Demand uncertainty and learning in fuzziness in a continuous review inventory model

Abstract

This paper presents a continuous review inventory model with backorders and lost sales with fuzzy demand and learning considerations. The imprecision in demand is characterized by triangular fuzzy numbers. The triangular fuzzy numbers, counts upon lead time, are used to construct fuzzy lead time demand. It is assumed that the imprecision captured by these fuzzy numbers reduce with time because of learning effect. This implies that the decision maker gathers information about the inventory system and builds up knowledge from the previous shipments. Learning process occurs in setting and estimating the fuzzy parameters to reduce errors and costs. Under these considerations, the proposed model offers a policy and a solution algorithm to calculate the number of orders and reorder level such that the total annual cost attains a minimum value. The results of the proposed model are compared with the continuous review inventory system with fuzzy demand with or without learning effect. It is shown that learning effect in fuzziness reduces the ambiguity associated with the decision making process. Finally, numerical examples are provided to illustrate the importance of using learning in fuzzy model. The convexity of the total cost function is also proved.

Keywords

Uncertainty continuous review inventory model possibilistic mean value learning in fuzziness

1 Introduction

Inventory systems are usually common and significant in all production processes, services, and business operations. Every organization relentlessly strives to uphold the optimum inventory to meet its requirements to avoid over or under inventory that can impact the financial figures. Inventory is always dynamic, which requires a very careful evaluation and control of external and internal factors through planning and reviews for timely replenishment. In this framework, continuous review inventory system is the most suitable mathematical model to deal with such type of problems as well as various types of uncertainties and imprecision are inherent in modeling such inventory problems. Uncertainties such as fuzziness or vagueness in inventory problems may be associated with demand, lead time, or various relevant costs. A great deal of research efforts have been devoted to it over the past few decades, see [1 –9]. Some other researchers explored fuzziness in some more multifaceted settings than those works mentioned above. Vijayan and Kumaran [10] investigated the mixed (Q, r) and (R, T) inventory models including trapezoidal fuzzy costs and examined the impact of imprecision on total system cost at each stage of the model. Dutta, Chakraborty, and Roy [11] developed a (Q, r) inventory system with uncertain demand using a fuzzy optimization approach. In a recent article, Dash and Sahoo [12] discussed optimal solution for a single period inventory model with fuzzy cost and demand as a fuzzy random variable. Sarkar and Mahapatra [13] developed a periodic review inventory model with fuzzy demand and costs under controllable lead time. The essence of all the papers discussed above is to examine the effect of imprecision in decision variables that affects the performance of the organization. There is a vast crisp inventory literatures under different conditions, the outline, which can be found in Cárdenas-Barrón [14 –16], Lin [17], and Sarkar and Moon [18], Moon et al. [19], Sarkar et al. [20 –22], Sarkar [23] and their references. In the present day scenario, these objectives are very difficult to accomplish owing to the great number of factors involved due to uncertainty in modeling parameters. For instance, due to changes in the supply chain environment, the demand of a commodity is ill-defined and may vary from one cycle to another. Thus, in order to select the best operating strategy, fuzzy inventory models with imprecise demand should be studied extensively.

A closer look at the existing literature reveals that inspite of various models developed in fuzzy environment, very few of them were devoted to model role of human factor in the problem, while the entire focus was mainly on modeling the fuzziness associated with the planning period. Also, the models developed till now treated human capabilities or the degree of fuzziness that decision maker encounters as a crisp value and ignored the fact that human capabilities are subject to change with time which can have a drastic affect in estimating the fuzzy parameters. Hence, the variation in information available to a decision-maker and its impact on the degree of fuzziness has not been considered yet. Thus, it is apparent that the developed inventory models present an imperfect picture of real-world’s inventory planning problems, which influence the planning outcome. In order to present a better representation of reality, this paper assumes that the degree of fuzziness on which a decision is subject to may change over time. For instance, in a real life situation, it is difficult for a decision maker to forecast the future demand of a commodity, when the only information available is the size of the first order. Due to lack of historical data available, the decision maker resorts to specifying the upper and lower bounds between which the customer demand lies. Gradually, with more orders placed by the customer, the decision maker will gain more information and gets acquainted with the customer’s demand behavior and ordering pattern, which enables them for an exact prediction of the demand pattern. Hence, the manager finally learns with time, thus reducing the degree of ambiguity that makes the decision process indistinct.

The concept of learning effect based reduction in fuzziness has immense applications in various fields. Since the pioneering work of Wright [24] on effect of learning in repetitive task, several researchers have studied and extended the effect of learning to optimal lot sizing problems. A closer look at the literature available reveals that works related to learning in the perspective of inventory models involving fuzzy parameters are very less. Readers may refer to the exceptional works of Bera, Mahapatra, and Maiti [25] and Pal, Maiti, and Maiti [26], who assumed that either setup costs or production costs reduce because of learning. Glock, Kurt, and Jaber [27] examined the effect of learning in an inventory model with fuzzy demand and learning in fuzziness. Kazemi et al. [28] developed a fuzzy economic order quantity model for imperfect quality items using the learning effect on fuzzy parameters. In a recent article, Shekarian et al. [29] developed an economic order quantity model taking into consideration different holding costs for imperfect quality items subject to fuzziness and learning.

This paper applies the concept of learning effect in fuzziness to a continuous review inventory model with backorders under fuzzy environment. We assume average annual demand to be imprecise quantity characterized by triangular fuzzy numbers. Any ambiguity in the annual demand will be apparently reflected in the lead time demand. Hence, triangular fuzzy number measures on lead time are used to build up fuzzy lead time demand. The imprecision captured by these fuzzy number counts reduces with the learning curve. The learning process predicts and monitors the performance of an entity, a group of individuals or an organization, which improves with time, e.g., Argote [30]. This relationship is captured by learning curves, which is a decrease in the effort required for producing each unit in a repetitive manufacturing operation. Factors contributing to this improved performance include more effective use of tools and machines, increased acquaintance with operation tasks and work environment, and enhanced management competence. This concept has been extensively used and applied in various sectors (logistics, production, inventory systems, quality control, negotiated purchasing, technology assessment, project management, banking, software training, health-care management, and more), (refer Jaber [31]).

