7; 0:9g. Moreover, to observe the effects of the parameters γ 1 and γ 2 on the optimal lot-size and the maximum profit, we suppose that both parameters belong to f1; 1:5; 2; 2:5g. The computed results are reported in Table 2. Note that, in all considered cases, the inventory system we obtain is profitable. Moreover, we find that, in these numerical examples, the smaller optimal profit is obtained when β ¼ 0:9, γ 1 ¼ 1 and γ 2 ¼ 2:5. In this case, we have s−p ¼ 12; which is greater than ðξ=ðξ−1ÞÞððξ−1ÞK ξ−1 =ΔÞ1=ξ ¼ 4:260; as Gðqn Þ ¼ 3:60, the inventory system is profitable, which agrees with part 1 of Theorem 2. Also, note that when γ 2 4 γ 1 we obtain a smaller optimal lot size and maximum average profit than if we exchange the values of the
299
parameters γ 1 and γ 2 . This is due to the fact that, in the first case, a bigger holding cost is generated. Moreover, in Fig. 3, we plot the variation of the optimal lot size and the maximum profit as functions of γ 1 and γ 2 when β ¼ 0:3. From these graphs, we can extract the following conclusions: 1. Fixed γ 2 , if the value of γ 1 is increasing, then three cases are possible (please see Fig. 3(1) and (2): (a) the optimal lot size qn and the optimal profit Gðqn Þ are strictly decreasing (this occurs, for example, when γ 2 ¼ 2), (b) the optimal profit and the optimal lot size are firstly strictly increasing and, later on, they decrease slightly (this occurs, e.g., when γ 2 ¼ 10) and (c) the
Table 2 Optimal values qn and Gðqn Þ ¼ Gn for different parameters β, γ 1 and γ 2 respectively. β
0
0.1
0.3
0.5
0.7
0.9
γ1
γ2
qn
Gn
qn
Gn
qn
Gn
qn
Gn
qn
Gn
qn
Gn
1 1 1 1 1.5 1.5 1.5 1.5 2 2 2 2 2.5 2.5 2.5 2.5
1 1.5 2 2.5 1 1.5 2 2.5 1 1.5 2 2.5 1 1.5 2 2.5
6.32 4.07 3.11 2.59 4.07 3.24 2.76 2.45 3.11 2.76 2.51 2.33 2.59 2.45 2.33 2.22
8.84 7.90 7.17 6.60 7.90 7.37 6.93 6.56 7.17 6.93 6.70 6.48 6.60 6.56 6.48 6.38
10.0 5.02 3.52 2.81 5.21 3.77 3.07 2.65 3.65 3.10 2.75 2.50 2.90 2.69 2.51 2.37
10.1 8.48 7.43 6.67 8.56 7.72 7.10 6.62 7.51 7.13 6.79 6.50 6.72 6.64 6.51 6.36
38.4 8.47 4.68 3.37 10.7 5.58 3.95 3.17 5.75 4.20 3.42 2.96 3.94 3.41 3.04 2.77
16.6 10.5 8.13 6.82 11.3 8.89 7.59 6.74 8.78 7.78 7.08 6.55 7.22 6.92 6.61 6.32
326 16.8 6.59 4.16 37.6 9.77 5.48 3.94 12.8 6.59 4.61 3.67 6.68 4.87 3.94 3.39
53.7 14.8 9.11 6.84 20.9 11.3 8.31 6.77 12.4 9.11 7.51 6.53 8.59 7.52 6.77 3.68
3 104 43.2 10.0 5.29 591 23.3 8.53 5.19 76.6 13.9 7.09 4.88 21.4 8.98 5.85 4.49
1440 25.6 10.0 6.29 122 17.4 9.09 6.35 33.6 12.3 7.98 6.13 14.5 8.93 6.87 5.74
6 1013 187 17.2 7.05 3 109 114 16.3 7.44 4 105 64.0 14.1 7.33 5038 34.8 11.5 6.86
3 1011 52.5 7.73 3.60 7 106 35.1 7.48 3.88 2 104 21.5 6.61 3.86 560 12.5 5.48 3.63
Fig. 3. Optimal lot size and maximum profit curves in function of γ 1 and γ 2 for β ¼ 0:3.
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optimal lot size and the maximum profit are strictly increasing simultaneously (if γ 2 ¼ 30, this is what happens). 2. The same comments as before can be made about the optimal lot size and the optimal average profit when γ 1 is fixed (please see Fig. 3(3) and (4)).
Fig. 4 depict the curves of the optimal lot size and the maximum average profit when γ 2 ¼ 2. We can now establish the following comments about these results. 1. Given γ 1 ≤4, as the parameter β increases, the optimal lot size increases. For fixed γ 1 44, the optimal order quantity is a slightly decreasing function of β (please see Fig. 4(1)). 2. Fixed γ 1 and regarding the maximum profit, three cases can occur if the value of the elasticity of demand rate is increasing: ~ such that the optimal average profit (a) there is a point, say β, ~ and it decreases as β Gðqn Þ increases as β increases into ½0; βÞ ~ 1Þ (this occurs, for example, when γ 1 o 4), increases into ðβ; (b) the maximum profit is strictly decreasing (this occurs, for ^ such that example, when γ 1 ¼ 4) and (c) there is a point, say β, the optimal average profit Gðqn Þ decreases as β increases into ^ and it increases as β increases into ðβ; ^ 1Þ (this occurs, for ½0; βÞ example, when γ 1 4 4). We also observe that, in all these cases, the maximum profit converges to zero when β goes to 1 (please see Fig. 4(2)). 3. For fixed β, as the parameter γ 1 increases, the optimal lot size and the maximum average profit decrease (please see Fig. 4 (3) and (4). Moreover, we see that, contrary to what happened in Table 2, the system can be non-profitable for certain values of the parameters β and γ 1 (this occurs, for example, when β ¼ 0:7 and γ 1 ¼ 13).
Finally, in Fig. 5, we plot the variation of the optimal lot size and the maximum profit as functions of β and γ 2 when γ 1 ¼ 2. From these graphs we can extract the following conclusions:
1. With γ 2 fixed, if the value of β is increasing, then the optimal order quantity is strictly increasing, but this increase is very slight when γ 2 ≥4 (please see Fig. 5(1)). 2. Fixed γ 2 and regarding the maximum profit, we have the following cases if the value of β is increasing: (a) it grows to ~ such that the infinity when γ 2 ¼ 1, (b) there is a point, say β, n ~ optimal average profit Gðq Þ increases as β increases into ½0; βÞ ~ 1Þ (this occurs when and it decreases as β increases into ðβ; 1 o γ 2 o 4) and (c) the maximum profit is strictly decreasing (this occurs, for example, when γ 2 ≥4). Moreover, we see that, if γ 2 4 1, the maximum profit converges to zero when β goes to 1 (please see Fig. 5(2)). 3. Lastly, with β fixed, if the value of γ 2 is increasing, Fig. 5(3) and (4) show that the optimal lot size and the maximum average profit decrease (but is always positive, i.e., all the systems are profitable).
4.1. Differences between the optimal solutions for maximum profit and minimum cost To end this section, we illustrate graphically the difference between the optimal solution for the problem of maximum profit and the optimal solution for the problem of minimum cost. Obviously, the total cost of the inventory system per unit time may be calculated by considering only the opposite of the two negative terms in (8) and it is given by the expression CðqÞ ¼ αλKq−α þ ðαλ=ΔÞqξ−α . It is clear
Fig. 4. Optimal lot size and maximum profit curves in function of β and γ 1 for γ 2 ¼ 2.
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301
Fig. 5. Optimal lot size and maximum profit curves in function of β and γ 2 for γ 1 ¼ 2.
that this function can be minimized by taking s¼p in problem (10) and, consequently, using Solution Algorithm with u¼0. Thus, the optimal lot size is qnmin ¼ v1=ξ . Now, supposing that γ 1 ¼ γ 2 ¼ 1:5 and β ¼ 0:3 (that is, α ¼ 0:7 and ξ ¼ αγ 1 þ γ 2 ¼ 2:55), while the other parameter values remain as at the beginning of this section, in Fig. 6, we plot the functions G(q), C(q), ðξ−αÞHCðqÞ=T, ðαK þ ð1−αÞΨ ðqÞÞ=T and αK=T. Note that: 1. In the problem of maximum profit, the optimal lot size occurs when the function ðξ−αÞHCðqÞ=T intersects the function ðαK þ ð1−αÞΨ ðqÞÞ=T, as follows from (17). In this case, we see that the optimal lot size is qn ¼ 5:58 and the maximum profit is Gðqn Þ ¼ 8:89. 2. In the same way, for the problem of minimum cost, the optimal lot size that minimizes the total inventory cost per unit time, C(q), is obtained when the function ðξ−αÞHCðqÞ=T intersects the function αK=T. In this case, we obtain qnmin ¼ 3:28 as the optimal lot size with Cðqnmin Þ ¼ 4:20 as the minimum total inventory cost per unit time. 3. Also, we observe that qn ¼ 5:58 is greater than qnmin ¼ 3:28. 4. In addition, if after solving the problem of minimum cost we add the revenues and the purchasing costs, then the profit obtained is Gðqnmin Þ ¼ 7:80, which is lower than the optimal profit per unit time Gðqn Þ ¼ 8:89. Therefore, when the maximum profit policy is achieved, a 14% improvement in the profit would be obtained with the minimum cost solution.
