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DOI:10.3233/IFS-141519 IOS Press
Development of a fuzzy economic order
quantity model for imperfect quality items
using the learning effect on fuzzy parameters
Nima Kazemi, Ezutah Udoncy Olugu
∗, Salwa Hanim Abdul-Rashid and Raja Ariffin Bin Raja Ghazilla
Center for Product Design and Manufacturing (CPDM), Department of Mechanical Engineering,Faculty of Engineering, University of Malaya, Kuala Lumpur, Wilayah Persekutuan, Malaysia
Abstract. This paper develops an inventory model for items with imperfect quality in a fuzzy environment by assuming that learning occurs in setting the fuzzy parameters. This implies that inventory planners collect information about the inventory system and build up knowledge from previous shipments, and thus learning process occurs in estimating the fuzzy parameters. So, it is hypothesized that the fuzziness associated with all fuzzy inventory parameters is reduced with the help of the knowledge acquired by the inventory planners. In doing so, the study developed a total profit function with fuzzy parameter, where triangular fuzzy number is used to quantify the fuzziness of the parameters. Next, the learning curve is incorporated into the fuzzy model to account for the learning in fuzziness. Subsequently, the optimal policy, including the batch size and the total profit are derived using the classical approach. Finally, numerical examples and a comparison among the fuzzy learning, fuzzy and crisp cases are provided to highlight the importance of using learning in fuzzy model.
Keywords: EOQ model, fuzzy set theory, imperfect quality, inventory control, learning
1. Introduction
In today’s highly competitive market, where compa-nies are under pressure to reduce the time they need for distributing their products and services, the impor-tance of inventory management has become highly inevitable for organizations. In order to be successful in facing this challenge, organizations need to continu-ally adopt proper techniques of keeping and managing their inventories. Choosing the right policy for man-aging inventories has always been a great challenge to manufacturing companies. Since the early of 20th century, when the foundation of the earlier economic order quantity (EOQ) and economic production quantity models (EPQ) was laid, numerous mathematical models
∗Corresponding author. Ezutah Udoncy Olugu, Center for
Prod-uct Design and Manufacturing (CPDM), Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Wilayah Persekutuan, Malaysia. Tel.: +60379675212; Fax: +60379675317; E-mail: [email protected].
emerged with the objective to assist organizations in better planning of their inventories. Although there has been a significant endeavor by scholars and practitioners to provide more practical versions of inventory mod-els, these models have significant shortcomings with real-world problems. For example, one of the assump-tions of the classical models is that the supply process is of perfect quality, whereas it is a common occurrence that buyers receive batches containing certain fraction of imperfect quality items (see, e.g., [1, 2]). Thus, it is essential to develop models that release the unrealistic assumptions of the conventional models and consider imperfect quality in produced or received batches.
Another unworkable assumption of the most inven-tory systems is that the data available to the decision makers are constant during the planning horizon. How-ever, the available data may vary from time to time, which makes the inventory system’s modeling more cumbersome. That is, one or some inventory values may not remain constant during planning horizon, and could
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adopt different values from cycle to cycle. This happensfrequently in real-world inventory problem when, for example, companies have modification in their product, or they encounter a situation that market demand alters regularly. Hence, this raises the degree of uncertainty that inventory planners should take into account when planning. Having this dynamic situation makes inven-tory planning more troublesome, as decision makers are unable to define exact values for input data. In such cases, it is possible that the decision objects have a fluc-tuation from their bases or could be defined orally, such as: “ordering cost is substantially less than x” or “set up cost is approximately of value y”. Fuzzy theory has been recognized as a useful tool to tackle this kind of imprecision that allows converting the oral expression or approximate estimation to a mathematical relation (e.g., [3, 4]). These mathematical expressions could be combined into the inventory problems and could be helpful in providing a flexible model, which facilitates modeling imprecise data.