The aim and impetus behind this paper is to examine the impact of learning to reduce fuzziness and also to study how this reduction affects the operating strategy to reduce the total system cost of a continuous review inventory in a fuzzy environment. With this objective, this paper applies the concept of learning in fuzziness to a continuous review inventory model with backorders under fuzzy environment. The impact of learning in fuzziness on the inventory policy is analyzed by comparing the results of the proposed model with the fuzzy continuous review inventory model (with no learning). The results of the paper have profound inferences for the managers operating in an uncertain environment and whose knowledge base is expected to improve with a passage of time because of learning. As the learning phenomenon is believed to have economic and decision-making repercussions in an inventory management system, hence this paper contributes to the area of fuzzy inventory by assuming that the decision makers can use their knowledge in setting the fuzzy parameters thereby reducing the ambiguity associated with inventory system. It is for the first time that learning in fuzziness on a continuous review inventory system with fuzzy demand is examined.

The rest of the paper is structured as follows: Section 2 outlines the preliminaries. Section 3 develops a continuous review inventory system under a fuzzy knowledge base. Using possibilistic mean value for defuzzification, the estimation of expected total cost is derived in the fuzzy sense. In Section 4, the model developed in a fuzzy framework to incorporate learning effect in fuzziness. Section 5 elucidates the proposed model with numerical examples, results, and analysis. Section 6 concludes the paper and lines of further studies are outlined.

2 Preliminary concepts

Definition 1. A triangular fuzzy number $\tilde{X}$ defined on the space of real numbers $ℝ$ can be characterized by a triplet (a - Δ₁, a, a + Δ₂) and is defined by the membership function $μ_{\tilde{X}}$ , as $μ_{\tilde{X}} (x) = {\begin{matrix} \frac{x - a + Δ_{1}}{Δ_{1}} & for a - Δ_{1} \leq x \leq a \\ \frac{a + Δ_{2} - x}{Δ_{2}} & for a \leq x \leq a + Δ_{2} \\ 0 & otherwise \end{matrix}$ (1)

Definition 2. The α-cut of $\tilde{X} = (a_{1}, a_{2}, a_{3}) = (a - Δ_{1}, a, a + Δ_{2})$ where 0 ≤ α ≤ 1 is X (α) = [X_L (α) , X_R (α)], where X_L (α) and X_R (α) are the left and right end points of X (α) and are defined as follows:

$\begin{matrix} X_{L} (α) & = & a - Δ_{1} (1 - α) and \\ X_{R} (α) & = & a + Δ_{2} (1 - α) \end{matrix}$ (2)

Definition 3. For a given fuzzy number $\tilde{X}$ , the interval-valued possibilistic mean is defined as $M (\tilde{X}) = [M_{L} (\tilde{X}), M_{R} (\tilde{X})]$ where $M_{L} (\tilde{X})$ and $M_{R} (\tilde{X})$ are the lower and upper limits of $\tilde{X}$ are, respectively, defined by Carlsson and Fullěr [32] $M_{L} (\tilde{X}) = \frac{\int_{0}^{1} α X_{L} (α) d α}{\int_{0}^{1} α d α} and M_{R} (\tilde{X}) = \frac{\int_{0}^{1} α X_{R} (α) d α}{\int_{0}^{1} α d α}$

The possibilistic mean of $\tilde{X}$ is then defined as $\bar{M} (\tilde{X}) = \frac{M_{L} (\tilde{X}) + M_{R} (\tilde{X})}{2}$

In other words, it can be written as $\bar{M} (\tilde{X}) = \int_{0}^{1} α (X_{L} (α) + X_{R} (α)) d α$ (3)

3 Mathematical modeling and analysis

3.1 Fuzzy continuous review inventory model (Model I)

Notation

Decision variables

number of orders per year (integer number)

reorder point (units)

lead time (in weeks)

Parameters

\tilde{D}

annual demand (Triangular fuzzy number)

{\tilde{d}}_{L}

demand during lead time (Triangular fuzzy number)

Lotsize or order quantity (units)

ordering cost per order ($/order)

inventory holding cost per unit per year

shortage cost per unit ($/unit shortage)

fraction of demand backordered during stock out period, 0 < β < 1

lost sales cost per unit short ($/unit short)

M (d_L)

expected lead timedemand

M (d_L - r) ⁺

expected shortage quantity at the end of each cycle

Assumptions

Inventory is continuously reviewed. Replenishments are made, whenever the inventory level falls to the reorder point r.

Reorder point r is positive. The safety stock = r – expected lead time demand, is positive.

Annual demand and demand during lead time are imprecise in nature and represented by $\tilde{D} = (D - Δ_{1}, D, D + Δ_{2})$ and ${\tilde{d}}_{L} = L \otimes \tilde{D}$ , respectively.

The lead time L has m mutually independent components each having a different crashing cost for reducing lead time. The ith component has a minimum duration a_i and normal duration b_i, and a crashing cost per unit time c_i. Furthermore, it is assumed that c₁ ≤ c₂ ≤ … ≤ c_m.

Let $L_{0} = \sum_{j = 1}^{m} a_{j}$ and L_i be the length of the lead time with components 1, 2, …, i crashed to their minimum duration. Then L_i can be written as $L_{i} = L_{0} - \sum_{j = 1}^{i} (b_{j} - a_{j})$ and the lead time crashing cost per cycle C (L) is expressed for L ∈ [L_i, L_i-1] as

$\begin{matrix} C (L) & = & c_{i} (L_{i} - L) + \sum_{j = 1}^{i - 1} c_{j} (b_{j} - a_{j}) and \\ C (L_{0}) & = & 0 \end{matrix}$ (4)

The expression for annual variable cost in a crisp sense under the model Moon and Choi [33] is given by

$\begin{matrix} TC (Q, r, L) \\ = ordering cost + holding cost + stockout cost \\ + lead time crashing cost \\ = \frac{AD}{Q} + h [\frac{Q}{2} + r - M (d_{L}) + (1 - β) B (r)] \\ + \frac{D}{Q} [s + π (1 - β)] B (r) + \frac{D}{Q} C (L) \end{matrix}$ (5) where B (r) is expected number of shortages per cycle.

As the demand and demand during lead time are taken as imprecise quantities, thus the value of lead time demand may have a little variation in an uncertain environment. Usually, the demand estimation during this short lead time period is based on the subjective management decision and it can be imprecisely expressed. To capture reality in a more efficient way, this paper assumes demand and demand during lead time as fuzzy numbers. When these parameters are considered as fuzzy numbers, the total annual cost also becomes a fuzzy number. Here we characterize all the fuzzy observations of demand and demand during lead time are characterized as triangular fuzzy numbers. Thus, the expression given in (5) under a fuzzy framework can be formulated as $\begin{matrix} \tilde{TC} (Q, r, L) \\ = \frac{\tilde{D}}{Q} A + h (\frac{Q}{2} + r - M ({\tilde{d}}_{L}) + M {({\tilde{d}}_{L} - r)}^{+} (1 - β)) \\ + \frac{\tilde{D}}{Q} M {({\tilde{d}}_{L} - r)}^{+} [s + π (1 - β)] + \frac{\tilde{D}}{Q} C (L) \end{matrix}$ (6) where $M {({\tilde{d}}_{L} - r)}^{+}$ is the expected or possible shortages during each cycle.

