1. Introduction
In the present socio-economic scenario, there is a consistent tug-of-war between the optimum order quantity and the shipment cost of items produced in any production setup. Considering the changing customers’ requirements, production firms are constantly updating their shipment policies to optimize economic order quantities to enhance net profits. In this scenario, production shipment policy acts as an imperative parameter for any firm to survive in a high-paced competitive environment. This research focuses on a comprehensive shipment policy, considering the most significant parameters, including backorders in a shipment, imperfections in operations, and a reworked item in a production system. To minimize costs, different researchers have modeled several optimal batch problems considering different production conditions. For instance, Harris [
1] is among the pioneers who developed the Economic Orders Quantity (EOQ) inventory model. The second most important model was developed by Taft [
2]. This model is called the Economic Production Quantity (EPQ) model. Subsequently, these models were modified and expanded by other researchers [
3,
4,
5,
6]. The research has shown that small perturbations in the parameters of the EOQ and EPQ models do not impose any significant impact on the solution of a problem [
7,
8,
9]. Owing to this, the Economic Production Quantity (EPQ) model emerged as an optimal substitute that shows promising results for a production environment when applied with some assumptions.
In an actual production environment, the system runs with some imperfections. The imperfections in a production system produce low-quality items for several reasons—namely, defects in raw materials, changes in machine capabilities, backorders, rework, and differences in the experience of the operators. Some studies are available in the literature in which the proposed models have considered these imperfections. For example, Jamal et al. [
3] studied the EPQ model to obtain the optimum batch size. The proposed model considers a re-work process after several production cycles. Expanding the contributions of Jamal et al. [
3], Sarkar et al. [
4] formulated the same problem with additional terms of backorders. The model proposed by Cardenas-Barron et al. [
7] encompasses numerous parameters. The model addresses the reworked production quantities and other production system defects. Chang et al. [
8] proposed a mathematical model for total time consumption in a production system involving manual and automatic operations. Wee et al. [
5] adopted the same methodology and developed a model which considered the development of refurnished products with non-conformities. It was concluded that in repeated manufacturing cycles, there is an effective way to reprocess faulty and defective products. The data obtained confirmed that the critical aspects could be more related to the manufacturing cost and the service expenditures of the process.
An identical model was presented by Sarkar et al. [
10], which focused on the inflation effect. It was shown that the prolonged use of the manufacturing units could potentially damage the smooth operating of the system, i.e., it could produce defects in the system. The focus of the research was on how to overcome the defects produced during smooth operations and to reprocess the defective products. The demands of overtime on the workers could be a cause of defects in the system, or it could be other unknown reasons. The focus is on the prolonged reproducibility of the system using the reliability of the system. The decision variable of system reliability was used to hypothesize a new model consisting of the integrated entities that were organized to maintain a smooth operating system as well as to instantly remove defects. It was found that the model reduced the operating system’s total costs. By means of an algebraic model, it was shown that three parallel distribution systems can be operated to attain considerably lower production costs.
Swenseth et al. [
11] derived a mathematical model for reducing overall production costs considering a large shipment volume. Ertogral et al. [
9] established a model to deliver products to retailers considering the unit transportation cost. Ghasemy Yaghin et al. [
12] developed a comprehensive model for an optimum number of lot sizes and quantities of various values. Ekici et al. [
13] assumed certain limitations on the production batch size and proposed a model for varying customer demands under several manufacturing settings. Geri et al. [
14] developed a model for stock policy in which the single manufacturer shipped single types of product to the retailer in a fixed lot size. Tseng et al. [
15] reported that transportation cost is the key to economic growth, and it represents 6.5 percent of the market revenues. Rodigue et al. [
16] assessed the conditions driving the global forms of production, distribution, and transport mainly by looking at the levels of geographical and functional integration of global production networks given the high level of disintegration within them.
Cardenas-Barrón [
17] presented a simple derivation of the work presented by Jamal et al. [
3]. Differential calculus was used to find the optimal solution to the problem, and the derivation was based on algebraic derivation. The results obtained were equivalent to the results obtained by Jamal et al. [
3]. Cárdenas-Barrón et al. [
18] extended the work of Chiu et al. [
19,
20] by determining both the optimal number of shipments and the optimal replenishment lot size. The solution of Cárdenas-Barrón et al. [
18] presented better results than Chiu et al. [
19,
20]. Kun-Jen Chung [
21] studied the work of Cárdenas-Barrón et al. [
7] and presented a solution procedure for finding the optimal solution of the total cost. Goyal et al. [
22] considered the problem of determining the economic production and shipment policy of a product supplied by a vendor to a single buyer to minimize the total costs incurred by the vendor and the buyer. Sana et al. [
23] developed a framework of production policy to find the optimal safety stock, production lot size, and optimal production rate. Sarkar et al. [
6] derived mathematical models to obtain the optimal cycle time to minimize the annual total relevant cost.
To reduce the production of imperfect products, the production systems must be highly reliable to cope with the changing shipment demands and supply. Some substantial studies have considered production and system reliability parameters. For instance, Sarkar et al. [
24] considered the production cost, development cost, and material cost as dependent reliability parameters. Sarkar et al. [
25] developed a production-inventory model for item deterioration in a two-echelon supply chain management. The objectives of the study were to obtain minimum cost and optimum lot size for three different models with an integer number of deliveries. Sarkar et al. [
26] presented an economic manufacturing quantity model for stock-dependent demand in an imperfect production process in which unit production cost was employed as a function of production rate and reliability parameters. Extending their work, a new model presented by Sarkar et al. [
27] adopted considered preventive and corrective maintenance, and safety stock for repair times.