5. Conclusions Most of the works on inventory control presented in the literature consider that the holding costs are linear in both time and quantity of items held in stock. Also, some works assume that the holding costs are linear in the stock level, but non-linear in
Fig. 6. Optimal solutions for maximum profit and minimum cost with γ 1 ¼ γ 2 ¼ 1:5 and β ¼ 0:3.
time, or that they are linear in time, but non-linear in quantity of items. However, there are very few papers on the subject with holding costs simultaneously non-linear in time and stock level. To further explore this issue, a general inventory system is developed including the above cases and, moreover, assuming that the demand depends on the inventory level. This last assumption forces us to distinguish between the problems of minimizing costs and the problems of maximizing profits. Usually, the first include in their costs only the sum of the ordering costs and the holding costs, while in the second, it is necessary to add, at least, the costs associated with the purchase of the items and the income from the sale. Both problems may arise in the practical life of companies.
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In this work, we study the problems of maximizing profit when the demand rate is a concave potential function of the inventory level and the holding cost is a potential function depending on both the quantity of items and the time they are stored. The problems of minimizing costs may be considered the particular cases of the corresponding problem of maximizing profit if we assume that the unit purchasing cost equals the unit selling price. Taking into account the above assumptions, we develop a general procedure to determine the optimal lot size and the maximum inventory profit. Moreover, an important question for inventory managers is to establish conditions to ensure that the inventory system will generate profits. Thus, in this paper, some properties about the profitability of the inventory system are given. As a consequence of these properties, we present a rule to check if a replenishment policy is the optimal one. This rule is given considering a certain balance between the three components of the system: ordering cost, holding cost and gross profit, and may be interpreted as an extension of the well known result associated to the classical lot size model: the economic order quantity happens when the holding cost is exactly equal to the ordering cost. Numerical examples show that, if the demand rate depends potentially on the inventory level, maximizing the profit is not equivalent to minimizing the total inventory cost, and the improvement in profits with respect to the solution of the minimum cost can be important. The proposed model can be extended in several ways. For example, it may seem interesting to consider the existence of a reorder point before the stock is depleted, which could improve the profit due to the increased demand caused by a higher level of stock. Another way would be the incorporation of changes in selling price, variations in demand rate, discounts in purchasing cost, deteriorating items, etc. Also, we might consider other bivariate mathematical functions for the cumulative holding cost and extend the deterministic demand function to stochastic fluctuating demand patterns. Acknowledgment This work is partly supported by the Spanish Ministry of Science and Innovation through the research project MTM2010-18591.
Table 4 Cumulative holding cost Hðt; xÞ. t
x
1 2 3 4 5 6
25
50
100
200
12.5 25 47.5 70 92.5 115
25 50 95 140 185 240
50 130 210 310 410 520
110 280 460 670 890 1120
We assume that the regular storage cost is 0.1 currency units per item and per unit time. Thus, for example, if x ¼100 and t¼ 4, the cumulative holding cost is the sum of the regular storage cost plus the insurance cost, that is (0.1)(4)(100)+270¼310 currency units. In a similar way, the cumulative holding costs given in Table 4 are obtained. The parameters h, γ 1 and γ 2 can be estimated with the values in Table 4 using a non-linear regression. Thus, the values h ¼0.31, γ 1 ¼ 1:28 and γ 2 ¼ 1:12 are obtained with a determination coefficient R2 ¼ 0:998. Hence, the function Hðt; xÞ ¼ 0:31t 1:28 x1:12 can adequately represent the cumulative holding cost for this inventory system.
Appendix B This appendix shows the evaluation of the holding cost per inventory cycle in a situation where it is simultaneously non-linear in time and stock level. γ Let Hðt; xÞ ¼ ht 1 xγ 2 be the cumulative holding cost for x units that have been stored during t units of time and x ¼ IðtÞ the decreasing inventory level curve with Ið0Þ ¼ q and IðTÞ ¼ 0. Furthermore, let: (i) A be the region of the plane enclosed by the curve x ¼ IðtÞ and the coordinate axes; (ii) P ¼ ft o ; t 1 ; …; t n g be the partition of the interval ½0; T into n equal pieces with t i ¼ iðT=nÞ and (iii) A(P) be the staggered region defined by the union of rectangles Ai ¼ ½t i−1 ; t i ½0; Iðt i Þ, for 1≤i≤n−1. It is clear that A is the limit of the regions A(P) as n tends to infinity and the holding cost δi for each rectangle Ai , with i ¼ 1; …; n−1, can be evaluated as: γ
¼ This appendix provides an example, which shows the utility of the γ function Hðt; xÞ ¼ ht 1 xγ 2 to evaluate the holding cost in an inventory model and how the parameters h, γ 1 and γ 2 can be estimated. We consider an inventory system for luxury items (such as expensive jewelry or designer watches) which needs to be insured while remaining stored. Table 3 gives the cumulative insurance cost for various units of time and quantity, showing that it is not linear in both, time and quantity. Table 3 Cumulative insurance cost. t
γ −1 hγ 1 θi 1 ðt i −t i−1 Þ½Iðt i Þγ 2
with t i−1 oθi o t i by the Mean Value Theorem of differential γ γ1 calculus applied in the factor ðt i 1 −t i−1 Þ. Therefore, the holding cost associated with the region A(P) is γ −1
½Iðt i Þγ 2 ðt i −t i−1 Þ n−1 γ 1 −1 ¼ hγ 1 ∑i ¼ 1 t i ½Iðt i Þγ 2 ðt i −t i−1 Þ γ 1 −1 γ −1 −hγ 1 ∑n−1 −θi 1 Þ½Iðt i Þγ 2 ðt i −t i−1 Þ i ¼ 1 ðt i
n−1 1 ∑n−1 i ¼ 1 δi ¼ hγ 1 ∑i ¼ 1 θ i
50
100
200
n−1
20 40 80 120 160 210
40 110 180 270 360 460
90 240 400 590 790 1000
γ −2
hγ 1 ðγ 1 −1Þ ∑ ηi 1 i¼1
10 20 40 60 80 100
ð18Þ
Again, using the Mean Value Theorem of differential calculus γ −1 γ −1 in factor ðt i 1 −θi 1 Þ, the second term in this expression can be rewritten as
x 25
1 2 3 4 5 6
γ
1 Þ½Iðt i Þγ 2 δi ¼ Hðt i ; Iðt i ÞÞ−Hðt i−1 ; Iðt i ÞÞ ¼ hðt i 1 −t i−1
Appendix A
½Iðt i Þγ 2 ðt i −θi Þðt i −t i−1 Þ
ð19Þ
with t i−1 o θi oηi o t i . Taking into account that ηi oT, Iðt i Þ o q, t i −θi o t i −t i−1 and t i −t i−1 ¼ T=n, it follows that (19) is bounded above by hγ 1 ðγ 1 −1ÞT γ 1 qγ 2 ðn−1Þ=n2 . Then, it converges to zero as n tends to infinity. Finally, taking the limit of the first term in (18) as n tends to infinity, we find that the holding cost for region A, that
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303
On the other hand, if u þ v ¼ 1; we have qo ¼ 1 and Pðqo Þ ¼ 0. That is, qn ¼ 1 is the unique positive root of PðqÞ ¼ 0. Note that, in this case, the sequence fqi g∞i ¼ 0 is constantly equal to qn , because qi ¼ qi−1 for i ¼ 1; 2; …. □
References
Fig. 7. General shape of the function P(q).
