A cloudy fuzzy economic order quantity model for imperfectquality items with allowable proportionate discounts
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Abstract
In the traditional economic order quantity/economic production quantity model, most of the items considered are of perfect type. But this situation rarely takes place in practice. Thus, in this paper, an economic order quantity model with imperfectquality items is developed. \(100\%\) screening process is performed, and the items of imperfect quality are sold as a single batch. A proportionate rate of discount for the items of imperfect quality has also been studied. Moreover, a case study has been incorporated to comprehend the model. To nullify the issues of nonrandom uncertainties of demand rate in business scenario, cloudy fuzzy method has been utilized here. Numerical study reveals that cloud model along with its new defuzzification methods can give maximum profit of the model all the time instead of deterministic ones. Finally, sensitivity analysis and graphical illustrations are made to justify the novelty of the model.
Keywords
EOQ Screening cost Imperfect quality Cloudy fuzzy number New defuzzification method OptimizationIntroduction
In the beginning of twentieth century, Harris (1913) first studied a classical EOQ model. Wilson (1934) contributed a statistical approach to find order points where demand rate is assumed to be constant. Since then, a great amount of effort has been paid by several researchers in formulating more realistic lot sizing model by considering imperfectquality items. As the classical EOQ model is oldest and simplest, the academicians as well as the business persons are experiencing with some new ideas that genuinely improve the weaknesses of the existing research. In fact, a common assumption of traditional EOQ model was that all produced items would be of perfect quality. But this assumption was not logical for several reasons, including faulty production process, failure in the process of transportation, instant power cut during production, etc. Thus, it is difficult to produce or purchase items with \(100\%\) perfect type. Therefore, the screening of lot of items becomes emergent. These practices have received attention from many researchers, and many authors have addressed to this issue of lot sizing decision for imperfectquality items recent times. Consequently, a vast literature on imperfect type inventory production model has come up, by generalizing the EOQ model in numerous directions.
Karlin (1958) was one of the first to address on the EOQ model of imperfect type items. He studied the assumptions underlying the structure of the inventory cost components and presented three singlestage newsvendor models to characterize the optimal ordering policy under random supply. Porteus (1986) incorporated the effect of defective items into the original EOQ model. For the production process, he introduced ‘\(p\)’ as the percentage of defective and also assumed that the percentage of defective random variable \(p\) obeys the geometric distribution. Rossenblatt and Lee (1986) assumed that the time between the beginnings of the production run until the process goes out of control is exponential and that defective items can be reworked instantaneously at a low cost, and they conclude that the presence of defective products motivates customers to buy smaller lot sizes. Subsequently, Lee and Rossenblatt (1987) considered process inspection during the production run so that the shift to outofcontrol state can be detected and restoration made earlier. A joint lot sizing and inspection policy is studied under an EOQ model where random proportions of units are defective. These units can be discovered only through expensive inspections. Hence, the problem is bivariate; both normal lot size and fractional lot size for inspection are to be chosen. Gerchak et al. (1988) analyzed a singleperiod production problem where the production process was characterized by uncertain demand and variable yield. The singleperiod model was then extended to nperiod model. Cheng (1991) proposed an EOQ model with demand dependent on unit production cost and imperfect production processes. He formulated this inventory decision problem as a geometric program (GP), and it is solved to obtain closedform optimal solutions. Urban (1992) modeled the defect rate of a production process as a function of the run time length and derived closedform solutions for the model. This model accounted for either positive or negative learning effects in production process. BenDaya (1999) proposed multistage lot sizing models for imperfect production processes. Rezaei (2005) extended the model by assuming shortages in a cycle resulting from defective items are completely backordered in the beginning of each cycle and he also determined an optimal lot size of the order and the backorder. Salameh and Jaber (2000) developed an extended EOQ model where imperfectquality items are salvaged at a constant discounted price. Two years later, Goyal and CárdenasBarrón (2002) provided a new simple approach to the inventory model of Salameh and Jaber (2000). Chan et al. (2003) presented a new EPQ model in which three different situations of reselling, reworking, or rejecting imperfect products are incorporated.