Although fuzzy set theory provides an efficient tool to either model various types of uncertainties mathe-matically or deal with various sources of uncertainties in inventory management, the uncertainty associated with the inventory system could decrease performance of the system, blur data estimation process and increase the complexity of planning. On the other hand, with ref-erence to the literature on inventory management, the higher level of uncertainty could be so costly for firms and could thus increase their total cost of inventory sys-tem, which is the reason why firms undoubtedly try to avoid it. These intelligibly indicate that it is crucial to develop inventory models to tackle the uncertainty of inventory systems in an appropriate manner. This will definitely aid organizations to avoid making wrong and costly decisions. However, this research topic still remains as one of the challenging problems in the inven-tory management literature.
Learning concept, which occurs in every process as a consequence of practice, is found to be a useful tool to improve performance (see [5]). One of the areas that learning could help to improve the performance is the decision making process, particularly when decision makers should reconsider or revise their decision in the latter steps according to the knowledge they acquired from the earlier steps. As stated above, in fuzzy inventory models, there are abundant situations in which an exact value cannot be determined for parameters of the model, but instead a specific range (termed as spread values) can be defined so that inventory parameters fluctuate. At the initial planning stage, estimating the varying
val-ues (which shows the amount of fuzziness) is not simple since decision makers are not either acquainted well with the data or do not have much information about the process. However, during the planning period, as the time passes and more information about the properties of decision data are collected and analyzed, the decision makers could then be able to enhance the accuracy of data estimation, and thus could reduce the amount of uncer-tainty they are facing. It is obvious that learning is a very useful tool for decision makers to reduce the impact of imprecision on the quality of their decisions.
A closer look at the literature shows that in spite of several models which were developed under fuzzy con-ditions, only a few numbers of studies were devoted to model human factor role in the problem, while the entire focus was mainly on modeling the fuzziness associated with the planning problem. Of the entire aforementioned studies, only one study can be found (see section 2) that modeled the human role and its impact on fuzziness modeling in inventory planning. However, the important role that human plays in an inventory planning pro-cess and the high proportion of human work either in gathering or collecting, processing and revising inven-tory data highlights the influence that human has on inventory systems. Thus, it is apparent that the devel-oped inventory models present an imperfect picture of real-world’s inventory planning problems, which influ-ence the planning outcome. In addition, by considering human interaction with the inventory system, it is obvi-ous that assuming a constant amount of fuzziness in every phase of the planning is completely an unrealis-tic assumption as the amount of uncertainty a decision maker encounters may vary over time. In order to present a better representation of reality and close this research gap, this paper develops a model with imperfect quality and fuzzy parameters which learning occurs in setting fuzziness values. For this purpose, it is assumed that annual demand, holding cost, set up cost, selling price of defective items and percentage of defective items are fuzzy numbers, and therefore their value fluctuates between a lower and upper bound. It is subsequently assumed that the amount of fuzziness is subjected to learning and the fuzzy total profit function with learning in fuzziness is thus developed using fuzzy mathematics.
2. Literature review
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tical aspects. However, some of their assumptions arenever met in practice [6]. This inspired several authors to present the improved version of the basic models to give them a touch of reality. One of the limitations of the EOQ or EPQ models, which has received considerable attention by researchers, is that the items in a produced or received batch are not of perfect quality. Several works can be found in the literature that addressed this problem. For example, Chan et al. [7] developed an EPQ model with imperfect quality items which are sold at a lower price at the end of the production period or cycle. Jamal et al. [8] developed an EPQ model that produces defec-tive items which are reworked to good ones during the same production cycle. C´ardenas-Barr´on [9] developed an EPQ model for determining the economic production quantity and size of backorders for a system that gen-erates imperfect quality (defective) items with planned backorders. Defective items are reworked to as-good-as-new condition in the same production cycle. One of the prominent works along this line of research is the model of Salameh and Jaber [10], who assumed that shipments contain a random percentage of defective items. Upon receiving batches, they undergo 100% inspections with a rate faster than demand rate and imperfect quality items are withdrawn and sold as a single batch by the end of the screening process. The work of Salameh and Jaber [10] has received increasing attention by researchers. Hence, lots of researchers have presented the extension or modification of this model. For an extensive review of the models that deal with the model of Salameh and Jaber [10] the readers are referred to Khan et al. [11]. In addition, some papers which also appeared after 2011 are shortly noted in Jaber et al. [12]. None of the above reviewed models considered uncertainty in their model. However, as discussed in the previous section, assuming deterministic values in an uncertain decision situation may lead to erroneous inventory policies.