is, the holding cost per inventory cycle, is Z T t γ 1 −1 ½IðtÞγ 2 dt hγ 1 0
Appendix C Proof of Theorem 1. The function PðqÞ ¼ qξ −uq−v has the following properties: (i) Pð0Þ ¼ −v o 0 and limq-∞ PðqÞ ¼ ∞, (ii) P′ðqÞ ¼ ξqξ−1 −u, P′ð0Þ ¼ −u o0 and r ¼ ðu=ξÞ1=ðξ−1Þ is the unique root of P′ðqÞ for q 4 0 and (iii) P″ðqÞ ¼ ξðξ−1Þqξ−2 4 0 for q 4 0. Consequently, P(q) is a strictly convex function on the interval ð0; ∞Þ with a unique positive root qn , which verifies PðqÞ o 0 for q∈ð0; qn Þ and PðqÞ 40 for q∈ðqn ; ∞Þ. Moreover, qn 4 r, P′ðqÞ o0 for q∈ð0; rÞ and P′ðqÞ 40 for q∈ðr; ∞Þ. The general shape of the function P(q) is shown in Fig. 7. Now, if u þ v o 1 then qo ¼ ðu þ vÞ1=ξ o 1 and Pðqo Þ ¼ ð1−qo Þu 4 0. Similarly, if u þ v 4 1 then qo ¼ ðu þ vÞ1=ðξ−1Þ 4 1 and Pðqo Þ ¼ ðqo −1Þv 40. That is, in both cases, we have Pðqo Þ 4 0 and, hence, qn o qo and P′ðqo Þ 4 0: Thus, when u þ v≠1, it results that q1 ¼ qo −Pðqo Þ=P′ðqo Þ o qo . Moreover, from the Mean Value Theorem, there exists a point η∈ðqn ; qo Þ with Pðqo Þ ¼ Pðqo Þ−Pðqn Þ ¼ P′ðηÞðqo −qn Þ o P′ðqo Þðqo −qn Þ and, hence, qn o qo −ðPðqo Þ=P′ðqo ÞÞ ¼ q1 oqo (please, see the graphical interpretation for value q1 in Fig. 7). Now, if qn o qi−1 o qi−2 ; a similar reasoning proves that qn o qi o qi−1 and the sequence fqi g∞i ¼ 0 is strictly decreasing and bounded below. Then the sequence fqi g∞i ¼ 0 is convergent and, if we write L for the value of its limit (that is L ¼ limi-∞ qi Þ, taking into account the definition for qi, L ¼ L−ðPðLÞ=P′ðLÞÞ should occur, which implies that PðLÞ ¼ 0 and thus, L ¼ qn .
Alfares, H.K., 2007. Inventory model with stock-level dependent demand rate and variable holding cost. International Journal of Production Economics 108, 259–265. Axsäter, S., 2000. Inventory Control. Kluwer Academic Publishers. Baker, R.C., Urban, T.L., 1988. A deterministic inventory system with an inventorylevel-dependent demand rate. Journal of the Operational Research Society 39, 823–831. Berman, O., Perry, P., 2006. An EOQ model with state-dependent demand rate. European Journal of Operational Research 171, 255–272. Dye, C.Y., Ouyang, L.Y., 2005. An EOQ model for perishable items under stockdependent selling rate and time-dependent partial backlogging. European Journal of Operational Research 163, 776–783. Ferguson, M., Jayaraman, V., Souza, G.C., 2007. Note: an application of the EOQ model with nonlinear holding cost to inventory management of perishables. European Journal of Operational Research 180, 485–490. Giri, B.C., Chaudhuri, K.S., 1998. Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost. European Journal of Operational Research 105, 467–474. Goh, M., 1994. EOQ models with general demand and holding cost functions. European Journal of Operational Research 73, 50–54. Harris, F.W., 1913. How many parts to make at once. Factory, The Magazine of Management 10, 135–136. (reprinted in: Operations Research 38, 947–950). Naddor, E., 1982. Inventory Systems. John Wiley, New York. Padmanabhan, G., Vrat, P., 1995. EOQ models for perishable items under stock dependent selling rate. European Journal of Operational Research 86 (2), 281–292. Pando, V., García-Laguna, J., San-José, L.A., 2012a. Optimal policy for profit maximising in an EOQ model under non-linear holding cost and stock-dependent demand rate. International Journal of Systems Science 43 (11), 2160–2171. Pando, V., San-José, L.A., García-Laguna, J., Sicilia, J., 2012b. Maximizing profits in an inventory model with both demand rate and holding cost per unit time dependent on the stock level. Computers and Industrial Engineering 62 (2), 599–608. Roy, A., 2008. An inventory model for deteriorating items with price dependent demand and time-varying holding cost. Advanced Modeling and Optimization 10 (1), 25–37. Scarpello, G.M., Ritelli, D., 2008. EOQ when holding costs grow with the stock level: well-posedness and solutions. Advanced Modeling and Optimization 10 (2), 233–240. Teng, J.T., Chang, C.T., 2005. Economic production quantity models for deteriorating items with price- and stock-dependent demand. Computers and Operations Research 32, 297–308. Urban, T.L., 2005. Inventory models with inventory-level-dependent demand: a comprehensive review and unifying theory. European Journal of Operational Research 162, 792–804. Urban, T.L., 2008. An extension of inventory models with discretely variable holding costs. International Journal of Productions Economics 114, 399–403. Weiss, H.J., 1982. Economic order quantity models with nonlinear holding costs. European Journal of Operational Research 9, 56–60. Wolfe, H.B., 1968. A model for control of style merchandise. Industrial Management Review 9, 69–82. You, S.P., 2005. Inventory policy for products with price and time-dependent demands. Journal of the Operational Research Society 56, 870–873.
Contents lists available at SciVerse ScienceDirect
Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
An economic lot-size model with non-linear holding cost hinging on time and quantity Valentín Pando a, Luis A. San-José b,n, Juan García-Laguna a, Joaquín Sicilia c a
Departamento de Estadística e Investigación Operativa, Universidad de Valladolid, Valladolid, Spain IMUVA, Instituto de Matemáticas, Universidad de Valladolid, Paseo de Belén 15, 47011 Valladolid, Spain c Departamento de Estadística, Investigación Operativa y Computación, Universidad de La Laguna, Tenerife, Spain b
art ic l e i nf o
a b s t r a c t
Article history: Received 25 May 2012 Accepted 26 April 2013 Available online 21 May 2013
This paper develops an economic lot size inventory model where the demand rate depends on the stock level and the cumulative holding cost is non-linear on both the quantity and the time they are stored. More concretely, it is supposed that the demand rate is a concave potential function of the inventory level and the holding cost is potential on both time and quantity. Moreover, shortages are not allowed. A general procedure to determine the optimal lot size and the maximum inventory profit is developed. Also, some results about the profitability of the inventory system are presented. This work extends several inventory models previously considered in the literature. Finally, numerical examples, which help us to understand the theoretical results, are also given. & 2013 Elsevier B.V. All rights reserved.
Keywords: Inventory management Lot-size models Stock-dependent and time-dependent holding cost Stock-dependent demand rate
1. Introduction Since Harris presented his EOQ model in 1913, many efforts have been made to adjust their assumptions to more realistic situations in inventory management. As is known, in order to obtain a simple mathematical model, Harris assumed that shortage was not allowed and both demand rate and holding costs per unit and per unit time were constant. Thus, the hypotheses of the classic model ensured that the incomes were independent of the stock level and, therefore, the optimal lot size was obtained by minimizing the sum of the carrying cost and the ordering cost per unit of time. However, as is also well-known, empirical evidence shows that these hypotheses are an excessive simplification of reality. So, many models developed later worked with the idea of relaxing these assumptions. In what follows, we will focus on inventory models which maintain the assumption of not allowing shortage, but the other hypotheses have been revised; that is, we do not consider that both demand rate and holding cost per unit and per unit of time are constant. Over time, some practitioners and researchers have found that an increase in exposed merchandise may bring about increased sales of some items. So, Wolfe (1968) presented empirical evidence of this relationship, showing that the sales of style merchandise, such as women's dresses or sports clothes, are proportional to the amount of displayed inventory. After this observation, mathematical
n
Corresponding author. E-mail addresses: [email protected] (V. Pando), [email protected] (L.A. San-José), [email protected] (J. García-Laguna), [email protected] (J. Sicilia). 0925-5273/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2013.04.050
models for inventory systems captured this idea by regarding the use of stock level dependent demand rate in their formulation. As a starting point, Baker and Urban (1988) defined a model in which the demand rate was supposed to be a concave potential function of the inventory level. This new approach to the problem brought about the loss of independence between revenues and total inventory costs, because a larger order size results not only in higher inventory costs but also in higher revenues. Therefore, in this situation, besides the ordering and holding costs, it becomes necessary to consider a third component for the inventory system: the gross profit from the sale of the item (the difference between the selling price and the purchasing cost). Moreover, in this case, the principal objective should be to maximize the profit of the inventory system per unit time instead of minimizing the total inventory cost per unit time. For this reason, Baker and Urban (1988) raised the issue from the perspective of maximizing the profit per unit time. Since then, many papers have appeared considering stock dependent demand rate, but some approach the problem from the perspective of minimizing the total inventory cost per unit time. Urban (2005) provides a detailed overview of the related literature published up to that date. Dye and Ouyang (2005) proposed an inventory model with a demand rate linearly dependent on the stock level and shortages with a time-proportional backlogging rate, which is an extension of the model developed by Padmanabhan and Vrat (1995). Teng and Chang (2005) considered an EPQ model with maximizing profits where the demand rate is simultaneously dependent on the stock level and the selling price but with a ceiling. Also, You (2005) and Roy (2008) analyzed EOQ models with price dependent demand.
