A multi-constrained supply chain model with optimal production rate in relation to quality of products under stochastic fuzzy demand
Introduction
The earliest economic order quantity (EOQ) models were formulated based on Harris (1913) inventory model. In the last century, researchers published mathematical models to optimize the orders and lot size with different assumptions, such as controllable lead time, partial and complete backorders, fixed production rates, and quality improvements, in huge numbers. In basic EOQ models, the product's deterministic demand is considered with isolated optimal solutions for the buyer. Similarly, for the production models, the vendor obtains his economic production/manufacturing quantity (EPQ/EMQ) independently without concerning other parties of the supply chain (SC). In spite of that, the obtained EOQ and EPQ might be different in sizes. As a consequence, the vendor and the buyer start negotiations to reach an agreement. The result of these negotiations highly depends on the relative bargaining powers of both the parties and the solution is neither optimal for the vendor nor the buyer Zavanella, Marchi, Zanoni, and Ferretti (2019a). The individual or decentralized models can achieve local optima with the perspective of each supply chain member. It generally happens because of the possible conflicting individual objectives of each SC member.
Within the same SC, business groups or firms should collaborate to foster a competitive advantage over their competitors and better satisfy customers’ needs. The collaborative decision making is crucial in the current business environment, which is characterized by rapidly growing technologies, highly demanding buyers, and high globalization. Aligning strategic decisions and operations among the different member firms of the SC becomes a prerequisite in the JELS (joint economic lot-size) model, which transfers the focus to the collaborative decision and overcome the limitations of the EOQ model of considering processes and production systems individually independent entities. Goyal (1977), for the first time, introduced a JELS model, which shifted the researchers’ focus from individual models to the coordinated models for the inventory and replenishment decisions. Researchers extended and modified Goyal’s proposed JELS model with several assumptions and aspects: multi-stage production, multi-product, and two-stage production, lead time and uncertain or stochastic demand, and process quality improvements, controllable setup cost with investments.
Banerjee (1986) extended Goyal (1977) JELS model for a lot-for-lot (LFL) policy with deterministic conditions. Further, Goyal (1988) extended Banerjee (1986) model by relaxing the assumption of LFL policy and establishing the relation between purchaser’s order quantity and vendor’s production lot size and achieved significant improvements in results. Woo, Hsu, and Wu (2001) studied a single-supplier multiple purchaser model to control ordering costs with additional investments. Biswajit Sarkar and Majumder (2013) analyzed an inventory model with optimal investments, lot size, and two different probability distributions, normal distribution and unknown distribution, and scenarios for lead time demand. Sarkar, Shaw, Kim, Sarkar, and Shin (2017) developed an integrated inventory model with the assumption of variable transportation costs, imperfect production, and inspection policy. Kim, Kim, Sarkar, Sarkar, and Iqbal (2018) studied an integrated model with investments for setup cost reductions and quality improvements, imperfect products, and partial backorders. Cheikhrouhou et al. (2018) developed a supplier-retailer model for the inspection errors in terms of misclassification errors. They developed two different policies to send back the defective items to the supplier.
In conventional SC and production models, the production rate has been considered a rigid and constant parameter. However, with the developments in technology, the advanced machines have the capability of operating at different speeds, which makes the production rate a variable or flexible quantity (Sarkar and Chung (2019)). Frequently, the higher production rates are required to fulfill the increasing demand levels, forcing production systems to operate at higher production speeds. This causes these production systems to deteriorate quickly compared to one that operates slower with a constant production rate. The production system deterioration gradually increases the percentage of products not meeting the quality specifications, or in simple words, it reduces the quality of the products. Hence, there exists a trade-off between maintenance activities and different operating levels, as reflected by product quality, production speeds, and production process deterioration. The problem of determining optimal production rates, unit production costs, and operating levels under fuzzy demand is a complex decision-making problem.
Khouja and Mehrez (1994) were the first to introduce the concept of flexible production in terms of variable production rates and its relation to the process quality. Glock (2010) analyzed a single product two-stage production-inventory model with variable production rates and multiple equal and unequal sized shipments from one stage to the subsequent stages. Glock (2011) extended the previous model (Glock, 2010) for multi-stage production with the consideration of variable production rates at each production stage, deterministic demand, and equal and unequal sized batch shipments. Majumder, Jaggi, and Sarkar (2018) studied single vendor multiple buyer model with variable production rate, stochastic demand, lead time, and partial backordering for unsatisfied demands at the buyer’s end. Sarkar, Majumder, Sarkar, Kim, and Ullah (2018) investigated an integrated model for a single vendor and multiple buyers with three different deteriorating quality functions of process because of the increasing variable production rate and random demand during controllable lead time. Recently, Sarkar and Chung (2019) introduced an integrated model for work-in-progress inventory and production with variable production rate, investments for quality enhancement and setup cost reduction, and stochastic lead time demand. They concluded that the flexible production systems with variable production rates help control the number of produced defective products, reduce the holding cost, and minimize the SC cost. Few more studies on this topic can be found in Wang et al., 2015, Song et al., 2013. From the previous studies on variable production rates, one can note that no study has considered any constraint for the vendor. However, in industrial and practical scenarios, the vendor and the buyer face constraints or limits on different resources like budget, space, raw materials, and service level for customers’ demands.