Fuzzy set theory is recognized as a proper method in dealing with uncertainty. Since the development of fuzzy set theory by Zadeh [13], it has been one of the interesting area of research for scholars [14–16]. Reviewing the literature shows that numerous studies have thus far been conducted to investigate the appli-cation of fuzzy set theory in inventory management. One of the main research streams in this line of thought has been extending the classical models into a fuzzy environment enabling the models to tackle the fuzzi-ness. For example, Chang [17] presented an extended version of the work of Salameh and Jaber [10] in two different models, where the first one fuzzified the per-centage of defective items and the second one fuzzified
the percentage of defective items and demand simul-taneously. Vijayan and Kumaran [18] studied another form of the EOQ model termed as the Economic Order Time (EOT) model and performed two different poli-cies of fuzziness. In the first policy, they assumed that all the parameters of the EOT model were imprecise, but could be described by trapezoidal and triangular fuzzy numbers and then obtained the optimal policy using the Lagrangian method. In the next step, they examined the parameters and variable of the model under fuzzy sense and again trapezoidal and triangular fuzzy num-bers were applied to model the fuzziness. Bj¨ork [19] developed an EOQ model with backorders where the lead time (and consequently the maximum inventory level) and total demand were assumed to be triangular fuzzy numbers. Using an analytical solution, an opti-mal order quantity is derived for the model proposed. A similar topic was treated by Kazemi et al. [20], who investigated the classical model with backorders with a different defuzzification method to that of Bj¨ork [19]. The author proposed an analytical solution for solv-ing the fully-fuzzy model, where the model was tested for triangular and trapezoidal fuzzy numbers. Bj¨ork [21] considered a simpler problem to that of Kazemi et al. [20] and proposed a multi-item EOQ model with fuzzy cycle time. In a recent paper, Shekarian et al. [22] developed a fuzzified version of a lot-sizing model for a single-stage production system with defective items and rework, which defective items are immediately reworked within the same cycle. They assumed that the rate of defects and demand rate were triangular fuzzy number and therefore used two defuzzification methods to derive the crisp total cost function.
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parameters, and the fuzziness associated withinven-tory data could in turn be reduced. To this aim, this paper applies the concept of learning in fuzziness to an EOQ model with imperfect quality, which has enjoyed increasing attention in recent years. To the best of our knowledge, there is no inventory model with imperfect quality in the literature that applies learning in fuzzi-ness. This is the limitation that this paper addresses. The model developed in this paper is the extension of Salameh and Jaber [10] to the fuzzy-learning environ-ment. The next section will review some basics and definitions of fuzzy set theory, which will be applied through the paper.
3. Preliminaries
A fuzzy number ˜Ais called a triangular fuzzy number (TFN) and is denoted by (l, m, n) if it has the following piecewise linear membership function
µA˜(x)=
To perform fuzzy arithmetical operations by TFN, Function Principle proposed by Chen [23] is used. Function Principle is a suitable method for performing the operations of complex models to prevent arriving at a degenerated solution. This method will be so help-ful in handling the fuzzy operations, especially when the crisp model comprises terms of multiple opera-tions of fuzzy numbers. Furthermore, the type of fuzzy membership function will be kept constant during the operations, which helps to avoid facing further com-plexity by arithmetical operations (e.g., [24]). Now, assume ˜A=(l1, m1, n1) and ˜B=(l2, m2, n2) are two positive TFNs andαbe a real number. Based on the Functional Principle, the operations of the fuzzy numbers ˜Aand ˜Bare as the following:
In this paper, Graded Mean Integration Representa-tion (GMIR) method is applied to transform the fuzzy total profit function to its corresponding crisp func-tion. The main reason is due to the non-linear nature of the function that was used in this paper. In fact, the fuzzy total profit function in this paper consists a cou-ple of fuzzy multiplication and division terms, and since GMIR method keeps the shape of membership function, it is a proper choice to defuzzify the fuzzy profit func-tion of the model. In the following secfunc-tion, the GMIR method introduced by Chen and Hsieh [25] is described. For the fuzzy number ˜Ain Equation (1), letθ−1and π−1 be the inverse functions ofθandπ, respectively.