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Another issue presented by inventory managers was that, sometimes, the holding cost per unit of item is not proportional to the time held in stock, and/or the holding cost per unit time is also not proportional to the amount held in inventory. This question raised the need to consider other structures for the holding cost in the mathematical models of inventory systems. Under the hypothesis of a constant demand rate, Naddor (1982) proposed three inventory models based on the holding cost function per cycle for those situations. The first model considered that the holding cost per cycle was non-linear with time, but linear with the quantity of items (the perishable-goods system), the second one assumed that the holding cost per cycle was nonlinear with the quantity of items, but linear with time (the expensive-storage system) and the third model supposed that the holding cost per cycle was non-linear with both time and quantity of items (the general system). In this environment, Weiss (1982) introduced a deterministic model with constant demand rate, supposing that the holding cost per unit of product was a convex potential function of the time in stock. This situation can occur, for example, in the storage of perishable items such as food products. In this case, the longer the products are kept in storage, the more sophisticated the storage facilities and services are needed, and therefore, the higher the holding cost. In fact, this model proved to be equivalent to the perishable-goods system cited above, as noted by Ferguson et al. (2007). Later on, Goh (1994) generalized this situation to the case of stock-dependent demand rate, minimizing the total inventory cost per unit time. Alfares (2007) considered this same situation with two types of discontinuous step functions. Urban (2008) extended Alfares' work by using a profit maximization objective. Recently, Pando et al. (2012a) have studied the previous model of Goh (1994), but from the perspective of maximizing the profit. Inventory models with holding cost rate per unit time nonlinear in the stock level are scarcer. Nevertheless, this situation can be encountered in real inventories when the value of the item is very high and many precautionary steps are to be taken to ensure its safety and quality. This occurs, for example, when carrying luxury items like expensive jewelry and designer watches. From the perspective of minimizing the total inventory cost per unit of time and using a stock-dependent demand rate, Goh (1994) supposed that the holding cost rate per unit of time was a convex potential function of the quantity of items held in stock. Later on, Giri and Chaudhuri (1998) extended this model to the case of deteriorating items with a small constant fraction of deterioration. Berman and Perry (2006) also worked with stock-dependent holding cost and two types of demand rate functions. Scarpello and Ritelli (2008) gave some theoretical results for EOQ models when the holding cost grows with the stock level. Recently, Pando et al. (2012b) have analyzed one of the models of Goh (1994) from the perspective of maximizing the profit. To the best of our knowledge, the work of Naddor (1982) is the only paper dedicated to study inventory models for the situation in which the holding cost is non-linear with both time and quantity. This structure of the holding cost function is adequate for the representation of some real life situations. This occurs, for example, in the storage of very expensive products (then many precautionary steps are to be taken to ensure their safety) which deteriorates over time (so more sophisticated storage facilities and services are needed). However, in the work of Naddor (1982), the demand rate was considered constant. According to the previous comments, the objective of this paper is to study an inventory system with stock dependent demand rate and a structure of holding cost non-linear in both time and stock level from the perspective of maximizing profits per unit time. Thus, the main difference between our model and the one in Naddor (1982) is the assumption about the demand
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rate. So, instead of a constant demand rate, we consider the case where the demand rate is a concave potential function of the inventory level. Moreover, we develop the necessary theory to calculate the holding cost per cycle when it depends non-linearly on both time and stock level. The structure of this paper is as follows. Section 2 presents the assumptions and notation to be used throughout the paper. Section 3 deals with the mathematical formulation of the inventory model, the procedure to obtain the optimal policy and the main theoretical results. In Section 4, we provide some numerical examples to illustrate the theoretical results and show the difference between the optimal solutions for the problems of maximum profit and minimum cost. Finally, conclusions are drawn in Section 5.
2. Assumptions and notation The inventory system studied in this paper is developed on the basis of the following assumptions. The inventory is continuously reviewed and the planning horizon is infinite. The item is a single product and no shortage is allowed. The replenishment is instantaneous when the stock is depleted and the lead time is zero. The demand rate is a known function of the inventory level. The order cost is fixed, regardless of the lot size. The unit purchasing cost and the selling price are known and constant. The cumulative holding cost for x units of product that have been stored during t units of time is a potential function of t and x. Thus, this function may not be linear in any of these two variables. Table 1 summarizes the notation used in this paper. We suppose that the demand rate is described by the potential function of the inventory level DðtÞ ¼ λ½IðtÞβ
ð1Þ
with λ 4 0 and 0≤β o 1. With this functional form, as the inventory level decreases, so does the demand rate. Thus, at the beginning of a cycle, the inventory level decreases rapidly because the demand is bigger at a high level of stock. At time t¼ 0 the inventory level and the demand rate are at their highest level. As more inventory is depleted, the rate of decrease of the stock level slows down. The elasticity of the demand rate with respect to the stock level β represents the relative change in demand rate with respect to the corresponding relative change in the stock level (that is, β ¼ ð∂DðtÞ=∂IðtÞÞ=ðDðtÞ=IðtÞÞ). Note that, if the demand is inelastic (that is, β ¼ 0), we have DðtÞ ¼ λ and the model reverts to the wellknown inventory model with constant demand rate. Furthermore, it is supposed that the cumulative holding cost for x items stored during t units of time is given by the following Table 1 List of notation. q T t I(t) K p s Hðt; xÞ h γ1 γ2 D(t) λ β α ξ
Order quantity or lot size per cycle ( 40, decision variable) Length of the inventory cycle ð4 0Þ Time spent in inventory ð≤TÞ Inventory level at time t ð≤qÞ Ordering cost per order ð 4 0Þ Unit purchasing cost ð4 0Þ Unit selling price ð≥pÞ Cumulative holding cost for x items stored during t units of time ð 4 0Þ Scaling constant for holding cost ð4 0Þ Elasticity of the holding cost with respect to time ð≥1Þ Elasticity of the holding cost with respect to the stock level ð≥1Þ Demand rate at time t Scaling constant of demand rate ð 4 0Þ Elasticity of demand rate with respect to the stock level ð0≤β o 1Þ Auxiliary parameter, α ¼ 1−β ð0 o α≤1Þ Auxiliary parameter, ξ ¼ αγ 1 þ γ 2 ð 41Þ
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equation:
potential function of t and x: γ1 γ2
Hðt; xÞ ¼ ht x
ð2Þ
where h 40, γ 1 ≥1 and γ 2 ≥1. These three parameters can be estimated via a non-linear regression or a multiple linear regression between the logarithm of the cumulative holding cost and the logarithms of time and quantity. Appendix A provides a numerical example where this function is shown to be useful for evaluating the holding cost in an inventory model. Note that h ¼ Hð1; 1Þ, that is, the holding cost of one unit held in stock during one unit of time is the scaling parameter for the holding cost. The elasticity of the holding cost with respect to time γ 1 represents the relative change in this cost with respect to the relative change in time (that is, γ 1 ¼ ð∂H=∂tÞ=ðH=tÞ). Similarly γ 2 represents the relative change in this cost with respect to the relative change in the quantity (that is, γ 2 ¼ ð∂H=∂xÞ=ðH=xÞ). Also, if γ 1 ¼ 1 in formula (2), we obtain the same cumulative holding cost function considered by, among others, Naddor (1982, expensive-storage system), Berman and Perry (2006) and Pando et al. (2012b). Similarly, if γ 2 ¼ 1 in (2), we have the cumulative holding cost of the model by Naddor (1982, perishable-goods system), which was later used by Ferguson et al. (2007) and Pando et al. (2012a). We assume the general condition ðs≥pÞ, instead of s 4 p, because this assumption allows several inventory models presented by other authors to be included in our study. We refer to some models that have been developed from the perspective of minimizing the average cost, without considering the revenues and purchasing costs. Formally, they can be obtained from the model developed here by taking s ¼p and, in this article, we refer to them as inventory models of minimum cost. However, in the model studied here, our objective is to maximize the average profit and, therefore, the revenues and purchasing costs have to be included. Hence, we will name this class of models as inventory models of maximum profit. Consequently, those inventory models of minimum cost which do not include revenues or purchasing costs analyze only a part of the systems associated to the models of maximum profit; that is, those systems are subsystems of the system considered here. In a few inventory models (for example, in the EOQ models studied by Harris, 1913 and Naddor, 1982), resolving the first situation is equivalent to solving the second because the demand rate is constant, but this does not always occur. In fact, this last case is what occurs in the model studied here. More specifically, if we solve a model of minimum cost, as in Naddor (1982), and the revenues and the purchasing costs are added later on, then the profit obtained is smaller than or, eventually, equal to the profit obtained when the revenues and the purchasing costs are included in the formulation of the model, that is, if we consider the model of maximum profit. In this paper, as in the model of Naddor (1982, general system), a lot-size inventory model with non-linear holding cost in both time and stock level is studied, but here we are considering the more general situation with a stock-level dependent demand rate and the problem is handled from the proper perspective of maximizing profit per unit time. Therefore, taking into account the previous comments, we present a model which generalizes the models studied by Goh (1994). The following section includes the mathematical formulation of the model and the solution procedure to obtain the optimal policy.