Afterward, Papachristos and Konstantaras (2006) presented an extension to the work of Salameh and Jaber (2000). In their model, they considered another situation where the retailer sells these defective items at the end of replenishment interval but not at the end of screening process. Considering the unavoidable shortages in reallife situations, Eroglu and Ozdemir (2007) developed an EOQ inventory model with a random percentage of defective products every cycle. In their paper, shortages are assumed to be backordered, and the research results show that an increase in rate of defective leads to a decrease in the optimal total profit. Maddah and Jaber (2008) revisited the inventory model introduced by Salameh and Jaber (2000) and also given a new version to the EOQ model with imperfect items by considering that several batches of defective items are delivered in a single lot. CárdenasBarrón (2009) developed an EPQ model with planed backorders and reworking of imperfect items from the perspective of singlestage manufacturing system. Khan et al. (2011) provided a comprehensive literature review for the extensions of EOQ model with imperfect products. Meanwhile, Sarkar (2012) investigated an EOQ model in which defective products are assumed to occur every cycle under dependent demand and progressive payment scheme. Jaber et al. (2014) proposed a new variant for the inventory model of Salameh and Jaber (2000) by considering options of buying new ones or repairing imperfect ones for defective products. Taleizadeh et al. (2016) presented an EOQ model in which defective items are sent to a local repair store and the shortages are assumed to be partial backordered, which extended the EOQ inventory model discussed by Jaber et al. (2014). In most recent years, Pal and Mahapatra (2017) developed an inventory model with imperfect products for a threelevel supply chain, and three different ways of dealing with defective products were investigated in their model. From a sustainable point of view, Kazemi et al. (2018) studied EOQ model by considering carbon emissions and products with imperfect quality. Aghili and Hoseinabadi (2017) studied over the repairable items under fuzzy environment. The pricing and ordering policies of imperfect items in a supply chain have been developed by Taleizadeh et al. (2015). Also, Mondal et al. (2013) considered the inflation of money in productionrepairing inventory model in fuzzy rough systems. For the cases of pharmacological products Taleizadeh and Nooridaryan (2015, 2016) discussed over rework process of the items considering game theoretic approach in a supply chain.
Moreover, the essential and impractical assumptions are that all the inventory scenarios occur under a certain and deterministic environment. But in today’s competitive and dynamic business world, it is not possible to access all the necessary information. Hence, the information related to the inventory system is not welldefined as assumed in the traditional models. One of the effective methods to overcome these drawbacks is using fuzzy set theory, developed by Zadeh (1965), making possible to transform illdefined information to powerful mathematical expressions.
For more than half of a century, fuzzy set theory has been gaining a noticeable momentum by applying in many fields of operations research (Gholizadeh and Shekarian 2012; Shekarian and Gholizadeh 2013; Shekarian et al. 2016a) as well as inventory management (Shekarian et al. 2016b, c). Numerous variants and extensions of fuzzy inventory models have been offered on the evolution of classical inventory models including news vendor, reorder point, inventory control, joint economic lot size models, etc. Inventory management requires demand forecasts as well as parameters for inventoryrelated costs such as holding, replenishment, shortages and backorders (Kahraman et al. 2006). As the precise estimations of these model attributes are often difficult in practice, the inventoryrelated data can be calibrated using the fuzzy techniques, which facilitates dealing with the realworld cases in a more proper way.
Some notable recent works in fuzzy system like Das et al.(2015) presented an integrated production inventory model under interactive fuzzy credit period for deteriorating item with several markets. Due to randomness and fuzziness, Kumar and Goswami (2015) and Mahata and Goswami (2013) proposed a fuzzy random EPQ model for imperfectquality items with possibility and necessity constraints. Currently, Mahata (2017) investigated the learning effect of the unit production time on optimal lot size for the imperfect production process with partial backlogging of shortage quantity in fuzzy random environments. He assumed that the setup cost, the average holding cost, the backorder cost, the raw material cost, and the labor cost are characterized as fuzzy variables, and the elapsed time until the machine shifts from “incontrol” state to “outofcontrol” state is characterized as a fuzzy random variable. Articles on learning effect have been discussed wisely by Shekarian et al. (2016b). Alternatively, De and Beg (2016a, b) introduced dense fuzzy approach to capture the degree of learning experiences recent times. After that, the idea of dense fuzzy number was extended by De and Mahata (2017). To do this, they have developed a cloudtype fuzzy number incorporating the inventory cycle time to the measure of fuzziness. To resolve the difficulties and to defuzzify the cloudtype fuzzy number, they invented a new defuzzification method also. Recently, Karmakar et al. (2017) first established a pollutionsensitive dense fuzzy economic production quantity model with cycle timedependent production rate. Concurrently, De and Sana (2015) developed a backlogging model implementing a phi coefficient test for pentagonal fuzzy number. Chakraborty et al. (2015) investigated a supply chain model with stockdependent demand under fuzzy random and bifuzzy environments. Beyond this, researchers like De and Sana (2013), De et al. (2014), Karmakar et al. (2015), etc., have kept a remarkable destination over the fuzzy backlogging models.