The graded meanρ-level value of ˜Ais ρ(θ−1(ρ)+2π−1(ρ)) and the GMIR of the fuzzy number ˜Ais calculated as:
τ( ˜A)= mean integration representation of ˜Acan be calculated by formula (2) which is as
τ( ˜A)=
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Fig. 1. The behavior of the Wright’s learning curve.
fields like manufacturing, healthcare, energy, military, information technologies, education, design, and bank-ing [28, 29]. Since then, several learnbank-ing curve models were developed that have different forms [29–33]. All the models developed have commonly argued that the performance improves with the repetition of a task. Of all the available models, the Wright’s [27] learn-ing curve is found to be the most popular and basic model due to its mathematical simplicity and ability to fit well a wide range of learning data [28, 31, 33, 34]. For a comprehensive review of learning models and their application readers may refer to Jaber [35] and Anzanello and Fogliatto [29]. The learning curve, which is used in this paper, is that of Wright [27] which is of the form
yi =y1i−β (4)
where yi is the performance at the time of ith ship-ment, y1 is the performance at the beginning of the
planning period,iis the number of shipments andβis learning exponent. The learning exponent can be cal-culated using the equationβ= −log(δ)/log(2), where
δis learning rate and can be described as a percentage ranging from 50% to 100%. If the learning rate is 100%, the learning exponent adopts 0 in equation while in case the learning rate is 50% the learning exponent is equal to 1. Figure 1 illustrates the behavior of the learning curve in Equation (4).
4. The model
The following notations are being made for develop-ing the mathematical models:
p purchasing price of unit product
d screening cost per unit
A buyer’s ordering cost per cycle
γ fraction of defective items
Q lot size per cycle
h buyer’s holding cost per unit of time
x screening rate
D demand rate per unit of time
ν unit selling price of defective items
s unit selling price of non-defectives (good) items
f(γ) probability density function ofγ N(γ, y) number of good items in each order
t screening time
T cycle length
li the lower deviation value (spread) value for ith parameteri=D, h, γ, d, A ui the upper deviation value (spread) value
for ith parameteri=D, h, γ, d, A
4.1. Fuzzy modeling of the EOQ model with imperfect quality
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D, h, γ, d, Arepresent the values that a parameter can deviate from its base value, which are called fuzziness values or spread values. The fuzzy form of the EOQ model with imperfect quality in Salameh and Jaber [10] is given as
TP U˜ (Q)=D˜(s−ν+
Replacing triangular fuzzy numbers in Equation (6) will result in the following fuzzy total profit function
TP U˜ (Q)=(s−ν) (D−lD, D, D+uD)
Using Function Principle defined in Section 3.1, the graded mean integration (defuzzified) value of the fuzzy total profit function in Equation (7) is given by
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crisp value of fuzzy total profit per unit time. In the next section, the fuzzy EOQ model presented in this section will be developed to account for learning in fuzziness.4.2. Incorporating learning into fuzzy model
In this section, we model the decision maker’s learn-ing in estimatlearn-ing the fuzziness values and develop a model with the assumption that decision makers learn with time when adopting a fuzziness value for the parameters. In addition, we assume that decision maker could use their knowledge to reduce the fuzziness of the data. In order to investigate the effect of learning in fuzziness on inventory policy, we suppose that buildup of knowledge occurs with the number of shipments. We consider the learning to follow the Wright’s [27] power learning curve given in Equation (4). If learn-ing affects the fuzzy parameters and if their value changes according to the number of shipment, then for
j=D, A, h, d, γ the value ofjthupper and lower fuzziness parameter at the time ofithshipment will be as
Whereβ is the learning exponent defined in Equation (4). Equations (9) and (10) show that for the first ship-ment the fuzziness values take their initial values (lj,1
for lower fuzziness anduj,1 for upper one), which
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Before obtaining the optimal policy for the model, we need to verify that the total profit per unit time given in Equation (11) is concave. This can be proved by taking the first and second partial derivatives with respect toQ. Taking the partial derivatives with respect toQyields:
∂τL(TP U˜ (Q))
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+1
x(D−lD,1i
−β) (h− lh,1i−β)
−1
2(h−lh,1i
−β) (γ−
lγ,1i−β−1)]
−2
3[
Dh
x −
Dh x(1−γ)−
h(1−γ)
2 ]
+1
6[− 1
x(D+uD,1i
−β) (h+
uh,1i−β)]
+1
x(D+uD,1i −β
) (h+uh,1i−β)
1
1−γ−uγ,1i−β
+1
2(h+uh,1i
−β)−1
2(h+uh,1i
−β)
(γ+lγ,1i−β)
It can be noted that when there is no learning,β=0, the total profit function in Equation (11) and optimal order quantity in Equation (14) reduce to that of fuzzy model developed in section 4.1. Besides, when there is no imprecision in estimating the parameters, then deviation values will be equal to zero, that islj=0 anduj =0,j=D, A h d γ. In this case, the total profit per unit time and optimal lot size lead to optimal policy derived in Salameh and Jaber [10].