dIðtÞ ¼ −λ½IðtÞβ ; dt
ð3Þ
0≤t≤T
with the initial condition Ið0Þ ¼ q. The solution of (3) is IðtÞ ¼ ðqα −αλtÞ1=α
ð4Þ
where α ¼ 1−β. Taking into account that the replenishment is done when the stock is depleted, we have IðTÞ ¼ 0 and, therefore, the corresponding length of the inventory cycle T is given by T¼
qα αλ
ð5Þ
Therefore, (4) can be rewritten as 1=α t IðtÞ ¼ q 1− T
ð6Þ
Once the mathematical expression for the inventory level is obtained, the β parameter is better understood. Specifically, the cumulative demand in the second half of the inventory cycle is calculated as 1=ð1−βÞ Z T Z T 1 λ½IðtÞβ dt ¼ − dIðtÞ ¼ IðT=2Þ ¼ q 2 T=2 T=2 and the proportion of sales in this second half of the cycle with respect to the sales over the whole cycle is given by ρ ¼ ð12Þ1=ð1−βÞ . Evidently, as 0≤β o 1, it is always 0 o ρ≤ 12 which makes sense for this type of demand with higher sales in the first half of the cycle. Then, the β parameter can be estimated as a function of ρ as β ¼ 1 þ ðln 2Þ=ðln ρÞ. The realization of the inventory level is depicted in Fig. 1. With the assumptions given in the previous section, the total profit per cycle is calculated as the difference between the revenue per cycle and the sum of the ordering cost, the purchasing cost and the inventory holding cost per cycle. Obviously, the ordering cost is K, the revenue per cycle is sq and the purchase cost per cycle is pq. The holding cost per cycle can be calculated as (please see Appendix B) Z T HCðqÞ ¼ hγ 1 t γ1 −1 ½IðtÞγ 2 dt 0
Using (5) and (6), we have Z T γ1 −1 t t γ 2 =α dt HCðqÞ ¼ hγ 1 qγ 2 T γ 1 1− T T T 0 γ ¼ hγ 1 qγ 2 T γ 1 B γ 1 ; 2 þ 1 α hγ 1 B γ 1 ; γα2 þ 1 αγ 1 þγ 2 qξ ð7Þ ¼ q ¼ γ1 ðαλÞ Δ R1 where Bða; bÞ ¼ 0 za−1 ð1−zÞb−1 dz is the well-known beta function, ξ ¼ αγ 1 þ γ 2 and Δ ¼ ðαλÞγ 1 =hγ 1 Bðγ 1 ; ðγ 2 =αÞ þ 1Þ. Obviously, ξ≥1 þ α 4 1 and Δ 4 0. Therefore, the total profit per cycle is given by ðs−pÞq−K−ðqξ =ΔÞ and, by (5), the profit per unit time, or gain, is GðqÞ ¼ ½ðs−pÞq− K−ðqξ =ΔÞ=T, which can be expressed as GðqÞ ¼ αλðs−pÞq1−α −αλKq−α −
αλ ξ−α q Δ
ð8Þ
Thus, the problem consists in determining the decision variable q 4 0 such that the function G(q) given by (8) is maximized. Since GðqÞ ¼ −ðαλ=ΔÞgðqÞ where 3. The model
gðqÞ ¼ qξ−α −ðs−pÞΔq1−α þ KΔq−α
Considering the expression for the demand rate given in (1), it is clear that the stock level I(t) verifies the following differential
the optimal solution that minimizes g(q) is the same as the optimal solution that maximizes G(q).
ð9Þ
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length of the cycle time using (5). Moreover, from (8), (9) and (11) it follows that αλ n −α n ξ ðq Þ ½ðq Þ −ðs−pÞΔqn þ KΔ Δ αλ ¼ − ðqn Þ−α ½uqn þ v−ðs−pÞΔqn þ KΔ Δ
Gðqn Þ ¼ −
Substituting the values of u and v given in (12) and (13) respectively into the previous expression, the maximum average profit is given by ð1−αÞðs−pÞ αK −ðs−pÞ qn þ þK Gðqn Þ ¼ −αλðqn Þ−α ξ−α ξ−α αλ ðξ−1Þðs−pÞqn −Kξ ¼ ξ−α ðqn Þα
Fig. 1. Behavior of the inventory level over time.
Consequently, the maximum-profit problem is equivalent for solving the non-linear program MinimizefgðqÞ : q 4 0g
ð10Þ
3.1. Solution procedure The first two derivatives of the function g(q) are g′ðqÞ ¼ ðξ−αÞqξ−α−1 −ð1−αÞðs−pÞΔq−α −αKΔq−α−1 ξ−α ð1−αÞðs−pÞΔ αKΔ q− ¼ αþ1 qξ − ξ−α ξ−α q and g″ðqÞ ¼ ðξ−αÞðξ−α−1Þqξ−α−2 þ αð1−αÞðs−pÞΔq−α−1 þ αð1 þ αÞKΔq−α−2 Since g″ðqÞ 40 for q 40, it follows that g(q) is a strictly convex function; moreover, limq↓0 gðqÞ ¼ limq-∞ gðqÞ ¼ ∞. Therefore, the function g′ðqÞ has only one positive root which is the value qn where g(q) attains the global minimum, that is, the solution for the problem. Obviously, the equation g′ðqÞ ¼ 0 is equivalent to this other PðqÞ ¼ qξ −uq−v ¼ 0
ð11Þ
where u¼
ð1−αÞðs−pÞΔ ≥0 ξ−α
ð12Þ
v¼
αKΔ 40 ξ−α
ð13Þ
It is clear that if u ¼0 (that is, α ¼ 1 or s¼p) the solution is αKΔ 1=ξ qn ¼ v1=ξ ¼ ð14Þ ξ−α Therefore, it is only necessary to solve Eq. (11) when u 4 0 (that is, 0 o α o1 and s 4p). The following result answers this question Theorem 1. Let the function PðqÞ ¼ qξ −uq−v, with ξ 41, u 4 0, v 4 0 and q 40. Write qo ¼ maxfðu þ vÞ1=ξ ; ðu þ vÞ1=ðξ−1Þ g. For i ¼ 1; 2; 3; …, "consider the sequence qi ¼ qi−1 −ðPðqi−1 Þ=P′ðqi−1 ÞÞ ¼ ððξ−1Þqξi−1 þ vÞ=ðξqξ−1 i−1 −uÞ. Then P(q) has a unique" positive root qn which is the limit of the sequence fqi g∞i ¼ 0 . Proof. Please see Appendix C.
□
Using this theorem, we can numerically calculate the optimal lot-size qn with the desired precision and evaluate the optimal
ð15Þ
Although calculating the optimal lot size qn (resolving the equation PðqÞ ¼ 0) can be easily done with any solver software, we include below a simple algorithm to evaluate the solution using the Newton–Fourier method. It can be applied directly by the inventory manager without specific software. Solution algorithm. Obtaining the optimal policy and the optimal profit in the model. Step 1. Calculate Δ ¼ ðαλÞγ 1 =hγ 1 Bðγ 1 ; ðγ 2 =αÞ þ 1Þ, ξ ¼ αγ 1 þ γ 2 , u ¼ ð1−αÞðs−pÞΔ=ðξ−αÞ and v ¼ αKΔ=ðξ−αÞ. Step 2. If u ¼0 then the optimal lot-size is qn ¼ v1=ξ . Go to Step5. Step 3. If u þ v ¼ 1 then the optimal lot size is qn ¼ 1. Go to Step5. Step 4. Calculate qo ¼ maxfðu þ vÞ1=ξ ; ðu þ vÞ1=ðξ−1Þ g and r ¼ ðu=ξÞ1=ðξ−1Þ . (a) Select a positive tolerance value, say TOL 4 0, with 0 o TOL o r. (b) Calculate qi ¼ ððξ−1Þqξi−1 þ vÞ=ðξqξ−1 for i−1 −uÞ i ¼ 1; 2; 3; …; until Pðqi −TOLÞ o 0. (c) Take as optimal lot-size the value qn ¼ qi . Step 5. Compute the optimal inventory cycle T n ¼ ðqn Þα =αλ and the maximum profit Gðqn Þ ¼ ðαλ=ðξ−αÞÞ½ððξ−1Þðs−pÞqn −KξÞ=ðqn Þα Remark. In order to understand the previous algorithm and its criterion of stop, it is convenient to see the proof of Theorem 1 in Appendix C. Thus we have: (i) the function P(q) attains its minimum at point r; (ii) PðqÞ o 0 if and only if q∈ð0; qn Þ; (iii) the succession qi is strictly decreasing and r o qn o qi for all i ¼ 0; 1; 2; …; and (iv) as the tolerance TOL belongs to the interval ð0; rÞ, it follows that qi −TOL 40 for all i ¼ 0; 1; 2; …. Then, when the condition Pðqi −TOLÞ o 0 is satisfied, it follows qi −TOL o qn , that is, qi −qn o TOL. This allows us to conclude that Pðqi −TOLÞ o 0 is a suitable stopping rule. 3.2. Meaning and discussion of optimal policy An interesting question for inventory managers is to establish conditions to ensure that the inventory system will generate profits. If this happens using the optimal lot size qn , then Gðqn Þ 4 0 and it is called a profitable inventory system. However, if Gðqn Þ o 0, we say that it is a non-profitable inventory system, because, even using the optimal lot size, it generates losses. Finally, if Gðqn Þ ¼ 0 then, for the optimal situation, it matches costs and revenues and it is called a break-even inventory system. With these definitions, the inventory managers are interested in characterizing the system as profitable, non-profitable or breakeven without calculating the optimal lot size qn and the maximum profit Gðqn Þ. Obviously, if s ¼p then Gðqn Þ o0 and the system is non-profitable. The following result gives a characterization for the type of inventory system as a function of the input parameters when s 4p.