Likewise, Sarkar et al. [
28] also developed models for optimum batch quantity in a multi-stage system with a rework process for two operational policies. The first policy deals with rework within the same cycle with no shortage, and the second policy deals with the rework done after several cycles incurring shortages in each cycle. Taleizadeh et al. [
29] developed an economic production and inventory model in a three-layer supply chain for a single-product and general demand functions. Hayat et al. [
30] developed a shipment policy for an imperfect production setup with transportation costs taken into consideration. The model analyzed lot-sizing for manufacturers and retailers with imperfections in terms of equal-sized shipments.
Wanzhu et al. [
31] worked on optimum production lot sizing and implemented the strategy in a quick response manufacturing setup. The statistical evaluations induced optimized production scheduling and significant improvement in lead time, including shipment time and costs. Abdul et al. [
32] developed a transportation cost model to analyze the supply chain efficiency, including the shipment of products with different lot sizes and in-process inventory. They proposed a mathematical model based on metaheuristics, and the result was supportive in a closed-loop supply chain context. Asad et al. [
33] also highlighted some important parameters which are imperative for the components of supply chain integration considering shipment of products to retailers. Waqas et al. [
34] studied the prospective dimensions of production flexibility at their interface with the integrated functional units. The work showed that optimal production flexibility is important for developing a reliable shipment policy. With system flexibility, the implementation of an integrated advanced manufacturing planning and execution system, which could support shorter product cycles with real-time monitoring of shipment processes, is imperative [
35].
Guchhait et al. [
36] developed and optimized the cost model by addressing the defective products, backorders, and warranty policy. Taleizadeh et al. [
37] developed a stochastic inventory control model with partial backordering and introduced supply disruption. Snyder et al. [
38] proposed a model that considers the variance of the variable lead time-dependent demand. They estimated mean and variance through smoothing methods. Hsiao et al. [
39] also employed a stochastic demand and developed a model for a single vendors that considers a delay in transportation. Dominguez et al. [
40] presented a model with variable lead time that addresses the dynamic property of the closed-loop supply chain system and the multi-echelon supply chain. Haeussler et al. [
41] also introduced a model by employing optimization techniques. Malik and Sarkar [
42] introduced a game strategy and stochastic lead time demand to reduce the total expected cost. The contributions of researchers in similar areas are presented in
Table 1.
The proposed work expands the most recent study of Hayat et al. [
30] by developing a shipment policy for the defective manufacturing system. None of the previous studies mentioned in
Table 1 have considered the distributor in their model development for imperfect systems. However, in real-life cases, the distributor is as an important stakeholder and must not be ignored in the mathematical formulation. In this context, the proposed work extends the previous research with the inclusion of the distributor in the model development. For the proposed scenario, the inventory flow with rework and backorders is illustrated in
Figure 1.
The rest of this paper is organized as follows:
Section 2 specifies the case for model development and provides the parameters and assumptions to be taken in the model development. Mathematical models for two of the proposed cases are developed in
Section 3. For the purpose of having more insight into the developed models,
Section 4 presents numerical computation and sensitivity analysis. Finally, this work is summarized and future directions are provided in
Section 5.
5. Conclusions and Future Recommendations
The model developed for the optimal shipment of products and numerical findings contributes to the knowledge in the fields of production and supply chain management. The main contributions of the presented research are as follows:
In this research, a shipment size model for the manufacturer, distributor, and retailer with an equal-sized shipping policy is developed for the imperfect production system. The all-unit reduction transportation cost structure has been evaluated for the proposed model. The optimum solution methods are also developed and analyzed. From the solution procedure, it can be observed that the shipment decision varies as the transportation cost is incorporated into the system. The model presented in Case-II resulted in a reduced total system cost by 30.70, which means the total cost has been reduced by more than 1.00%. This shows the effectiveness of the proposed development for an imperfect production system. Similarly, in the model presented in Case-II, the total system cost has been reduced to 76.29, which means that the total cost has been reduced by more than 2.00%. This also highlights an advantage of the developed model. These reductions in total costs are just based on the numerical values assumed in the presented examples. The assumed numeric values are just for model validation purposes and in a real scenario are much lower (demand: just 300 units per year, production 550 per year, and so on) than the real cases values. These results can only be applied and compared with the real cases if the numeric values in the computations are taken as per actual data.
The numerical computations and sensitivity analysis are performed to point out the specifications of this work. From sensitivity analysis, it is found that increasing the values of certain parameters, like the fixed setup cost (K), ordering cost (Km, Kr, and Kd), unit inventory carrying cost (Cm, Cr, and Cd), fixed cost per backorder (Bf), linear backorder cost (BL), unit manufacturing cost (Mc), and demand (Dr) results in increased values of the total cost of the system. By decreasing the values of these parameters, the value of the total cost of the system is decreased as well. Different results (−50% to +50%) can be estimated for (L) and (N) by changing the shipment lot size (L) and the number of shipments (N). Furthermore, it is found that demand (Dr) and unit manufacturing cost (Mc) are the most sensitive parameters compared to all other parameters.
In this research, a single type of production system is considered to produce only one type of item. In actual production systems, several products can be produced simultaneously in multistage production systems. So, the proposed model can be extended considering multiple products and a multistage production system. In this model, we have considered one retailer, manufacturer, and distributor, while many retailers, manufacturers, and distributors can be considered to extend the scope of this model. In this shipment policy, we have considered a single manufacturer, single retailer, and single distributor. Future research will consider a situation where multiple suppliers may exist along with the manufacturer, retailer, and distributor. In addition, the research can be extended by considering the variable demand rate.