Some studies have considered the practical constraints for the vendor and the buyer in production and supply chain models. Taleizadeh, Niaki, and Barzinpour (2011) presented a buyer-vendor supply chain model for multiple products under the stochastic demand. They considered the limited capacity for the buyer to purchase products and warehouse storage space limitations for the vendor. Malik and Kim (2020) studied a supply chain model for a flexible production system with available budget, storage space, and service level constraints. Nobil, Sedigh, and Cárdenas-Barrón (2020) developed a constrained multi-product single-machine economic production quantity model with the discrete delivery order. They considered limited budget for the production cycle for the joint production policy. Zhang, Zhang, and Yao (2020) discussed a newsvendor problem for two substitutable products with the budget limitations to determines the optimal order quantity and selling price for each product by the Karush-Kuhn-Tucker (KKT) conditions. Recently, Malik and Sarkar (2020b) proposed a disruption management model in a multi-item production system by considering practical constraints like the available budget for the production cycle and storage space.
Porteus (1986), for the first time, introduced the quality improvements and setup cost reductions for the imperfect production systems with continuous investments. Further, Liao and Shyu (1991) proposed lead time reductions with discrete investments as an additional crashing cost. Recently, Malik and Sarkar (2018a) studied an inventory model for the setup cost reduction with continuous investments. The setup cost reductions with continuous investments were analyzed repeatedly in the literature under different conditions until the introduction of the discrete investments by Huang, Cheng, Kao, and Goyal (2011). The discrete investments for the setup cost reductions in production systems are closer to reality. The advantage of the discrete investments is management can start investments whenever required and can be stopped at any time. Recently, the lead time crashing with quality improvements and setup cost was analyzed under the stochastic-fuzzy conditions by Malik and Sarkar (2018b). Dey, Sarkar, and Pareek (2019) studied SC model for the setup time and cost reductions in a flexible production system. Sarkar and Chung (2019) analyzed SSMD policy for work-in-process inventory in a production system with quality improvements, setup cost, and lead time control. Further, Malik and Sarkar (2020a) proposed a coordination supply chain model for the flexible production systems by considering discrete investments and lead time reductions. They developed the coordination policy for the supply chain members based on the Nash bargaining model.
Mostly, studies consider the demand is fixed and certain. However, in practical scenarios, this is not the case and demand is uncertain. For this, some authors considered demand as the triangular fuzzy number and solved this problem. Pan and Yang (2008) investigated the JELS model with uncertain demand and an optimal number of deliveries to the buyer. De and Sana (2013) developed an EOQ (Economic Order Quantity) model with backorders and promotional index for fuzzy order and shortage quantity as the decision variables. Further, De and Sana (2013) model was extended by De, Goswami, and Sana (2014) for the time sensitive-backlogging. De and Sana (2014) discussed the managerial issues in a production plant with multi-manufacturers and demand uncertainties. They considered multiple constraints for the production system and made some comparisons between intuitionistic fuzzy optimization and general fuzzy optimization methodologies. De and Sana (2018a) analyzed an economic production-inventory quantity model for stochastic-uncertain demand with an order size, reorder point, and lead-time as decision variables.
Jamali, Sana, and Moghdani (2018) developed an efficient inventory control policy for stochastic demands to reduce unnecessary under and over-stock expenses and maintain a high customer service level. They employed the simulation-based optimization methods to achieve optimal solutions and proposed a genetic algorithm and hybrid improved cuckoo search algorithm to solve these types of problems. De and Sana (2018b) proposed a two-layer supply chain model under stochastic- fuzzy demand with unknown variance and finite mean. The supplier offers the Buyback policy to the retailer and shortages cost is related to loss of profit and goodwill dependent. They developed a decentralized system and a centralized system of decision making and solved the crisp model analytically and employed the fuzzy Hausdorff distance method for the fuzzy model. Bhattacharyya and Sana (2019) studied a production-inventory system for the green manufacturing industry with green technology implementation by considering the capital investment for setup, service level, and green technology as decision variables. They considered production lot size as an increasing function of the investment to set up the manufacturing system and green technology.