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Theorem 2. Assuming that s 4p, we have 1. The inventory is profitable if and only if s−p 4 ðξ=ðξ−1ÞÞððξ−1ÞK ξ−1 =ΔÞ1=ξ . 2. The inventory is a break-even system if and only if s−p ¼ ðξ=ðξ−1ÞÞððξ−1ÞK ξ−1 =ΔÞ1=ξ . 3. The inventory is non-profitable if and only if s−p o ðξ=ðξ−1ÞÞððξ−1ÞK ξ−1 =ΔÞ1=ξ . Proof. From (15) it is clear that Gðqn Þ 40 if and only if qn 4Kξ=ððξ−1Þðs−pÞÞ. Thus, by the properties of the function P(q) (please see Appendix C), this last inequality is equivalent to P½Kξ=ððξ−1Þðs−pÞÞ o 0. Now, as ξ Kξ Kξ KΔ ξð1−αÞ ¼ þα P − ðξ−1Þðs−pÞ ðξ−1Þðs−pÞ ξ−α ξ−1 ξ Kξ KΔ ¼ − ðξ−1Þðs−pÞ ξ−1 Fig. 2. Optimal policy for profit maximizing.
we deduce that the inventory is profitable if and only if ½Kξ=ððξ−1Þðs−pÞÞξ o KΔ=ðξ−1Þ, that is, s−p 4 ðξ=ðξ−1ÞÞððξ−1ÞK ξ−1 =ΔÞ1=ξ , which is assertion 1. The same reasoning applied to break-even and non-profitable systems proves assertions 2 and 3. □ As a consequence of this theorem, it is clear that s ¼ p þ ðξ=ðξ−1ÞÞððξ−1ÞK ξ−1 =ΔÞ1=ξ is the minimum selling price that the inventory manager should overcome to obtain a profitable system. Another interesting issue for practitioners is to have a rule to know if a replenishment policy is the optimal one. For this purpose, the following theorem allows us to characterize the optimal policy for the model under consideration. Theorem 3. Let the functions Ψ ðqÞ ¼ ðs−pÞq (gross profit per cycle with lot size q) and HCðqÞ ¼ qξ =Δ (holding cost per cycle with lot size q). Then, the lot size q is optimal if and only if ðξ−αÞHCðqÞ ¼ αK þ ð1−αÞΨ ðqÞ
ð16Þ
Proof. If q is optimal (that is, q ¼ qn ), from (7), (11), (12) and (13), we have HCðqn Þ ¼ ðqn Þξ =Δ ¼ ðð1−αÞðs−pÞ=ðξ−αÞÞqn þ αK=ðξ−αÞ ¼ ðð1−αÞ =ðξ−αÞÞΨ ðqn Þ þ ðα=ðξ−αÞÞK. That is, ðξ−αÞHCðqn Þ ¼ αK þ ð1−αÞΨ ðqn Þ. On the other hand, if q verifies ðξ−αÞHCðqÞ ¼ αK þ ð1−αÞΨ ðqÞ, subtracting this equation from the other one with qn , we have ðξ−αÞ½ðqn Þξ =ΔÞ−ðqξ =ΔÞ ¼ ð1−αÞðs−pÞðqn −qÞ; or, equivalently, from (12), ½ðqn Þξ −qξ ¼ uðqn −qÞ. Now, as qn is the solution of Eq. (11), we have ðqn Þξ ¼ uqn þ v and thus qξ −uq−v ¼ 0. Hence q ¼ qn because P (q) has a unique positive root. □ It is possible to obtain an interpretation of the above result if we consider the gross profit per unit time instead of the gross profit per cycle. Thus, dividing both sides of the equality (16) by the optimal cycle time T n , we have ðξ−αÞ
HCðqn Þ K Ψ ðqn Þ ¼ α n þ ð1−αÞ n T T Tn
ð17Þ
where HCðqn Þ=T n , K=T n and Ψ ðqn Þ=T n are the optimal holding cost, ordering cost and gross profit per unit time, respectively. Therefore, the function G(q) attains its maximum at the cut point of the functions ðξ−αÞðHCðqÞ=TÞ and αðK=TÞ þ ð1−αÞðΨ ðqÞ=TÞ. Fig. 2 illustrates this idea by plotting the three functions. Note that this interpretation is an extension of a well-known result associated to the classical lot size model: “for the optimal order quantity, the holding cost happens to be exactly equal to the ordering cost” (see, for instance, Axsäter, 2000, p. 32), but adapted here to the model under consideration. Specifically, if α ¼ γ 1 ¼ γ 2 ¼ 1, Eq. (17) leads to HCðqn Þ=T n ¼ K=T n which is the result previously cited.
3.3. Particular models Next, let us deduce that several lot size models studied by other authors turn out to be particular cases of the model developed in this paper. 1. When β ¼ 0 (that is, α ¼ 1) and choosing the notation h ¼ c1 , γ 1 ¼ n, γ 2 ¼ m, K ¼ c3 and λ ¼ r, we obtain the general system developed by Naddor (1982). Thus, putting a ¼ γ 1 Bðγ 1 ; γ 2 þ 1Þ, we have u¼0 and v ¼ c3 r n =ac1 ðm þ n−1Þ. Applying Solution Algorithm, we obtain qn ¼ ½c3 r n =ðac1 ðm þ n−1ÞÞ1=ðmþnÞ which coincides with the optimal policy shown by Naddor. Substituting this value into (15), we see that the maximum average profit is Gðqn Þ ¼ rðs−pÞ−ðm þ nÞ½ac1 r m ðc3 =ðm þ n−1ÞÞmþn−1 1=ðmþnÞ . Of course, the last term in Gðqn Þ is the minimum cost given by Naddor (1982, p. 108). Note that this model admits the following ones as particular cases: (i) If γ 1 ¼ 1 (that is, the holding cost per cycle is non-linear in quantity of items but linear in time), we obtain the model studied by Naddor (1982, expensive-storage system) and (ii) if γ 2 ¼ 1 (that is, the holding cost per cycle is non-linear in time but linear in quantity of items), then we revert to the model analyzed by Naddor (1982, perishable-goods system) which has also been studied by Weiss (1982). 2. If we suppose that γ 2 ¼ 1, we revert to the model developed by Pando et al. (2012a). Let us remember that, in this case, various particular models can be deduced: (i) If we set s¼ p, we obtain the model developed by Goh (1994, model A) and (ii) if we suppose that s 4 p and γ 1 ¼ γ 2 ¼ 1, we revert to the model proposed by Baker and Urban (1988) when the order point is zero. 3. If γ 1 ¼ 1, then the problem is equivalent to the one studied by Pando et al. (2012b). Thus, our optimal policy qn is equal to the one shown by them. Also, if we further assume that s ¼p, we obtain the model analyzed by Goh (1994, model B).