5. Numerical example
In this section, an example is given to illustrate the behavior of the model presented in Section 4.2. The input parameters of numerical example are adopted from Salameh and Jaber [10]. Consider an inven-tory model with crisp parameters having the following values: demand rateD=50,000 units/year, ordering
cost A= $100/cycle, holding cost h= $5/unit/year, screening ratex= 1 unit/min (equivalently,x= 175,200 units/year), screening cost d= $0.5/unit, purchase cost p= $25/unit, selling price of good-quality items s= $50/unit, and selling price of imperfect quality items ν= $20/unit. The values in which the fuzzy parameters can fluctuate are respectively set as (5000, 10000), (6, 5), (0.1, 0.35), (0.0005, 0.15), (1, 2) for
D, A, d, γ, h. In order to facilitate the computation process, the formulas were written in Microsoft Excel 2010. To investigate the optimal policy of the model with learning in fuzziness, the impacts of different learning rates on the optimal order quantity and total profit per unit time are examined, with the results sum-marized in Table 1. The learning exponents are set from 0.862 (very slow learning) to 0.074 (very fast learning). For discussion on various learning rates in industry reader may refer to Jaber [26]. The optimal policy was derived for different total number of ship-ments in a cycle, which adopts 5, 10, 15 and 20. The obtained results show that, comparing with the crisp model (see Table 2 and also Salameh and Jaber [10]), the optimal policy of the learning-fuzzy model is fairly sensitive to learning rate. Even though the change in policy is not much significant, it can be inferred from the result that using learning change the optimal policy of the model so that when learning increases both opti-mal values increase as a result. For example, when the number of orders is 5 and learning exponent shifts from 0.862 to 0.074, the optimal order quantity increases from 1438.11 to 1450.87. For other shipments, the same pattern can be observed. Comparing the total profit with crisp model also indicates that there is a slight change in the learning-fuzzy model; however learning-fuzzy model generates lower profit. Generally, the result may suggests that provided that learning occurs in inventory system, the deviation generated by the learning model could change the policy, and consequently ignoring
Table 1
Optimal policy with different learning rate
β i=5 i=10 i=15 i=20
Q∗ τ(TP U˜ (Q)) Q∗ τ(TP U˜ (Q)) Q∗ τ(TP U˜ (Q)) Q∗ τ(TP U˜ (Q))
0.862 1438.11 1,203,877.78 1436.40 1,202,830.03 1435.83 1,202,427.34 1435.54 1,202,210.87
0.737 1439.03 1,204,360.20 1437.08 1,203,271.60 1436.38 1,202,816.97 1436.02 1,202,560.02
0.621 1440.10 1,204,867.17 1437.94 1,203,784.39 1437.12 1,203,295.73 1436.67 1,203,006.22
0.515 1441.35 1,205,388.28 1439.04 1,204,363.41 1438.09 1,203,865.95 1437.55 1,203,557.80
0.415 1442.80 1,205,910.70 1440.40 1,204,997.64 1439.36 1,204,523.02 1438.75 1,204,216.28
0.322 1444.45 1,206,418.77 1442.10 1,205,667.98 1441.01 1,205,251.80 1440.34 1,204,971.77
0.234 1446.34 1,206,893.57 1444.19 1,206,344.58 1443.14 1,206,021.70 1442.46 1,205,796.04