From the above study, it is seen that not a single article for defective items has been developed utilizing the concept of nonrandomly uncertainty over demand rate as well as order quantity for cloudy fuzzy environment. Thus, an inventory model with imperfect quantity items with allowable proportionate discounts has been discussed over here. A new defuzzification method has been utilized to optimize the model. Based on new approach, numerical and graphical illustrations are made and end with a sensitivity analysis of the several parameters as well.
Preliminary concept (De and Beg 2016a, b; De and Mahata 2017)
Here few essential definitions and formulas which have been used frequently for solving the proposed model are discussed.
Normalized general triangular fuzzy number (NGTFN)
Note that the measures of fuzziness can be obtained from the following formula:
Yager’s (1981) ranking index
Note that the measures of fuzziness (degree of fuzziness \(d_{\text{f}}\)) can be obtained from the formula \(d_{\text{f}} = \frac{{U_{\text{b}}  L_{\text{b}} }}{2m}\), where \(L_{\text{b}}\) and \(U_{\text{b}}\) are the lower bounds and upper bounds of the fuzzy numbers, respectively, and \(m\) being their respective mode.
Cloudy normalized triangular fuzzy number (CNTFN)
Note that \(\mathop {\lim }\limits_{t \to \infty } a_{2} \left( {1  \frac{\rho }{1 + t}} \right) = a_{2}\) and \(\mathop {\lim }\limits_{t \to \infty } a_{2} \left( {1 + \frac{\sigma }{1 + t}} \right) = a_{2}\), so \(\tilde{A} \to \left\{ {a_{2} } \right\}\).
Ranking index over CNTFN
Note that, \(\alpha \;\;{\text{and}}\;\;t\) are independent variables.
Again (8) can be rewritten as \(I(\tilde{A}) = a_{2} \left[ {1 + \frac{\sigma  \rho }{4}\frac{{\log \left( {1 + T} \right)}}{T}} \right].\)
In general, for practical purpose the time horizon cannot be infinite so after defuzzification the indexed values do not come back to its crisp original even the restrictions have been removed in our assumptions.
Inventory process and cloudy fuzzy environment (De and Mahata 2017)
The measure of fuzziness depends upon what quantities are going to measure. In inventory process, the cycle time is one of the most important decision variable, so it is quite clear that fuzziness might have some relations on elapsed cycle time. In any inventory process, initially the uncertainties viewed are high and as the time progresses everything is began to clear for an inventory practitioner/decision maker (DM). As time progresses, the ambiguities underlying in the inventory system began to remove, and it is experiencing from the very ancient stage of any management system. When the inventory cycle time is low, the ambiguity becomes high and vice versa.
Let us discuss about the ambiguity over the demand rate, a most vital parameter of an inventory process. Here at the beginning the ambiguity over demand rate is high because, the people will usually take much time (no matter what offers have been declared or how attractive the getup of the system be) to accept and adopt the process.
If the cycle time ends prior to the “fully adopted” time period, then the cost becomes high. The basic insight over the public opinion is that ‘the system is less reliable’ as because the DM is hesitating to run the process for a longer time. This feeling must affects directly to the customers’ satisfactions level as well as on demand rate. However, as the cycle time becomes more the customers are began to get more satisfaction. A saturation on adoptability and reliability reaches. So the ambiguities have been removed from the process, and a grand paradigm shift on progress (financial development, cost minimization, achievement of large customer, etc.) of that system has been viewed.
Problem definition and case study
 (1)
How much amount of items to be ordered so that shortage will never come?
 (2)
What will be the inventory run time so as to keep maximum average profit all the time?
The several cost components associated with the retailer’s inventory system are studied as follows:
The inventory operation operates on \(8\) hours/day, so that for \(365\) days a year, the annual screening rate \(x = 1 \times 60 \times 8 \times 365 = 175200\;{\text{unit}}/{\text{year}}\) t[1 unit/min], screening cost \(\lambda = \$ 0.5\)/unit, yearly demand d = \(5000\) units, ordering cost \(k = \$ 200\)/cycle, holding cost per unit item per year \(h = \$ 5\), purchase cost \(c = \$ 25\)/unit, selling price of goodquality items \(s = \$ 50\)/unit.
Notation and assumptions
Assumptions
 (a)
Items, received or produced that are not of perfect quality and that are not necessarily defective, could be used in another production/inventory situation.
 (b)
A lot size is delivered instantaneously with a purchasing price of per unit and an ordering cost for the total lot.
 (c)
Each lot received contains percentage of defectives with a known probability density function.
 (d)
Goodquality items have a fixed selling price.
 (e)
Defective items are sold as a single batch at a proportionate discounted price.
 (f)
A
 (g)
\(100\%\) screening process of the lot is conducted.
 (h)
Shortages are not allowed.
 (i)
Lead time is zero.
Notations
 \(y\)