4. Computational results To illustrate the proposed model and its associated optimal policies, in this section, some numerical examples are presented. We consider the same numerical example proposed by Pando et al. (2012a), but adding values for the parameter which were not considered in their model. Therefore, we assume the following input parameters for the inventory system: λ ¼ 1, K ¼10, h¼0.5,
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s¼ 62, p¼ 50 and β∈f0; 0:1; 0:3; 0:5; 0:7; 0:9g. Moreover, to observe the effects of the parameters γ 1 and γ 2 on the optimal lot-size and the maximum profit, we suppose that both parameters belong to f1; 1:5; 2; 2:5g. The computed results are reported in Table 2. Note that, in all considered cases, the inventory system we obtain is profitable. Moreover, we find that, in these numerical examples, the smaller optimal profit is obtained when β ¼ 0:9, γ 1 ¼ 1 and γ 2 ¼ 2:5. In this case, we have s−p ¼ 12; which is greater than ðξ=ðξ−1ÞÞððξ−1ÞK ξ−1 =ΔÞ1=ξ ¼ 4:260; as Gðqn Þ ¼ 3:60, the inventory system is profitable, which agrees with part 1 of Theorem 2. Also, note that when γ 2 4 γ 1 we obtain a smaller optimal lot size and maximum average profit than if we exchange the values of the
299
parameters γ 1 and γ 2 . This is due to the fact that, in the first case, a bigger holding cost is generated. Moreover, in Fig. 3, we plot the variation of the optimal lot size and the maximum profit as functions of γ 1 and γ 2 when β ¼ 0:3. From these graphs, we can extract the following conclusions: 1. Fixed γ 2 , if the value of γ 1 is increasing, then three cases are possible (please see Fig. 3(1) and (2): (a) the optimal lot size qn and the optimal profit Gðqn Þ are strictly decreasing (this occurs, for example, when γ 2 ¼ 2), (b) the optimal profit and the optimal lot size are firstly strictly increasing and, later on, they decrease slightly (this occurs, e.g., when γ 2 ¼ 10) and (c) the
Table 2 Optimal values qn and Gðqn Þ ¼ Gn for different parameters β, γ 1 and γ 2 respectively. β
0
0.1
0.3
0.5
0.7
0.9
γ1
γ2
qn
Gn
qn
Gn
qn
Gn
qn
Gn
qn
Gn
qn
Gn
1 1 1 1 1.5 1.5 1.5 1.5 2 2 2 2 2.5 2.5 2.5 2.5
1 1.5 2 2.5 1 1.5 2 2.5 1 1.5 2 2.5 1 1.5 2 2.5
6.32 4.07 3.11 2.59 4.07 3.24 2.76 2.45 3.11 2.76 2.51 2.33 2.59 2.45 2.33 2.22
8.84 7.90 7.17 6.60 7.90 7.37 6.93 6.56 7.17 6.93 6.70 6.48 6.60 6.56 6.48 6.38
10.0 5.02 3.52 2.81 5.21 3.77 3.07 2.65 3.65 3.10 2.75 2.50 2.90 2.69 2.51 2.37
10.1 8.48 7.43 6.67 8.56 7.72 7.10 6.62 7.51 7.13 6.79 6.50 6.72 6.64 6.51 6.36
38.4 8.47 4.68 3.37 10.7 5.58 3.95 3.17 5.75 4.20 3.42 2.96 3.94 3.41 3.04 2.77
16.6 10.5 8.13 6.82 11.3 8.89 7.59 6.74 8.78 7.78 7.08 6.55 7.22 6.92 6.61 6.32
326 16.8 6.59 4.16 37.6 9.77 5.48 3.94 12.8 6.59 4.61 3.67 6.68 4.87 3.94 3.39
53.7 14.8 9.11 6.84 20.9 11.3 8.31 6.77 12.4 9.11 7.51 6.53 8.59 7.52 6.77 3.68
3 104 43.2 10.0 5.29 591 23.3 8.53 5.19 76.6 13.9 7.09 4.88 21.4 8.98 5.85 4.49
1440 25.6 10.0 6.29 122 17.4 9.09 6.35 33.6 12.3 7.98 6.13 14.5 8.93 6.87 5.74
6 1013 187 17.2 7.05 3 109 114 16.3 7.44 4 105 64.0 14.1 7.33 5038 34.8 11.5 6.86
3 1011 52.5 7.73 3.60 7 106 35.1 7.48 3.88 2 104 21.5 6.61 3.86 560 12.5 5.48 3.63
Fig. 3. Optimal lot size and maximum profit curves in function of γ 1 and γ 2 for β ¼ 0:3.
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optimal lot size and the maximum profit are strictly increasing simultaneously (if γ 2 ¼ 30, this is what happens). 2. The same comments as before can be made about the optimal lot size and the optimal average profit when γ 1 is fixed (please see Fig. 3(3) and (4)).
Fig. 4 depict the curves of the optimal lot size and the maximum average profit when γ 2 ¼ 2. We can now establish the following comments about these results. 1. Given γ 1 ≤4, as the parameter β increases, the optimal lot size increases. For fixed γ 1 44, the optimal order quantity is a slightly decreasing function of β (please see Fig. 4(1)). 2. Fixed γ 1 and regarding the maximum profit, three cases can occur if the value of the elasticity of demand rate is increasing: ~ such that the optimal average profit (a) there is a point, say β, ~ and it decreases as β Gðqn Þ increases as β increases into ½0; βÞ ~ 1Þ (this occurs, for example, when γ 1 o 4), increases into ðβ; (b) the maximum profit is strictly decreasing (this occurs, for ^ such that example, when γ 1 ¼ 4) and (c) there is a point, say β, the optimal average profit Gðqn Þ decreases as β increases into ^ and it increases as β increases into ðβ; ^ 1Þ (this occurs, for ½0; βÞ example, when γ 1 4 4). We also observe that, in all these cases, the maximum profit converges to zero when β goes to 1 (please see Fig. 4(2)). 3. For fixed β, as the parameter γ 1 increases, the optimal lot size and the maximum average profit decrease (please see Fig. 4 (3) and (4). Moreover, we see that, contrary to what happened in Table 2, the system can be non-profitable for certain values of the parameters β and γ 1 (this occurs, for example, when β ¼ 0:7 and γ 1 ¼ 13).
Finally, in Fig. 5, we plot the variation of the optimal lot size and the maximum profit as functions of β and γ 2 when γ 1 ¼ 2. From these graphs we can extract the following conclusions:
1. With γ 2 fixed, if the value of β is increasing, then the optimal order quantity is strictly increasing, but this increase is very slight when γ 2 ≥4 (please see Fig. 5(1)). 2. Fixed γ 2 and regarding the maximum profit, we have the following cases if the value of β is increasing: (a) it grows to ~ such that the infinity when γ 2 ¼ 1, (b) there is a point, say β, n ~ optimal average profit Gðq Þ increases as β increases into ½0; βÞ ~ 1Þ (this occurs when and it decreases as β increases into ðβ; 1 o γ 2 o 4) and (c) the maximum profit is strictly decreasing (this occurs, for example, when γ 2 ≥4). Moreover, we see that, if γ 2 4 1, the maximum profit converges to zero when β goes to 1 (please see Fig. 5(2)). 3. Lastly, with β fixed, if the value of γ 2 is increasing, Fig. 5(3) and (4) show that the optimal lot size and the maximum average profit decrease (but is always positive, i.e., all the systems are profitable).
4.1. Differences between the optimal solutions for maximum profit and minimum cost To end this section, we illustrate graphically the difference between the optimal solution for the problem of maximum profit and the optimal solution for the problem of minimum cost. Obviously, the total cost of the inventory system per unit time may be calculated by considering only the opposite of the two negative terms in (8) and it is given by the expression CðqÞ ¼ αλKq−α þ ðαλ=ΔÞqξ−α . It is clear
Fig. 4. Optimal lot size and maximum profit curves in function of β and γ 1 for γ 2 ¼ 2.
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301
Fig. 5. Optimal lot size and maximum profit curves in function of β and γ 2 for γ 1 ¼ 2.
that this function can be minimized by taking s¼p in problem (10) and, consequently, using Solution Algorithm with u¼0. Thus, the optimal lot size is qnmin ¼ v1=ξ . Now, supposing that γ 1 ¼ γ 2 ¼ 1:5 and β ¼ 0:3 (that is, α ¼ 0:7 and ξ ¼ αγ 1 þ γ 2 ¼ 2:55), while the other parameter values remain as at the beginning of this section, in Fig. 6, we plot the functions G(q), C(q), ðξ−αÞHCðqÞ=T, ðαK þ ð1−αÞΨ ðqÞÞ=T and αK=T. Note that: 1. In the problem of maximum profit, the optimal lot size occurs when the function ðξ−αÞHCðqÞ=T intersects the function ðαK þ ð1−αÞΨ ðqÞÞ=T, as follows from (17). In this case, we see that the optimal lot size is qn ¼ 5:58 and the maximum profit is Gðqn Þ ¼ 8:89. 2. In the same way, for the problem of minimum cost, the optimal lot size that minimizes the total inventory cost per unit time, C(q), is obtained when the function ðξ−αÞHCðqÞ=T intersects the function αK=T. In this case, we obtain qnmin ¼ 3:28 as the optimal lot size with Cðqnmin Þ ¼ 4:20 as the minimum total inventory cost per unit time. 3. Also, we observe that qn ¼ 5:58 is greater than qnmin ¼ 3:28. 4. In addition, if after solving the problem of minimum cost we add the revenues and the purchasing costs, then the profit obtained is Gðqnmin Þ ¼ 7:80, which is lower than the optimal profit per unit time Gðqn Þ ¼ 8:89. Therefore, when the maximum profit policy is achieved, a 14% improvement in the profit would be obtained with the minimum cost solution.
5. Conclusions Most of the works on inventory control presented in the literature consider that the holding costs are linear in both time and quantity of items held in stock. Also, some works assume that the holding costs are linear in the stock level, but non-linear in
Fig. 6. Optimal solutions for maximum profit and minimum cost with γ 1 ¼ γ 2 ¼ 1:5 and β ¼ 0:3.
time, or that they are linear in time, but non-linear in quantity of items. However, there are very few papers on the subject with holding costs simultaneously non-linear in time and stock level. To further explore this issue, a general inventory system is developed including the above cases and, moreover, assuming that the demand depends on the inventory level. This last assumption forces us to distinguish between the problems of minimizing costs and the problems of maximizing profits. Usually, the first include in their costs only the sum of the ordering costs and the holding costs, while in the second, it is necessary to add, at least, the costs associated with the purchase of the items and the income from the sale. Both problems may arise in the practical life of companies.