Also, the expected value of fuzzy lead time demand ${\tilde{d}}_{L}$ in the possibilistic sense is given by $\bar{M} ({\tilde{d}}_{L}) = \int_{0}^{1} α (d_{L}^{-} (α) + d_{L}^{+} (α)) d α$ (7)

The possibilistic mean of total costper year in fuzzy sense is a function of Q, r, and L. Assuming $\bar{M} (\tilde{TC})$ as the possibilistic mean value of $\tilde{TC} (Q, r, L)$ , it is calculated as $\bar{M} (\tilde{TC}) = \int_{0}^{1} α ({\tilde{TC}}_{α}^{-} + {\tilde{TC}}_{α}^{+}) d α for all α \in [0, 1]$ (8) where ${\tilde{TC}}_{α}^{-}$ and ${\tilde{TC}}_{α}^{+}$ are given by $\begin{matrix} {\tilde{TC}}_{α}^{-} (Q, r, L) \\ = h (\frac{Q}{2} + r - d_{L}^{+} (α) + M {({\tilde{d}}_{L} - r)}^{+} (1 - β)) \\ + M {({\tilde{d}}_{L} - r)}^{+} [s + π (1 - β)] \frac{{\tilde{D}}_{α}^{-}}{Q} \\ + (A + C (L)) \frac{{\tilde{D}}_{α}^{-}}{Q} \end{matrix}$ (9) $\begin{matrix} {\tilde{TC}}_{α}^{+} (Q, r, L) \\ = h (\frac{Q}{2} + r - d_{L}^{-} (α) + M {({\tilde{d}}_{L} - r)}^{+} (1 - β)) \\ + M {({\tilde{d}}_{L} - r)}^{+} [s + π (1 - β)] \frac{{\tilde{D}}_{α}^{+}}{Q} \\ + (A + C (L)) \frac{{\tilde{D}}_{α}^{+}}{Q} \end{matrix}$ (10)

Substituting (7) and (8) in (6) gives the expected total cost per year in the possibilistic sense and is given by $\begin{matrix} \bar{M} (\tilde{TC}) \\ = h (\frac{Q}{2} + r - \bar{M} ({\tilde{d}}_{L}) + (1 - β) M {({\tilde{d}}_{L} - r)}^{+}) \\ + \frac{A}{Q} (D + \frac{Δ_{2} - Δ_{1}}{6}) \\ + M {({\tilde{d}}_{L} - r)}^{+} [\frac{s + π (1 - β)}{Q} (D + \frac{Δ_{2} - Δ_{1}}{6})] \\ + \frac{C (L)}{Q} (D + \frac{Δ_{2} - Δ_{1}}{6}) \end{matrix}$ (11)

where $M {({\tilde{d}}_{L} - r)}^{+}$ is the expected shortage in the possibilistic sense, which has been further explained in Section 3.2.

To validate the viability of the model with respect to the number of orders, (11) can be re-formulated to account for the number of orders per year which follows $\begin{matrix} \bar{M} (n, r, L) \\ = h (\frac{D}{2 n} + r - \bar{M} ({\tilde{d}}_{L}) + (1 - β) M {({\tilde{d}}_{L} - r)}^{+}) \\ + n (A + C (L)) + \frac{n (A + C (L))}{D} (\frac{Δ_{2} - Δ_{1}}{6}) \\ + nM {({\tilde{d}}_{L} - r)}^{+} (s + π (1 - β)) (1 + \frac{Δ_{2} - Δ_{1}}{6 D}) \end{matrix}$ (12) where L ∈ [L_i, L_i-1] , i = 1, 2, … m. In order to obtain the optimal solution, it is needed to derive the expression of $M {({\tilde{d}}_{L} - r)}^{+}$ for a given r in $[\bar{M} ({\tilde{d}}_{L}), {\tilde{d}}_{L}]$ .

3.2 Calculation of expected shortages

Referring to the study of Dutta, Chakraborty, and Roy [4], the exact expression of $M {({\tilde{d}}_{L} - r)}^{+}$ can be obtained as follows:

Case 1. For L (D - Δ₁) ≤ r ≤ LD, we have the α - level set of lead time demand in under-stocking situation (where demand rate is greater than r) ${\tilde{d}}_{L} (α) = {\begin{matrix} [r, d_{L}^{+} (α)] for α \leq L (r) \\ [d_{L}^{-} (α), d_{L}^{+} (α)] for α > L (r) \end{matrix}$ which implies that $\begin{matrix} {({\tilde{d}}_{L} - r)}^{+} (α) \\ = {\begin{matrix} [0, d_{L}^{+} (α) - r] for α \leq L (r) \\ [d_{L}^{-} (α) - r, d_{L}^{+} (α) - r] for α > L (r) \end{matrix} \end{matrix}$

Using (3), the possibilistic mean of shortages is calculated as $\begin{matrix} M {({\tilde{d}}_{L} - r)}^{+} \\ = \int_{0}^{1} α [{({({\tilde{d}}_{L} - r)}^{+})}^{-} (α) + {({({\tilde{d}}_{L} - r)}^{+})}^{+} (α)] d α \\ = \int_{0}^{1} α (d_{L}^{+} (α) - r) d α + \int_{L (r)}^{1} α (d_{L}^{-} (α) - r + d_{L}^{+} (α) - r) d α \\ = \int_{0}^{1} α (d_{L}^{+} (α) - r) d α + \int_{L (r)}^{1} α (d_{L}^{-} (α) - r) d α \\ = \int_{0}^{1} α d_{L}^{+} (α) d α + \int_{L (r)}^{1} α d_{L}^{-} (α) d α - r (1 - 0.5 {(L (r))}^{2}) \\ = \frac{(LD - r)}{2} + \frac{L Δ_{2}}{6} + \frac{{(LD - r)}^{2}}{2 L Δ_{1}} - \frac{{(LD - r)}^{3}}{6 L^{2} Δ_{1}^{2}} \end{matrix}$ (13)

Case 2. For LD ≤ r ≤ L (D + Δ₂), the possibilistic mean of probable shortages is calculated as $\begin{matrix} M {({\tilde{d}}_{L} - r)}^{+} \\ = \int_{0}^{1} α [{({({\tilde{d}}_{L} - r)}^{+})}^{-} (α) + {({({\tilde{d}}_{L} - r)}^{+})}^{+} (α)] d α \\ = \int_{0}^{R (r)} α d_{L}^{+} (α) d α - 0.5 r {(R (r))}^{2} \\ = (\frac{L (D + Δ_{2}) - r}{6}) {(\frac{L (D + Δ_{2}) - r}{L Δ_{2}})}^{2} \end{matrix}$

Thus, $M {({\tilde{d}}_{L} - r)}^{+} = \frac{{(L (D + Δ_{2}) - r)}^{3}}{6 L^{2} Δ_{2}^{2}} .$ (14)

3.3 Optimal solution

This section provides the solution procedure for the problem of determining the optimal number of orders (n), optimal reorder point (r), and for optimal lead time (L) such that the total expected annual cost in the fuzzy sense has a minimum value.