0.152 1448.47 1,207,312.34 1446.75 1,206,983.26 1445.85 1,206,779.58 1445.25 1,206,632.26
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Table 2
Comparing crisp, fuzzy and fuzzy learning model
Crisp model Fuzzy model Fuzzy learning model
l u l u l u l u l u l u l u
D 0 0 5000 3000 5000 3000 5000 3000 5000 3000 5000 3000 5000 3000
A 0 0 6 8 6 8 6 8 6 8 6 8 6 8
d 0 0 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2 0.3 0.2
γ 0 0 0.01 0.03 0.01 0.03 0.01 0.03 0.01 0.03 0.01 0.03 0.01 0.03
h 0 0 1 2 1 2 1 2 1 2 1 2 1 2
i NA NA 5 5 5 5 5
β NA NA 0.862 0.621 0.415 0.234 0.074
Q∗ 1,434.61 1,414.66 1,429.06 1,426.58 1423.65 1,420.30 1,418.17
τ(TP U˜ (Q)) 1,212,235 1,192,435.13 1,199,290.56 1,198,256.81 1,196,968.71 1,195,389.97 119,4321.97
i NA NA 10 10 10 10 10
β NA NA 0.86 0.737 0.515 0.234 0.074
Q∗ 1,434.61 1,414.66 1,431.49 1,429.28 1,426.24 1,422.26 1,417.41
τ(TP U˜ (Q)) 1,212,235 1,192,435.13 1,200,255.54 1,199,380.64 1,198,111.14 1,196,329.95 1,193,926.07
i NA NA 15 15 15 15 15
β NA NA 0.86 0.737 0.515 0.234 0.074
Q∗ 1434.61 1414.66 1,432.39 1,430.43 1,427.47 1,423.30 1,417.86
τ(TP U˜ (Q)) 1,212,235 1,192,435.13 1,200,599.81 1,199,839.55 1,198,634.91 1,196,807.90 1,194,160.39
i NA NA 20 20 20 20 20
β NA NA 0.86 0.737 0.515 0.234 0.074
Q∗ 1434.61 1414.66 1432.87 1429.83 1428.24 1423.98 1418.17
τ(TP U˜ (Q)) 1,212,235 1,192,435.13 1,200,779.70 1,199,601.18 1,198,954.34 1,197,118.61 1,194,321.97
learning in inventory planning may result in improper policy. Additionally, the results in Table 1 can give general policy in learning-fuzzy model with imperfect quality. That is, when the learning rate is slow deci-sion maker should order smaller lot size to the supplier, which incur lower total profit. In contrast, faster learn-ing results in greater lot sizes with higher total profit. Therefore, learning in fuzziness makes the buyer order more and more, which tends to increase the total profit. The importance of learning here also suggests that orga-nizations should provide an environment to facilitate learning in their inventory systems. As to the sensitiv-ity of the model to the number of shipments, it is clear from Table 1 that both optimal values tend to decrease with number of shipment, at a constant learning rate. So, the optimal order quantity and total profit are expected to decrease when the number of shipments in a cycle increases. Figures 2 and 3, which are plotted based on the data tabulated in Table 1, depict the variation in opti-mal policy, while varying the other parameters of the model.
The behavior of the learning-fuzzy model developed in Section 4.2 is additionally investigated using several fuzzy numbers. Table 2 compares the result of crisp, fuzzy and fuzzy learning model for some arbitrary sets ofand different number of shipments, ranging from 5 to 20. For the learning-fuzzy model, the learning param-eters are respectively set at 0.862, 0.621, 0.415, 0.234,
5
Fig. 2. Three dimensional graph of the optimal order quantity, learn-ing exponent and number of shipments.
5
0.862 0.7370.621 0.515
0.415 0.322 0.234
Fig. 3. Three dimensional graph of the optimal total profit, learning exponent and number of shipments.