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In this work, we study the problems of maximizing profit when the demand rate is a concave potential function of the inventory level and the holding cost is a potential function depending on both the quantity of items and the time they are stored. The problems of minimizing costs may be considered the particular cases of the corresponding problem of maximizing profit if we assume that the unit purchasing cost equals the unit selling price. Taking into account the above assumptions, we develop a general procedure to determine the optimal lot size and the maximum inventory profit. Moreover, an important question for inventory managers is to establish conditions to ensure that the inventory system will generate profits. Thus, in this paper, some properties about the profitability of the inventory system are given. As a consequence of these properties, we present a rule to check if a replenishment policy is the optimal one. This rule is given considering a certain balance between the three components of the system: ordering cost, holding cost and gross profit, and may be interpreted as an extension of the well known result associated to the classical lot size model: the economic order quantity happens when the holding cost is exactly equal to the ordering cost. Numerical examples show that, if the demand rate depends potentially on the inventory level, maximizing the profit is not equivalent to minimizing the total inventory cost, and the improvement in profits with respect to the solution of the minimum cost can be important. The proposed model can be extended in several ways. For example, it may seem interesting to consider the existence of a reorder point before the stock is depleted, which could improve the profit due to the increased demand caused by a higher level of stock. Another way would be the incorporation of changes in selling price, variations in demand rate, discounts in purchasing cost, deteriorating items, etc. Also, we might consider other bivariate mathematical functions for the cumulative holding cost and extend the deterministic demand function to stochastic fluctuating demand patterns. Acknowledgment This work is partly supported by the Spanish Ministry of Science and Innovation through the research project MTM2010-18591.
Table 4 Cumulative holding cost Hðt; xÞ. t
x
1 2 3 4 5 6
25
50
100
200
12.5 25 47.5 70 92.5 115
25 50 95 140 185 240
50 130 210 310 410 520
110 280 460 670 890 1120
We assume that the regular storage cost is 0.1 currency units per item and per unit time. Thus, for example, if x ¼100 and t¼ 4, the cumulative holding cost is the sum of the regular storage cost plus the insurance cost, that is (0.1)(4)(100)+270¼310 currency units. In a similar way, the cumulative holding costs given in Table 4 are obtained. The parameters h, γ 1 and γ 2 can be estimated with the values in Table 4 using a non-linear regression. Thus, the values h ¼0.31, γ 1 ¼ 1:28 and γ 2 ¼ 1:12 are obtained with a determination coefficient R2 ¼ 0:998. Hence, the function Hðt; xÞ ¼ 0:31t 1:28 x1:12 can adequately represent the cumulative holding cost for this inventory system.
Appendix B This appendix shows the evaluation of the holding cost per inventory cycle in a situation where it is simultaneously non-linear in time and stock level. γ Let Hðt; xÞ ¼ ht 1 xγ 2 be the cumulative holding cost for x units that have been stored during t units of time and x ¼ IðtÞ the decreasing inventory level curve with Ið0Þ ¼ q and IðTÞ ¼ 0. Furthermore, let: (i) A be the region of the plane enclosed by the curve x ¼ IðtÞ and the coordinate axes; (ii) P ¼ ft o ; t 1 ; …; t n g be the partition of the interval ½0; T into n equal pieces with t i ¼ iðT=nÞ and (iii) A(P) be the staggered region defined by the union of rectangles Ai ¼ ½t i−1 ; t i ½0; Iðt i Þ, for 1≤i≤n−1. It is clear that A is the limit of the regions A(P) as n tends to infinity and the holding cost δi for each rectangle Ai , with i ¼ 1; …; n−1, can be evaluated as: γ
¼ This appendix provides an example, which shows the utility of the γ function Hðt; xÞ ¼ ht 1 xγ 2 to evaluate the holding cost in an inventory model and how the parameters h, γ 1 and γ 2 can be estimated. We consider an inventory system for luxury items (such as expensive jewelry or designer watches) which needs to be insured while remaining stored. Table 3 gives the cumulative insurance cost for various units of time and quantity, showing that it is not linear in both, time and quantity. Table 3 Cumulative insurance cost. t
γ −1 hγ 1 θi 1 ðt i −t i−1 Þ½Iðt i Þγ 2
with t i−1 oθi o t i by the Mean Value Theorem of differential γ γ1 calculus applied in the factor ðt i 1 −t i−1 Þ. Therefore, the holding cost associated with the region A(P) is γ −1
½Iðt i Þγ 2 ðt i −t i−1 Þ n−1 γ 1 −1 ¼ hγ 1 ∑i ¼ 1 t i ½Iðt i Þγ 2 ðt i −t i−1 Þ γ 1 −1 γ −1 −hγ 1 ∑n−1 −θi 1 Þ½Iðt i Þγ 2 ðt i −t i−1 Þ i ¼ 1 ðt i
n−1 1 ∑n−1 i ¼ 1 δi ¼ hγ 1 ∑i ¼ 1 θ i
50
100
200
n−1
20 40 80 120 160 210
40 110 180 270 360 460
90 240 400 590 790 1000
γ −2
hγ 1 ðγ 1 −1Þ ∑ ηi 1 i¼1
10 20 40 60 80 100
ð18Þ
Again, using the Mean Value Theorem of differential calculus γ −1 γ −1 in factor ðt i 1 −θi 1 Þ, the second term in this expression can be rewritten as
x 25
1 2 3 4 5 6
γ
1 Þ½Iðt i Þγ 2 δi ¼ Hðt i ; Iðt i ÞÞ−Hðt i−1 ; Iðt i ÞÞ ¼ hðt i 1 −t i−1
Appendix A
½Iðt i Þγ 2 ðt i −θi Þðt i −t i−1 Þ
ð19Þ
with t i−1 o θi oηi o t i . Taking into account that ηi oT, Iðt i Þ o q, t i −θi o t i −t i−1 and t i −t i−1 ¼ T=n, it follows that (19) is bounded above by hγ 1 ðγ 1 −1ÞT γ 1 qγ 2 ðn−1Þ=n2 . Then, it converges to zero as n tends to infinity. Finally, taking the limit of the first term in (18) as n tends to infinity, we find that the holding cost for region A, that
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On the other hand, if u þ v ¼ 1; we have qo ¼ 1 and Pðqo Þ ¼ 0. That is, qn ¼ 1 is the unique positive root of PðqÞ ¼ 0. Note that, in this case, the sequence fqi g∞i ¼ 0 is constantly equal to qn , because qi ¼ qi−1 for i ¼ 1; 2; …. □
References
Fig. 7. General shape of the function P(q).
is, the holding cost per inventory cycle, is Z T t γ 1 −1 ½IðtÞγ 2 dt hγ 1 0
Appendix C Proof of Theorem 1. The function PðqÞ ¼ qξ −uq−v has the following properties: (i) Pð0Þ ¼ −v o 0 and limq-∞ PðqÞ ¼ ∞, (ii) P′ðqÞ ¼ ξqξ−1 −u, P′ð0Þ ¼ −u o0 and r ¼ ðu=ξÞ1=ðξ−1Þ is the unique root of P′ðqÞ for q 4 0 and (iii) P″ðqÞ ¼ ξðξ−1Þqξ−2 4 0 for q 4 0. Consequently, P(q) is a strictly convex function on the interval ð0; ∞Þ with a unique positive root qn , which verifies PðqÞ o 0 for q∈ð0; qn Þ and PðqÞ 40 for q∈ðqn ; ∞Þ. Moreover, qn 4 r, P′ðqÞ o0 for q∈ð0; rÞ and P′ðqÞ 40 for q∈ðr; ∞Þ. The general shape of the function P(q) is shown in Fig. 7. Now, if u þ v o 1 then qo ¼ ðu þ vÞ1=ξ o 1 and Pðqo Þ ¼ ð1−qo Þu 4 0. Similarly, if u þ v 4 1 then qo ¼ ðu þ vÞ1=ðξ−1Þ 4 1 and Pðqo Þ ¼ ðqo −1Þv 40. That is, in both cases, we have Pðqo Þ 4 0 and, hence, qn o qo and P′ðqo Þ 4 0: Thus, when u þ v≠1, it results that q1 ¼ qo −Pðqo Þ=P′ðqo Þ o qo . Moreover, from the Mean Value Theorem, there exists a point η∈ðqn ; qo Þ with Pðqo Þ ¼ Pðqo Þ−Pðqn Þ ¼ P′ðηÞðqo −qn Þ o P′ðqo Þðqo −qn Þ and, hence, qn o qo −ðPðqo Þ=P′ðqo ÞÞ ¼ q1 oqo (please, see the graphical interpretation for value q1 in Fig. 7). Now, if qn o qi−1 o qi−2 ; a similar reasoning proves that qn o qi o qi−1 and the sequence fqi g∞i ¼ 0 is strictly decreasing and bounded below. Then the sequence fqi g∞i ¼ 0 is convergent and, if we write L for the value of its limit (that is L ¼ limi-∞ qi Þ, taking into account the definition for qi, L ¼ L−ðPðLÞ=P′ðLÞÞ should occur, which implies that PðLÞ ¼ 0 and thus, L ¼ qn .
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