Dey (2019) formulated a stochastic fuzzy JELS model with defective items, safety factor, and investment-based improvements in process quality. In a recent study, Malik and Sarkar (2019) analyzed the seller-buyer model for stochastic fuzzy demand, lead time, and setup cost reductions with the consideration of reliable and unreliable sellers. Moghdani, Sana, and Shahbandarzadeh (2019) contributed to solving an EPQ model by considering multiple deliveries and uncertain demands. They used GA, PSO, GWO, and ICA, well-known metaheuristic algorithms, to solve the problem and provided that GWO is more efficient than other algorithms. Further studies on uncertain demand and fuzzy sets can be found in De et al., 2014, Ameri et al., 2019, Birjandi et al., 2019, and Akhyani, Birjandi, Sheikh, and Sana (2020). From the above literature, no study considered flexible or variable production rates to tackle the uncertain demands and make sure to improve the customers' service level. There exists another research gap that needs to be considered in SC modeling.
One can see from the above literature review and Table 1, only some researchers have considered variable production rate of machine and quality of the production process in relation and proposed various single-vendor single-buyer supply chain models. However, no one has developed a supply chain model considering uncertain-stochastic demand in any of the above SC models for imperfect production and variable production rate. The studies mentioned above for variable production rates considered the lead time a normally distributed stochastic demand. However, in practice, it is hard to find the exact distribution of the lead time. Therefore, it is appropriate to consider the distribution-free approach for the stochastic lead time demand.
Additionally, the authors noted that the researchers had missed the point of different practical constraints like the available budget for the vendor, the available space for inventory at the buyer’s setup, and the service level for the end customers’ demand. From the literature, the authors identify a significant research gap that should be addressed. This proposed research fulfills the identified research gap. It presents interesting insights for industry managers by considering the impact of these, above mentioned, three real-life based constraints and the uncertainty of the customer’s demand at the buyer’s place. The discrete investments are made for the setup cost reduction for the vendor to minimize the SC cost. In the next part of the paper, the authors will introduce notation and assumptions. Next, the authors will develop a mathematical model for the above-described research gap and provide numerical analyses to validate the mathematical model. In the last part of the article, the authors provide sensitivity analysis and conclusions with future research directions.
Section snippets
Notation, and assumptions
The list of mathematical notation for the mathematical model is as followsDecision variables order quantity for the buyer (units/order) P production rate for the vendor (units/year) L lead time (weeks) n number of shipments delivered from vendor to the buyer (integer value) J investment for setup cost reduction to achieve setup cost S per production cycle ($) Parameters A ordering cost for the buyer ($/per order) S0 initial setup cost for the vendor ($/setup) S(J) setup cost for the vendor at investment J
Mathematical model
First of all, authors are going to formulate a cost model for the buyer and the vendor with crisp values. Next, authors will modify the model to the fuzzy mathematical model and to solve it; authors will defuzzify it by using the signed distance method.
Solution methodology
The presented cost function in Eq. (11) with space constraint in Eq. (12) and budget constraint in Eq. (13) is a non-linear constrained problem. For these kinds of non-linear problems, the Kuhn-Tucker optimization method is the best approach to get the global optimal solutions. In literature, the Lagrangian multiplier is the only available technique to convert constrained problems into unconstrained problems. Thus, to solve the model, authors first convert it to an unconstrained problem and
Numerical example, results, and discussion
In this section, authors present three numerical examples to test and a comparative study to validate the optimality of the above proposed model.
Sensitivity analysis
In this section, a cost parameters based sensitivity analysis is presented for the above example. The value of each parameter is changed from −50(%) to +50(%), while all the other parameters are fixed. The variations in TC (Δ) are calculated for different values of parameters (A, S0, HV, HB, R) and shown in Table 10.
From the sensitivity analysis table (Table 10), following insightful observations can be drawn. For all three cases, the increase and decrease in considered five parameter values
Conclusions and future research
In this paper, authors extend Khouja and Mehrez (1994) model with quality function, inspection, discrete investments, and three different constraints under a single vendor single buyer SC model. This study establishes the relationship between the production rate and quality of products under the stochastic and uncertain demand. A mathematical function is established for the unit production cost, which depends on the production rate and varies with the change in the optimal production rate. The
CRediT authorship contribution statement
Asif Iqbal Malik: Conceptualization, Methodology, Validation, Formal analysis, Writing - original draft. Byung Soo Kim: Supervision, Conceptualization, Writing - review & editing, Project administration.
Acknowledgments
This work was supported by Research Assistance Program (2019) in the Incheon National University and this research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning, South Korea (Grant number: NRF-2019R1F1A1056119).
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