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Table 3
The impact of increasing the amount of fuzziness for demand on optimal policies
Lower and upper limits Fuzzy learning model Fuzzy model
lA uA ld ud lγ uγ lh uh lD uD i β Q∗ τ(TP U˜ (Q)) Q∗ τ(TP U˜ (Q)) 3 4 0.04 0.03 0.003 0.004 0.4 0.3 300 400 15 0.621 1435.32 1,201,497.36 1438.69 1,201,852.33 3 4 0.04 0.03 0.003 0.004 0.4 0.3 700 800 15 0.621 1435.32 1,201,497.09 1438.76 1,201,844.50 3 4 0.04 0.03 0.003 0.004 0.4 0.3 1400 1700 15 0.621 1435.42 1,201,647.35 1439.37 1,202,639.43 3 4 0.04 0.03 0.003 0.004 0.4 0.3 2300 3200 15 0.621 1435.69 1,202,098.96 1441.00 1,205,047.72 3 4 0.04 0.03 0.003 0.004 0.4 0.3 3400 4500 15 0.621 1435.78 1,202,248.96 1441.68 1,205,834.82 3 4 0.04 0.03 0.003 0.004 0.4 0.3 4700 6900 15 0.621 1436.28 1,203,077.15 1444.60 1,210,256.88 3 4 0.04 0.03 0.003 0.004 0.4 0.3 5600 8400 15 0.621 1436.55 1,203,528.76 1446.22 1,212,665.18 3 4 0.04 0.03 0.003 0.004 0.4 0.3 6200 8900 15 0.621 1436.51 1,203,452.98 1446.08 1,212,249.12 3 4 0.04 0.03 0.003 0.004 0.4 0.3 7800 10200 15 0.621 1436.38 1,203,225.79 1445.62 1,211,004.84
Table 4
The impact of increasing the amount of fuzziness for ordering cost on optimal policies
Lower and upper limits Fuzzy learning model Fuzzy model
lD uD ld ud lγ uγ lh uh lA uA i β Q∗ τ(TP U˜ (Q)) Q∗ τ(TP U˜ (Q)) 4300 3400 0.04 0.03 0.003 0.004 0.4 0.3 1 2 15 0.621 1434.89 1,200,741.01 1436.53 1,197,732.78 4300 3400 0.04 0.03 0.003 0.004 0.4 0.3 4 3 15 0.621 1434.46 1,200,743.15 1438.00 1,197,725.50 4300 3400 0.04 0.03 0.003 0.004 0.4 0.3 5 7.5 15 0.621 1435.25 1,200,739.21 1439.22 1,197,719.47 4300 3400 0.04 0.03 0.003 0.004 0.4 0.3 6 9 15 0.621 1435.37 1,200,738.62 1440.06 1,197,715.36 4300 3400 0.04 0.03 0.003 0.004 0.4 0.3 8 11 15 0.621 1435.39 1,200,738.55 1440.44 1,197,713.46 4300 3400 0.04 0.03 0.003 0.004 0.4 0.3 9 12 15 0.621 1435.39 1,200,738.52 1440.63 1,197,712.51 4300 3400 0.04 0.03 0.003 0.004 0.4 0.3 13 18 15 0.621 1435.87 1,200,736.16 1443.96 1,197,696.06 4300 3400 0.04 0.03 0.003 0.004 0.4 0.3 17 21 15 0.621 1435.67 1,200,737.14 1443.45 1,197,698.58 4300 3400 0.04 0.03 0.003 0.004 0.4 0.3 19 25 15 0.621 1436.14 1,200,734.85 1446.39 1,197,684.06
Table 5
The impact of increasing the amount of fuzziness for fraction of defective items on optimal policies
Lower and upper limits Fuzzy learning model Fuzzy model
lD uD lA uA ld ud lh uh lγ uγ i β Q∗ τ(TP U˜ (Q)) Q∗ τ(TP U˜ (Q)) 4300 3400 3 4 0.04 0.03 0.4 0.3 0.001 0.002 15 0.621 1434.90 1,200,741.54 1436.86 1,197,748.08 4300 3400 3 4 0.04 0.03 0.4 0.3 0.004 0.003 15 0.621 1434.84 1,200,758.81 1436.57 1,197,826.21 4300 3400 3 4 0.04 0.03 0.4 0.3 0.008 0.006 15 0.621 1434.81 1,200,766.63 1436.48 1,197,841.90 4300 3400 3 4 0.04 0.03 0.4 0.3 0.009 0.008 15 0.621 1434.84 1,200,757.20 1436.69 1,197,779.85 4300 3400 3 4 0.04 0.03 0.4 0.3 0.009 0.01 15 0.621 1434.91 1,200,739.00 1437.06 1,197,674.54 4300 3400 3 4 0.04 0.03 0.4 0.3 0.01 0.011 15 0.621 1434.91 1,200,738.65 1437.09 1,197,664.45 4300 3400 3 4 0.04 0.03 0.4 0.3 0.011 0.012 15 0.621 1434.91 1,200,738.29 1437.11 1,197,654.17 4300 3400 3 4 0.04 0.03 0.4 0.3 0.012 0.013 15 0.621 1434.91 1,200,737.93 1437.14 1,197,643.69 4300 3400 3 4 0.04 0.03 0.4 0.3 0.013 0.013 15 0.621 1434.82 1,200,764.16 1436.66 1,197,770.87
model are closer to that of crisp model, irrespective of the rate of learning rate. This result is expected since the decision maker retains knowledge from previous ship-ments, and thus is able to estimate the parameters with more preciseness. This finding proves that learning in fuzziness is an appropriate tool to reduce the amount of uncertainty. Another conclusion from sample data is that the learning-fuzzy model produces greater lot size with higher total profit in comparison to fuzzy model. This could be also expected and be interpreted when learning occur. In other words, learning in fuzziness diminishes the amount of fuzziness decision makers encounter, and therefore helps in making decision with lower imprecision.