Let S is a set of L, where L ∈ [L_i, L_i-1] , i = 1, 2 … , m, then the simultaneous equations for solving n and r and the necessary conditions to obtain optimality are derived as follows:

Case 1. For L (D - Δ₁) ≤ r ≤ LD, one has $\begin{matrix} \bar{M} (n, r, L) \\ = \int_{0}^{1} α ({TC}_{α}^{-} + {TC}_{α}^{+}) d α \\ = h (\frac{D}{2 n} + r - \bar{M} ({\tilde{d}}_{L}) + M {({\tilde{d}}_{L} - r)}^{+} (1 - β)) \\ + n (A + C (L)) + \frac{n (A + C (L))}{D} (\frac{Δ_{2} - Δ_{1}}{6}) \\ + nM {({\tilde{d}}_{L} - r)}^{+} (s + π (1 - β)) \\ (1 + (\frac{Δ_{2} - Δ_{1}}{6 D})) \end{matrix}$ (15)

Now, the optimal values of n and r have to be calculated. For fixed L ∈ [L_i, L_i-1] , i = 1, 2, … m, the partial derivatives of (15) with respect to n and r are taken by assuming n as continuous variable and these results are as follows: $\begin{matrix} \frac{\partial \bar{M} (n, r, L)}{\partial n} \\ = - h (\frac{D}{2 n^{2}}) + A + C (L) \\ + (A + C (L)) (\frac{Δ_{2} - Δ_{1}}{6 D}) + M {({\tilde{d}}_{L} - r)}^{+} \\ (s + π (1 - β)) (1 + (\frac{Δ_{2} - Δ_{1}}{6 D})) \end{matrix}$ (16) $\begin{matrix} \frac{\partial \bar{M} (n, r, L)}{\partial r} \\ = h + h (- \frac{1}{2} - \frac{LD - r}{L Δ_{1}} + \frac{{(LD - r)}^{2}}{2 L^{2} Δ_{1}^{2}}) (1 - β) \\ + (- \frac{1}{2} - \frac{LD - r}{L Δ_{1}} + \frac{{(LD - r)}^{2}}{2 L^{2} Δ_{1}^{2}}) \\ n (s + π (1 - β)) (1 + \frac{Δ_{2} - Δ_{1}}{6 D}) \end{matrix}$ (17)

Equating $\frac{\partial \bar{M} (n, r, L)}{\partial n} = 0,$ one has $n^{2} = \frac{hD}{2 [((A + CL) (1 + \frac{Δ_{2} - Δ_{1}}{6 D}) + M {({\tilde{d}}_{L} - r)}^{+} (s + π (1 - β))) (1 + \frac{Δ_{2} - Δ_{1}}{6 D})]}$ (18)

Equating $\frac{\partial \bar{M} (n, r, L)}{\partial r} = 0,$ the expression for n becomes $n = \frac{[h + h (1 - β) X]}{X (s + π (1 - β)) (1 + \frac{Δ_{2} - Δ_{1}}{6 D})}$ (19) where $X = \frac{1}{2} + \frac{LD - r}{L Δ_{1}} - \frac{{(LD - r)}^{2}}{2 L^{2} Δ_{1}^{2}}$ .

By equating the square of (19) with (18), the value of r (say r⁽⁰⁾ (L)) can be calculated using any numerical programming tools. By substituting r⁽⁰⁾ (L) into (19), one can calculate the value of n (sayn⁽⁰⁾).

Next, we use following condition to determine integer value of n (say n⁽⁰⁾ (L)) $\begin{matrix} \bar{M} (n^{(0)} (L), r^{(0)} (L), L) \\ = min {\bar{M} (n_{1}, r, L), \bar{M} (n_{2}, r, L)} \end{matrix}$ where n₁ =⌊ n⁽⁰⁾ ⌋ and n₂ = ⌊ n⁽⁰⁾ ⌋ + 1.

Case 2. For LD ≤ r ≤ L (D + Δ₂), one has $\begin{matrix} \bar{M} (n, r, L) \\ = \int_{0}^{1} α ({TC}_{α}^{-} + {TC}_{α}^{+}) d α \\ = h (\frac{D}{2 n} + r - \bar{M} ({\tilde{d}}_{L}) + M {({\tilde{d}}_{L} - r)}^{+} (1 - β)) \\ + n (A + C (L)) + \frac{n (A + C (L))}{D} \\ (\frac{Δ_{2} - Δ_{1}}{6}) + M {({\tilde{d}}_{L} - r)}^{+} \\ (n (s + π (1 - β)) + (s + π (1 - β)) \frac{n}{D} (\frac{Δ_{2} - Δ_{1}}{6})) \end{matrix}$ (20)

For fixed L ∈ [L_i, L_i-1] , i = 1, 2, … m, one can take the partial derivatives of (20) with respect to n and r and the results are as follows: $\begin{matrix} \frac{\partial \bar{M} (n, r, L)}{\partial n} \\ = - \frac{hD}{2 n^{2}} + (A + C (L)) (1 + \frac{Δ_{2} - Δ_{1}}{6 D}) \\ + M {({\tilde{d}}_{L} - r)}^{+} (s + π (1 - β)) \\ + \frac{M {({\tilde{d}}_{L} - r)}^{+} (s + π (1 - β)) (Δ_{2} - Δ_{1})}{6 D} \end{matrix}$ (21) $\begin{matrix} \frac{\partial \bar{M} (n, r, L)}{\partial r} \\ = h - \frac{h (1 - β) {(L (D + Δ_{2}) - r)}^{2}}{2 L^{2} Δ_{2}^{2}} \\ - \frac{{(L (D + Δ_{2}) - r)}^{2} n (s + π (1 - β))}{2 L^{2} Δ_{2}^{2}} \\ - \frac{{(L (D + Δ_{2}) - r)}^{2} n (s + π (1 - β)) (Δ_{2} - Δ_{1})}{12 L^{2} Δ_{2}^{2} D} \end{matrix}$ (22)