Order size, is a decision variable
 \(d\)

Demand rate
 \(x\)

Screening rate, \(x > d\)
 \(c\)

Unit purchasing price ($)
 \(k\)

Fixed cost of placing an order ($)
 \(p\)

Percentage of defective items in \(y\)
 \(f\left( p \right)\)

Probability density function of \(p\)
 \(s\)

Unit selling price of items of good quality ($)
 \(\lambda\)

Unit screening cost ($)
 \(T\)

Cycle length (years), is a dependent variable
Formulation of mathematical model
Let d be the demand rate per unit time of an inventory process. An order of size \(y\) is placed whenever the inventory level reaches zero. The items are received instantaneously with a unit purchasing price \(c\), fixed ordering cost \(k\) ,and the unit inventory holding cost h per unit time. Each lot received contains percentage defectives. A \(100\%\) screening process of the lot is conducted at a rate of \(x\) units per unit time with unit screening cost \(\lambda\). After completion of the screening process, items of poor quality are sold as a single batch at proportionate discounted price. However, the price of a goodquality item is \(s\) per unit.
With the above assumptions, the total cost per cycle for the modified EOQ model for imperfectquality items is:
In the proposed model, a proportionate rate of discount is introduced depending on the quality of the imperfect type items. Out of the imperfect type items, the best item is sold with minimum discount. Subsequently, depending on the quality of all other imperfect type items, a proportionate rate of discount has been introduced.
Numerical Example 1
Utilizing these data in Eqs. (18), (21), and (22), the optimum value of \(y\) is given by: \(y^{*} = 1556 {\text{units}}\), the optimal cycle time is \(T^{*} = 0.3015\) years, and the maximum profit per year is given as \({\text{ETPU}}\left( {y^{*} } \right) = 1230057\)/year.
Comparison with the traditional EPQ/EOQ model
Comparison with Salameh and Jaber (2000) model
The optimal lot size in proposed model is greater than the lot size in Salameh and Jaber’s (2000) model.
Fuzzy mathematical model
Moreover, the \(\alpha\)cuts of \(\mu \left( {\tilde{d},T} \right)\) and \(\mu \left( {\tilde{y},T} \right)\) are obtained by using (33) and (34), and they can be put as \(\left[ {d_{2} \left( {1  \frac{\rho }{1 + T}} \right) + \frac{{\alpha \rho d_{2 } }}{1 + T}, d_{2} \left( {1 + \frac{\sigma }{1 + T}} \right)  \frac{{\alpha \sigma d_{2 } }}{1 + T}} \right]\) and \(\left[ {\frac{{d_{2} T}}{1  p}\left( {1  \frac{\rho }{1 + T}} \right) + \frac{{\alpha \rho Td_{2 } }}{{\left( {1 + T} \right)\left( {1  p} \right)}}, \frac{{d_{2} T}}{1  p}\left( {1 + \frac{\sigma }{1 + T}} \right)  \frac{{\alpha \sigma Td_{2 } }}{{\left( {1 + T} \right)\left( {1  p} \right)}} } \right],\) respectively.
 (1)
If \(\left( {\rho  \sigma } \right) \to 0\) then \(I\left( {\tilde{z}} \right) \to \left[ {\frac{1}{{\frac{{d_{2} B}}{{\left( {1  p} \right)}}\left[ {\frac{\tau }{2}} \right] + 1}}} \right] \times \left[ {d_{2} \left[ {\begin{array}{*{20}c} {\frac{{d_{2} A}}{{\left( {1  p} \right)}}\left[ {\frac{\tau }{2}} \right]} \\ {  2kA^{/} } \\ \end{array} } \right]  \frac{{A^{//} d_{2}^{2} }}{{\left( {1  p} \right)^{2} }}\left[ {\frac{\tau }{2}} \right]^{2} } \right]\)
\(\Rightarrow \left[ {\frac{1}{By + 1}} \right] \times \left[ {d\left( {Ay  2kA^{/} } \right)  A^{//} y^{2} } \right] \Rightarrow z\), which is similar to crisp objective function.
Numerical Example 2
Optimal solution for the imperfect EOQ model
Model  \(T^{*}\) (days)  \(y^{*}\) (Unit)  \(z_{*}\) ($)  CI \(= \frac{{log\left {1 + T^{*} } \right}}{{T^{*} }}\) 