AUTHOR COPY
learning exponent, considering the higher amount offuzziness in the fuzzy learning model leads to smaller lot size in comparison to the fuzzy model. This can be inferred from all three parameters that are analyzed. Moreover, looking at Table 3 it can be observed that when the amount of fuzziness in demand is increased fuzzy model produce more profit in comparison to the fuzzy learning model. The computations for the other two parameters give different results to that of demand. In other words, the total profit generated by fuzzy learn-ing model is higher than fuzzy model when the amount of fuzziness in ordering cost and defective items is increased.
6. Summary and conclusion
This paper extended a fuzzy EOQ model with imperfect quality by assuming that learning occurs in estimating fuzzy parameters, which is retained by the number of shipments. The models that have touched upon the area of fuzzy inventory management have not considered learning in fuzzy parameters, which is a research gap that this paper covered. Wherefore, learn-ing is used in settlearn-ing triangular fuzzy numbers and it is hypothesized that the fuzziness reduced in conformance with a learning curve, following the learning curves proposed by Wright [27]. The model, consequently, proposes optimal planning for learning in fuzziness situation. To gain further insights on how learning in fuzziness affects the model, the behavior of the model studied is numerically investigated and compared to the fuzzy and crisp case. Examining the effect of learn-ing on the model showed that, as learnlearn-ing increases, both optimal lot size and optimal profit increases. A further insight from this example is that when the learn-ing rate is slow smaller lot size should be ordered, however, when the learning becomes faster the opti-mal policy is to order higher lot size from the supplier. The comparison between the crisp, fuzzy and fuzzy model demonstrated that the model with learning in fuzziness could appropriately handle the imprecision associated with the model since it could result to closer values to the crisp case by reducing the amount of uncer-tainty. This might be an enticement to benefit from the usefulness of learning in inventory planning when the available data are imprecise.
The model developed in this paper could be extended in several directions. It would be interesting to modify the model to account for forgetting. If this is the case, the model could be tested against the learn-forget curve
model (for review of forgetting model see Anzanello and Fogliatto [29]), which are direct extensions of the learning curves of Wright [27]. Moreover, a more com-prehensive study is needed to analyze the model with the data of learning process gained from real-world inventory problems. This seems to be an interesting research, which makes the model more applicable in real situations. It would also be beneficial to study the effect of alternative learning curves on the model stud-ied in this paper to account for the fact that different forms of learning may occur in practice that cannot be tackled by using Wright’s learning curve. Finally, the fuzzy model with learning in fuzziness presented in this paper could be applied to other inventory models with fuzzy condition.
Acknowledgments
The authors wish to express their gratitude to Uni-versity of Malaya for funding their research (Grant no. RP018a-13aet).
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