Equating $\frac{\partial \bar{M} (n, r, L)}{\partial r} = 0,$ the result becomes $n = \frac{h (L^{2} ((β - 1) (D^{2} + 2 D Δ_{2}) + Δ_{2}^{2} (1 + β) + 2) - 2 rL (D + Δ_{2}) (β - 1) + r^{2} (β - 1))}{{(L (D + Δ_{2}) - r)}^{2} (s + π (1 - β)) (1 + \frac{Δ_{2} - Δ_{1}}{6 D})} . (24)$ (24)

By equating the square of (24) with (23), the value of r (say r⁽⁰⁾ (L)) can be calculated using any numerical programming tools. By substituting r⁽⁰⁾ (L) into (24), one can calculate the value of n (say n⁽⁰⁾).

Now, the following condition can be used to determine integer value of n (say n⁽⁰⁾ (L)) $\begin{matrix} \bar{M} (n^{(0)} (L), r^{(0)} (L), L) \\ = min {\bar{M} (n_{1}, r, L), \bar{M} (n_{2}, r, L)} \end{matrix}$ where n₁ =⌊ n⁽⁰⁾ ⌋ and n₂ = ⌊ n⁽⁰⁾ ⌋ + 1.

For fixed i∈ { 1, 2, … m }, by numerical analysis method, one can find $L_{i}^{0} \in [L_{i}, L_{i - 1}]$ such that

$\begin{matrix} \bar{M} (n^{(0)} (L_{i}^{(0)}), r^{(0)} (L_{i}^{(0)}), L_{i}^{(0)}) \\ = min_{L \in [L_{i}, L_{i - 1}]} \bar{M} (n^{(0)} (L), r^{(0)} (L), L) . \end{matrix}$ (25)

Moreover, for each i = 1, 2, … m, the value of $\bar{M} (n^{(0)} (L_{i}^{(0)}), r^{(0)} (L_{i}^{(0)}), L_{i}^{(0)})$ is evaluated and set as $\begin{matrix} \bar{M} (n_{f}^{*}, r_{f}^{*}, L^{*}) \\ = min_{1 \leq i \leq m} \bar{M} (n^{(0)} (L_{i}^{(0)}), r^{(0)} (L_{i}^{(0)}), L_{i}^{(0)}) \end{matrix}$

Thus, the optimal number of orders is $n_{f}^{*}$ , the optimal reorder point is $r_{f}^{*}$ , and the optimal lead time is L^*.

3.4 Sufficient Condition to obtain the minimum of

\bar{M} (n, r, L)

A sufficient condition for $n_{f}^{*}$ and $r_{f}^{*}$ to attain the minimum of $M {({\tilde{d}}_{L} - r)}^{+}$ is the convexity of $\bar{M} (n, r, L)$ . For $\bar{M} (n, r, L)$ to be a convex function for fixed L ∈ [L_i, L_i-1], its Hessian matrix must be positive definite at $(n_{f}^{*}, r_{f}^{*})$ , which requires $\frac{\partial^{2} \bar{M} (n, r)}{\partial n^{2}} > 0$ and $\frac{\partial^{2} \bar{M} (n, r)}{\partial n^{2}} \frac{\partial^{2} \bar{M} (n, r)}{\partial r^{2}} - {(\frac{\partial^{2} \bar{M} (n, r)}{\partial n \partial r})}^{2} > 0$ , [See Appendix 1].

For both cases discussed above, the sufficient condition as well as the restriction on the range of $r^{*} \geq \bar{M} ({\tilde{d}}_{L})$ must be satisfied simultaneously, and hence at $(n_{f}^{*}, r_{f}^{*})$ $\frac{\partial {\bar{M}}^{2} (n, r)}{\partial n^{2}} = h (\frac{D}{n^{3}})$ is always positive. The sufficient condition reduces to $\frac{\partial^{2} \bar{M} (n, r)}{\partial n^{2}} \frac{\partial^{2} \bar{M} (n, r)}{\partial r^{2}} - {(\frac{\partial^{2} \bar{M} (n, r)}{\partial n \partial r})}^{2} > 0$ , [See Appendix 2].

As the Hessian of the objective function is positive definite at $(n_{f}^{*}, r_{f}^{*})$ , thus a minimum at $(n_{f}^{*}, r_{f}^{*})$ is obtained (refer Dutta [4]).

The detailed expressions for both the cases can be found in Appendix 1.

4 A fuzzy continuous review inventory system and learning in fuzziness (Model II)

This section extends the fuzzy model developed in previous section to justify for the effect of learning in fuzziness. Learning in fuzziness is assumed to follow the mathematical function proposed by Wright [24] and extensively used by many researchers (e.g., Jaber [34]; Yelle [35]), which follows theform $y (t) = y (1) t^{- l}$ (26) where y (t) is the performance at time t, y (1) is the performance at the beginning of the planning period, t is the time count, and l is the learningexponent.

Let us assume that n orders are placed annually (i.e., an order every 365/n days). If learning occurs as a function of the number of orders placed and if it affects the fuzzy variables Δ₁ and Δ₂ subject to the same learning rate, then the value of the fuzzy parameter j at the time of the ith order is given by the expression (Glock, Kurt, and Jaber [27]) $Δ_{j, i} = {\begin{matrix} Δ_{j, i}, & i = 1 \\ Δ_{j, i} {((i - 1) \frac{365}{n})}^{- l}, & otherwise \end{matrix}$ (27)

Equation (27) indicates that learning takes place only after the first order has been issued. Thus, Δ_1,1 and Δ_2,1 are assigned to the fuzzy parameters for the first and lower values for all the following orders.