Crisp  90  651.406  122465.80  
General fuzzy  46  643.205  119362.80  0.0840 
Cloudy fuzzy  101  720.38  122816.4  0.0454 
Sensitivity analysis
Sensitivity analysis of the imperfect cloudy fuzzy EOQ model
Parameters  % change  \(T^{*}\) (Days)  \(y^{*}\) (Unit)  \(z^{*}\) ($)  \({\text{RC}} = \frac{{z^{*}  z_{*} }}{{z_{*} }} \times 100\)%  \({\text{CI}} = \frac{{\log \left {1 + T^{*} } \right}}{{T^{*} }}\) 

\(d_{0}\)  + 50  84  887.830  185260.5  + 51.07  0.0529 
+ 30  90  824.751  160335.5  + 30.75  0.0502  
− 30  121  599.137  85691.02  − 30.12  0.0396  
− 50  143  503.365  60889.55  − 50.35  0.0349  
\(\rho\)  + 50  137  927.337  123839.8  + 0.99  0.0360 
+ 30  121  830.442  123466.8  + 0.68  0.0398  
− 30  88  640.738  122562.5  − 0.05  0.0510  
− 50  81  599.444  122314.9  − 0.26  0.0540  
\(\sigma\)  + 50  86  624.323  122467.1  − 0.13  0.0521 
+ 30  92  658.411  122661.3  + 0.03  0.0495  
− 30  115  799.378  123337.5  + 0.58  0.0412  
− 50  126  864.252  123601.6  + 0.79  0.0384  
p  + 50  100  722.634  124863.1  + 1.82  0.0460 
+ 30  101  721.662  124093.4  + 1.2  0.0459  
− 30  104  719.306  121914.1  − 0.58  0.0450  
− 50  105  718.703  121229.0  − 1.14  0.0440  
h  + 50  81  570.295  122205.9  − 0.34  0.0540 
+ 30  88  618.550  122494.4  − 0.11  0.0511  
− 30  125  894.138  123562.2  + 0.76  0.0383  
− 50  157  1109.482  124044.1  + 1.16  0.0322  
\(\lambda\)  + 50  102  719.390  121693.3  − 0.76  0.0450 
+ 30  102  719.787  122208.0  − 0.34  0.0450  
− 30  102  720.974  123751.9  + 0.92  0.0454  
− 50  102  721.371  124266.6  + 1.34  0.0453  
\(k\)  + 50  123  868.140  122364.3  − 0.13  0.0390 
+ 30  115  812.520  122597.2  − 0.03  0.0413  
− 30  87  613.338  123419.7  + 0.65  0.0515  
− 50  75  529.062  123761.8  + 0.93  0.0577  
c  + 50  95  673.484  58655.8  − 52.17  0.0480 
+ 30  98  691.642  84383.83  − 31.19  0.0470  
− 30  106  751.067  161581.1  + 31.76  0.0440  
− 50  109  772.692  187318.1  + 52.75  0.0430  
s  + 50  118  831.242  251671.2  + 105.27  0.0410 
+ 30  111  783.786  200187.5  + 63.25  0.0420  
− 30  94  664.689  45792.57  − 62.66  0.0480  
− 50  No…  Feasible..  Solution.  ….  … 
Discussion on sensitivity analysis
From the sensitivity analysis Table 2, it is seen that the parameters like the initial demand \(d_{0}\), the unit purchasing price c, and the unit selling price s are highly sensitive parameters. The range of these changes assumes values between − 62.66 and + 105% as a whole. The other parameters have negligible effect on the proposed model for the changes of the parameters \(\left\{ {\rho , \sigma , p, k, h, \lambda } \right\}\) within − 50 to + 50%. Also the study reveals that for the case of highly sensitive parameters the optimum cycle time varies within the range 84–118 days with respect to the variation of maximum order quantity 831.242–887.830 units. Throughout the whole table, the minimum cloud index (0.0322) appears whenever the parameter h has been changed to − 50%, but for cloud index 0.0410 (> 0.0322) the profit curve assumes value 1.05 times with respect to the crisp optimum.
Graphical illustrations
Conclusion
 (1)
Cloudy fuzzy model always gives average maximum profit value of the model.
 (2)
Lesser ambiguities (less fuzziness) do not mean more profit of the model.
 (3)
All cost parameters are not similarly responsible for the enhancement of profit curve.
 (4)
Choice of perfect order quantity and the specific cycle time can change the overall decision of an inventory process.
Notes
Acknowledgements
The authors are thankful to the anonymous reviewers for their valuable suggestions to improve the quality of the article.
Compliance with ethical standards
Conflict of interest
The authors declare that there is no conflict of interest regarding the publication of the article.
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