For fixed L ∈ [L_i, L_i-1] , i = 1, 2, … m, the expected total cost for order i with i > 1 is thus given as

$\begin{matrix} {\bar{M}}_{i} (n, r, L) = (A + C (L)) \\ + \frac{h}{n} (\frac{D}{2 n} + r - DL + (1 - β) M {({\tilde{d}}_{L} - r)}^{+}) \\ + M {({\tilde{d}}_{L} - r)}^{+} (s + π (1 - β)) \\ + \frac{1}{6 Dn} (Δ_{2, 1} - Δ_{1, 1}) {((i - 1) \frac{365}{n})}^{- l} \\ \times (A + C (L) + hD + (s + π (1 - β)) \\ M {({\tilde{d}}_{L} - r)}^{+}) \end{matrix}$ (28) where the expression for $M {({\tilde{d}}_{L} - r)}^{+}$ can be obtained from (13) or (14) respectively, as

$\begin{matrix} M {({\tilde{d}}_{L} - r)}^{+} \\ = \frac{(LD - r)}{2} + \frac{L Δ_{2, 1} {((i - 1) \frac{365}{n})}^{- l}}{6} \\ + \frac{{(LD - r)}^{2}}{2 L Δ_{1, 1} {((i - 1) \frac{365}{n})}^{- l}} \\ - \frac{{(LD - r)}^{3}}{6 L^{2} {(Δ_{1, 1} {((i - 1) \frac{365}{n})}^{- l})}^{2}} \end{matrix}$ (29) if $L (D - Δ_{1, 1} {((i - 1) \frac{365}{n})}^{- l}) \leq r \leq LD$ or

$\begin{matrix} M {({\tilde{d}}_{L} - r)}^{+} \\ = \frac{{(L (D + Δ_{2, 1} {((i - 1) \frac{365}{n})}^{- l}) - r)}^{3}}{6 L^{2} {(Δ_{2, 1} {((i - 1) \frac{365}{n})}^{- l})}^{2}} \\ if LD \leq r \leq L (D + Δ_{2, 1} {((i - 1) \frac{365}{n})}^{- l}) \end{matrix}$ (30)

Hence, the expected total cost for n orders with learning in fuzziness is given by $\begin{matrix} \bar{M} (n, r, L) \\ = {\begin{matrix} {\bar{M}}_{1} (n, r, L) if L (D - Δ_{1, 1} {((i - 1) \frac{365}{n})}^{- l}) \leq r \leq LD \\ {\bar{M}}_{2} (n, r, L) if LD \leq r \leq L (D + Δ_{2, 1} {((i - 1) \frac{365}{n})}^{- l}) \end{matrix} \end{matrix}$

where the expressions for ${\bar{M}}_{1} (n, r, L)$ and ${\bar{M}}_{2} (n, r, L)$ are provided in Appendix 2.

The following algorithmic procedure is followed to determine the optimal solutions for n^*, r^* and L^*.

4.1 Computational algorithm

Step 1. For each L ∈ [L_i, L_i-1] , i = 1, 2, 3 … m, perform step (i) to (iv)

Choose two initial trial values of n (say $n = n_{f}^{*} and n_{f}^{*} - 1$ ). Obtain value for r (denoted by r (L)) by solving $\partial {\bar{M}}_{1} (n, r, L) / \partial r = 0$ or $\partial {\bar{M}}_{2} (n, r, L) / \partial r = 0$ and compute the corresponding values of $\bar{M} (n, r, L) and \bar{M} (n - 1, r, L)$ , respectively.

If $\bar{M} (n, r, L) \geq \bar{M} (n - 1, r, L)$ , then compute $\bar{M} (n - 2, r, L), \bar{M} (n - 3, r, L) \dots$ until the result becomes $\bar{M} (k - 1, r, L) \geq \bar{M} (k, r, L) .$ Set n (L) = k.

If $\bar{M} (n, r, L) \leq \bar{M} (n - 1, r, L)$ , then compute $\bar{M} (n + 1, r, L), \bar{M} (n + 2, r, L) \dots$ until it becomes $\bar{M} (k, r, L) \leq \bar{M} (k + 1, r, L) .$ Set n (L) = k.

Set $\bar{M} (n (L_{i}^{(0)}), r (L_{i}^{(0)}), L_{i}^{(0)}) = \underset{L \in [L_{i}, L_{i - 1}]}{Min} \bar{M} (n (L), r (L), L)$ .

Step 2. Set

\bar{M} (n^{*}, r^{*}, L^{*}) = \underset{i = 1, 2, \dots, m}{Min} \bar{M} (n (L_{i}^{(0)}), r (L_{i}^{(0)}), L_{i}^{(0)})

5 Numerical experiments and managerial insights

Example 1. In order to validate the applicability of the above models in fuzzy environment, let us consider an inventory system where the annual customer demand of a product is ill-defined, i.e., the value of $\tilde{D}$ is linguistic and is expressed as “demand is around 600 units”. This demand can be quantified by a triangular fuzzy number with the left and right spreads as Δ₁ = 35, Δ₂ = 60, respectively. The values of the other parameters for Model I in appropriate units are: A = $200/order, h = $20/unit/year, s = $50/unit/year, π = $150/unit, β = 0.5. The lead time has three components with the data shown in Table 1.

Table 1
Lead time for different components

Lead time component i Normal duration b_i Minimum duration b_i - a_i Unit crashing

(days) a _i (weeks) cost, c_i ($/day)

(days)

1 20 6 2 0.4

2 20 6 2 1.2

3 16 9 1 5.0

Lead time component i	Normal duration b_i	Minimum duration	b_i - a_i	Unit crashing
1	20	6	2	0.4
2	20	6	2	1.2
3	16	9	1	5.0

For i = 1, L ∈ [L₁, L₀] = [42, 56], the results, obtained for the given dataset for the continuous review inventory system in a fuzzy environment with no learning (Model I), are given in Table 2.

Table 2

Results of solution procedure for Model I for L ∈ [L₁, L₀] = [42, 56]

L (days)	C (L)	n	r	$\bar{M} (d_{L})$	$\bar{M} (n, r, L)$
56	0	5	99.23	92.69	2347.95
55	0.4	5	97.46	91.04	2347.45
54	0.8	5	95.69	89.38	2346.95
53	1.2	5	93.91	87.73	2346.44
52	1.6	5	92.14	86.07	2345.94
51	2.0	5	90.37	84.42	2345.43
50	2.4	5	88.60	82.76	2344.93
49	2.8	5	86.83	81.11	2344.43
48	3.2	5	85.05	79.45	2343.92
47	3.6	5	83.28	77.80	2343.42
46	4.0	5	81.51	76.14	2342.91
45	4.4	5	79.74	74.49	2342.41
44	4.8	5	77.97	72.83	2341.90
43	5.2	5	76.19	71.18	2341.40
42	5.6	5	74.42	69.52	2340.90

“Bold” values within Table 2 indicate the optimum values.

From Table 2, we find that the minimum value of $\bar{M} (n^{(0)} (L), r^{(0)} (L), L)$ for L ∈ [L₁, L₀] is $2340.90, which occurred at L = 42 days. From (25), this solution is denoted by $L_{1}^{(0)} = 42$ , $n^{(0)} (L_{1}^{(0)}) = 5$ , $r^{(0)} (L_{1}^{(0)}) = 74.42$ , and $\bar{M} (n^{(0)} (L_{1}^{(0)}), r^{(0)} (L_{1}^{(0)}), L_{1}^{(0)}) = 2340.90$ .

For i = 2, L ∈ [L₂, L₁] = [28, 42], we use same procedure and obtain the following results: $L_{2}^{(0)} = 42$ , $n^{(0)} (L_{2}^{(0)}) = 5$ , $r^{(0)} (L_{2}^{(0)}) = 74.42$ , and $\bar{M} (n^{(0)} (L_{2}^{(0)}), r^{(0)} (L_{2}^{(0)}), L_{2}^{(0)}) = 2340.90$ .

By comparing the values of $\bar{M} (n^{(0)} (L_{i}^{(0)}), r^{(0)} (L_{i}^{(0)}), L_{i}^{(0)}),$ for i = 1, 2, 3, we get the minimum value $\bar{M} (n^{(0)} (L_{1}^{(0)}), r^{(0)} (L_{1}^{(0)}), L_{1}^{(0)}) = \bar{M} (n^{(0)} (L_{2}^{(0)}), r^{(0)} (L_{2}^{(0)}), L_{2}^{(0)}) = 2340.90$ . Thus, the optimal lead time is $L^{*} = L_{1}^{(0)} = L_{2}^{(0)} = 42$ days (=6 weeks), the optimal number of orders $n_{f}^{*} = 5$ , and the optimal reorder point is $r_{f}^{*} = 74.42$ units. The behavior of the objective function can be seen in the Fig. 1.

Fig.1

Cost function with respect to n and r when L = 6.

Example 2. This example considers an inventory situation for Model II to examine the effect of learning rate on the decision parameters. l = 0.05 is considered and rest of the parametric values are same as in Example 1.

Following the algorithmic procedure developed in Section 4.1, the optimal total inventory cost $\bar{M} (n, r, L)$ can be found for different values of n and results are presented in Table 3.

Table 3

Results of solution procedure for Model II

L (in weeks)	n	r	$\bar{M} (n, r, L)$
6	4	73.38	2410.90
6	5	73.50	2317.80
8	6	98.13	2321.07

“Bold” values within Table 3 indicate the optimum values.

From Table 3, we find the minimum value of $\bar{M} (n^{*}, r^{*}, L^{*})$ is $2317.80 which occurs at L^* = 6, n^* = 5, r^* = 73.50, respectively.

By comparing the optimal results of Model II (with learning in fuzziness) with Model I (without learning), it is clear that the optimal policy is derived for different number of shipments and cost. Following inferences can be drawn from the results:

Incorporating learning in fuzziness leads to a reduction in total annual cost. Hence, compared to Model I (with no learning), the total annual cost of Model II (with learning) is less.

The results of our proposed model have important implications for managers working in an uncertain environment for selecting an optimal policy. In a real life situation, a firm can face a significant degree of impreciseness in predicting its customer demand, but if it can be assumed that the employees responsible for issuing the orders get acquainted with the customer preference over a period of time, then ordering in smaller lots and increasing its order frequency is better. Thus the decision maker ultimately learns with time reducing the ambiguity that affects the decision making process. Hence, learning in fuzziness reduces the ambiguity associated with the future orders as the decision-maker gains information and gradually learns about the customer’s demand ordering pattern and their product preferences.

Although this process may impose higher ordering costs, the computational results suggest that it is advisable and favorable for an organization to order a higher stock of demand later on rather than to accomplish them in the initial cycles by increasing the order frequency.

Thus, the model in this paper is ideal for decision makers to design marketing strategies to stay ahead of the challenges that their products are likely to face in a fuzzy environment and thereby gain competitive edge in business and drive customer satisfaction. The significance of learning here also suggests that organizations should provide an environment to facilitate learning in their inventory systems thereby reducing the uncertainty associated with inventory system.

6 Conclusions

This model extended the basic continuous review inventory model with shortages under fuzzy demand and learning in fuzziness. Based on the existing literature, this was the first model where these assumptions were considered. After the pioneer attempt by Jaber [31] and Glock, Kurt, and Jaber [27], this model studied the fuzziness of the learning effect along with fuzzy demand in the continuous review inventory model. The triangular fuzzy number was used for the demand and demand during lead time. A possiblistic mean approach was used to calculate expected demand in fuzzy sense during lead time. The effect of learning was another realistic assumption for this model. One important computational algorithm was used to obtain the numerical result which indicates that learning improves the knowledge base of the decision maker over time, thereby helps in making precise predictions and reduce errors and costs.

The proposed model in this paper can be extended in several ways. For instance, we can generalize the model by considering other model parameters like ordering cost, inventory carrying charges as fuzzy variables subject to learning. In such a situation, studying the impact of different learning rates on the model parameters seems to be an interesting extension. Furthermore, it would be interesting to study the effect of alternative learning curves and different forms of learning which cannot be dealt using Wright’s learning curve on the model under study. Also, a more exhaustive study is needed to examine other inventory models with the data of learning process obtained from real-world inventory systems to make them more applicable in real situations. The model under study can also be extended by studying the impact of learning and forgetting in fuzziness on the total inventory costs.

Footnotes

Appendix 1.

Appendix 2. Acknowledgments

The authors are very much thankful for the important comments by the Associate Editor to improve the earlier version of this paper. The authors are also thankful to the reviewers for their valuable comments which helped in a significant improvement of the manuscript.

References

Gen

, Tsujimura

and Zheng

P.Z.

, An application of fuzzy set theory to inventory control models, Comput Ind Eng33(3-4) (1997), 553–556.

Ouyang

L.Y.

and Yao

J.S.

, A minimax distribution free procedure for mixed inventory model involving variable lead time with fuzzy demand, Comput Oper Res29(5) (2002), 471–487.

Chang

H.C.

, Yao

J.S.

and Ouyang

L.Y.

, Fuzzy mixture inventory model involving fuzzy random variable lead time demand and fuzzy total demand, Eur J Oper Res169(1) (2006), 65–80.

Dutta

, Chakraborty

and Roy

A.R.

, Continuous review inventory model in mixed fuzzy and stochastic environment, Appl Math Comput188(1) (2007), 970–980.

Dey

and Chakraborty

, A fuzzy random continuous review inventory system, Int J Prod Econ132(1) (2011), 101–106.

Joshi

and Soni

, (Q, R) inventory model with service level constraint and variable lead time in fuzzy-stochastic environment, Int J Ind Eng Comput2(4) (2011), 901–912.

Wang

, Fu

Q.L.

and Zeng

Y.R.

, Continuous review inventory models with a mixture of backorders and lost sales under fuzzy demand and different decision situations, Expert Syst Appl39(4) (2012), 4181–4189.

Shah

N.H.

and Soni

H.N.

, Continuous review inventory model with fuzzy stochastic demand and variable lead time, Int J Appl Ind Eng1(2) (2012), 7–24.

Gil

M.A.

, Miguel

L.D.

and Ralescu

D.A.

, Overview on the development of fuzzy random variables, Fuzzy Sets Syst157 (2006), 2546–2557.

10.

Vijayan

and Kumaran

, Models with a mixture of backorders and lost sales under fuzzy cost, Eur J Oper Res189(1) (2008), 105–119.

11.

Dutta

, Chakraborty

and Roy

A.R.

, Uncertain demand in (Q, r) inventory systems: A fuzzy optimization approach, J Fuzzy Math20(3) (2012), 501–514.

12.

Dash

J.K.

and Sahoo

, Optimal solution for a single period inventory model with fuzzy cost and demand as a fuzzy random variable, J Intell Fuzzy Syst28(3) (2015), 1195–1203.

13.

Sarkar

and Mahapatra

A.S.

, Periodic review fuzzy inventory model with variable lead time and fuzzy demand, Int Tran Oper Reas (2015), 1–31. DOI: 10.1111/itor.12177

14.

Cárdenas-Barrón

L.E.

, An easy method to derive EOQ and EPQ inventory models with Backorders, Comput Math Appl59(2) (2010), 948–952.

15.

Cárdenas-Barrón

L.E.

, The derivation of EOQ/EPQ inventory models with two backorders costs using analytic geometry and algebra, Appl Math Model35(5) (2011), 2394–2407.

16.

Cárdenas-Barrón

L.E.

, Chung

K.J.

and Treviño

, Garza, Celebrating a century of the economic order quantity model in honor of Ford Whitman Harris, Int J Prod Econ155 (2014), 1–7.

17.

Lin

H.J.

, Reducing lost-sales rate on the stochastic inventory model with defective goods for the mixture of distributions, Appl Math Model37(5) (2013), 3296–3306.

18.

Sarkar

and Moon

, Improved quality, setup cost reduction, and variable backorder costs in an imperfect production process, Int J Prod Econ155(1) (2014), 204–213.

19.

Moon

, Shin

and Sarkar

, Min–max distribution free continuous-review model with a service level constraint and variable lead time, Appl Math and Compt299 (2014), 310–315.

20.

Sarkar

, Gupta

, Chaudhuri

and Goyal

S.K.

, An integrated inventory model with variable lead time, defective units and delay in payments, Appl Math and Compt237 (2014), 650–658.

21.

Sarkar

, Chaudhuri

and Moon

, Quality improvement and setup cost reduction for the distribution free continuous-review inventory model with a service level constraint, 34 (2014), 74–82.

22.

Sarkar

, Mandal

and Sarkar

, Quality improvement and backorder price discount under controllable lead time in an inventory model, J Manuf Sys35 (2015), 26–36.

23.

Sarkar

, Supply chain coordination with variable backorder, inspections, and discount policy for fixed lifetime products, Math Prob Engin2016 (2016), 14. Article ID 6318737.

24.

Wright

T.P.

, Factors affecting the cost of airplanes, J Aeronaut Sci3(4) (1936), 122–128.

25.

Bera

U.K.

, Mahapatra

N.K.

and Maiti

, An imperfect production-inventory model over afinite time horizon under the effect of learning, Int J Math Oper Res1(3) (2009), 351–371.

26.

Pal

, Maiti

M.K.

and Maiti

, An EPQ model with price discounted promotional demand in an imprecise planning horizon via genetic algorithm, Comput Ind Eng57(1) (2009), 181–187.

27.

Glock

C.H.

, Kurt

and Jaber

M.Y.

, An EOQ model with fuzzy demand and learning in fuzziness, Int J Serv Oper Manag12(1) (2012), 90–100.

28.

Kazemi

, Olugu

E.U.

, Abdul-Rashid

S.H.

and BinGhazilla

R.A.

, Development of a fuzzy economic order quantity model for imperfect quality items using the learning effect on fuzzy parameters, J Intell Fuzzy Syst28(5) (2015), 2377–2389.

29.

Shekarian

, Olugu

E.U.

, Abdul-Rashid

S.H.

and Kazemi

, An economic order quantity model considering different holding costs for imperfect quality items subject to fuzziness and learning, J Intell Fuzzy Syst30(5) (2016), 2985–2997.

30.

Argote

, Group and organizational learning curves: Individual, system and environmental components, Brit J Soc Psychol32(1) (1993), 31–51.

31.

Jaber

M.Y.

, Learning Curves: Theory, Models, and Applications, ed. 2011, CRC/Taylor & Francis, Boca Raton.

32.

Carlsson

and Fullĕr

, On possibilistic mean value and variance of fuzzy numbers, Fuzzy Sets and Syst122(2) (2001), 315–326.

33.

Moon

and Choi

, A note on lead time and distributional assumptions in continuous review inventory models, Comput Oper Res25(11) (1998), 1007–1012.

34.

Jaber

M.Y.

, Learning and forgetting models and their applications, In Handbook of Industrial and Systems Engineering, edited by

A.B.

Badiru , 30.1–30.27, CRC/Taylor & Francis, Boca Raton2006.

35.

Yelle

L.E.

, The learning curve: Historical review and comprehensive survey, Decision Sci10(2) (1979), 302–328.

Demand uncertainty and learning in fuzziness in a continuous review inventory model

Abstract

Keywords

1 Introduction

2 Preliminary concepts

3.1 Fuzzy continuous review inventory model (Model I)

Notation

Decision variables

Parameters

Assumptions

4 A fuzzy continuous review inventory system and learning in fuzziness (Model II)

5 Numerical experiments and managerial insights

Table 1 Lead time for different components Lead time component i Normal duration b i Minimum duration b i - a i Unit crashing (days) a i (weeks) cost, c i ($/day) (days) 1 20 6 2 0.4 2 20 6 2 1.2 3 16 9 1 5.0

Footnotes

Appendix 1.

Appendix 2.

Acknowledgments

References

Table 1
Lead time for different components

Lead time component i Normal duration b_i Minimum duration b_i - a_i Unit crashing

(days) a _i (weeks) cost, c_i ($/day)

(days)

1 20 6 2 0.4

2 20 6 2 1.2

3 16 9 1 5.0