A Probabilistic Inventory Problem for Imperfect Quality with Partial Shortage Backordering, Carbon Emission, and Its Computation Using Genetic Algorithm in Python

In the manufacturing process, items produced in factories under the supervision of manufacturers are not always in perfect condition in terms of quality, and this assumption serves as a fundamental premise in contemporary inventory modeling. The presence of imperfect items can be described either with deterministic or probabilistic approaches. Many research studies have been conducted focusing on inventory models that consider the presence of imperfect items. Inventory models dealing with probabilistically imperfect items and incorporating controlled lead times have been explored [1], which followed a lead-time demand distribution approach. Building on this research, inventory models were further investigated, while also considering the concept of shortage backordering [2]. Additionally, an inventory model for products with imperfect quality has been examined, taking into account the assumption of quality improvement over lead time [3]. Other relevant research related to the aspect of imperfect quality also has been conducted by some researchers [4-8]. The development of an inventory model relies on predefined assumptions. Nevertheless, there are still numerous conditions and scenarios within supply chain management that have not yet been incorporated into inventory models dealing with imperfect-quality products.

Recent inventory management research has increasingly focused on environmental concerns, specifically the carbon emissions (greenhouse gases) generated by production machinery, equipment, loading/unloading operations, and vehicles. Several studies have included carbon emission factors in their inventory models [9-19]. The shortage condition is a situation that is frequently encountered in practice. Interestingly, this condition is not actively sought to be circumvented by participants within the supply chain system. Various strategies are employed to minimize the occurrence of product shortages, with one such approach being the adoption of a partial backordering policy. This policy, in turn, influences the quantity of product shipments from manufacturers to fulfill retailer demands. Therefore, the partial backordering policy can also be integrated into the analysis of carbon emission management stemming from loading and unloading equipment and vehicles. The relationship between carbon emission policies and partial backordering also has been explored [20].

Furthermore, the presence of imperfect quality products can accelerate the depletion of product inventory beyond the planned timeframe, as imperfect quality items cannot be sold. This aligns with the partial backordering policy, which seeks to address such shortages. Therefore, in this context, there is a hypothesis that the existence of imperfect quality products and the partial backordering policy can be analyzed concurrently within the framework of carbon emission management policies. To date, no prior research has explored this particular combination of factors. From the existing literature, it is evident that no prior research has addressed the construction of a mathematical model for product with imperfect quality while simultaneously considering greenhouse gas emissions, reorder procedures, and policies for managing shortages through backordering as part of the objective function formulation.

The optimization analysis of inventory models involving multiple variables and parameters is generally quite complex. It is not uncommon for the optimum analytical solution to become infeasible to express explicitly. Numerical approaches often provide a realistic choice, where solutions can be obtained through computation based on algorithms designed to determine the optimal solution. One of the programming languages suitable for developing algorithms to handle complex computational cases is Python.

Based on the above description, it can be explained that this research involves 2 research questions (RQ):

RQ 1. How does a multi-player inventory model consider aspects such as imperfect quality, shortage backordering, and carbon emission management?

RQ 2. How does a genetic algorithm using the Python programming language determine the optimal solution of the inventory model computationally?

The research results, which are the answers to the research questions, are innovative in that they introduce a new multi-player inventory model that has not existed before. Previous research has not included a green inventory model that integrates aspects of imperfect quality, shortage backordering, and carbon emission in one mathematical model. Furthermore, innovation lies in the genetic algorithm developed to determine the optimal solution for the green inventory model computationally. Genetic algorithms are rarely used in inventory model literature, and the use of the Python programming language to develop this genetic algorithm in the context of inventory models has not been done before this research. In this research, a new green inventory model will be formulated, considering the existence of imperfect quality products and incorporating a partial backordering policy. The optimization analysis will be constructed based on integra- tion and synchronization schemes. Subsequently, the numerical solutions will be determined using a genetic algorithm developed using Python.

The rest of this explanation is structured as follows. In Section 2 general assumptions, formulation of mathematical model, and analytical analysis for the optimum value are formulated. The algorithm and numerical computation of the inventory model are discussed in Section 3. Finally, some conclusion from our works and provide Several recommendations for future research are presented in the final section.

3.1 Mathematical inventory model

A mathematical inventory model will be developed for an inventory system comprising a single manufacturer with multiple retailers. The existence of imperfect quality will be taken into account, and a partial backordering policy will be adopted by all retailers. Before delving into further details, let’s first introduce the notations and assumptions being utilized.

Assumptions

1. The inventory management system pertains to a single item product. Planning horizon is assumed to be infinite (without any time constraints).

2. The order level is known, constant, and continuous.

3. There is no lead time and the replenishment rate of product is infinite.

4. The expense (cost) related to carbon emissions is taken into account, and the demand rate remains consistent and well-known.

5. Shortage conditions are permissible and are addressed through a partial backordering policy. We adhere to Mahato and Mahata [20]’ approach, where a fixed rate β is used to backorder a portion of the shortages.

6. Imperfect items a present in a lot of sizes q. The fraction of these imperfect items, denoted as γ_i, follows a probability density function represented by f(γ_i). To guarantee that the manufacturer possesses adequate production capacity to meet retailer demand, it is postulated that the expected value of $E\left[\gamma_i\right]<1-\frac{D}{P}$.

7. The entirely of the lot quantity sorting procedure is finished at the retailer’s site before the commencement of each cycle duration T. In this scenario, the time taken for sorting is included in the delivery lead time. Any products displaying imperfect quality will be sent back to the manufacturer during the subsequent lot shipment via a return process.

8. There is an assumption that no extra shipping expenses are incurred for this return process. The manufacturer offers compensation of ω for each imperfect-quality product identified. Additionally, the manufacturer plans to resell these imperfect-quality products through a secondary market.

9. The principles of synchronization and integration are agreed upon by the manufacturer and all retailers for the determination of optimal solution.

Next, the inventory mathematical modeling process will be carried out with the following steps:

1. Formulation of Assumptions: Based on the assumptions used, a mathematical inventory model will be developed. This inventory model includes the total cost formulation needed by each player in the supply chain system, namely the retailers and the manufacturer.

2. Determination of the Total Cost Function: The total cost functions for the retailers and the manufacturer are determined based on the identification of the cost components required according to the previously defined assumptions and inventory levels. The inventory levels are presented in a graph that depicts the inventory condition of each retailer and manufacturer within one cycle.

3. Formation of the Combined Retailer Total Cost Function: This function is obtained by summing up the total cost functions of all individual retailers.

4. Formation of the Joint Total Cost Function: This total cost function is derived based on the integration scheme assumption by summing the total cost functions of all retailers and manufacturer.

Once these four stages are completed, the inventory modeling process is finished. The next step is optimization to obtain the optimal values of the decision variables, q and n, using optimization principles and theories applied to the supply chain system total cost function obtained in step 4.

The construction of an inventory model for imperfect-quality products, involving a single manufacturer and multiple retailers, will now be explained, based on the previously outlined assumptions. The inventory model consists of a total cost function that encompasses both the manufacturer and retailers, which is subsequently combined into a total cost function for the entire inventory system through an integration process. The objective function for retailers in one cycle is composed of several components, encompassing ordering expenses, product transportation charges, product sorting expenditures, holding costs, backordering costs, and carbon emission handling cost.

The ordering cost C_i^b is the standard cost that each retailer must pay to the manufacturer to enable the production of products needed by each retailer. The products produced by the manufacturer will be shipped to each retailer, and the transportation cost F_i is borne by the retailer. Since it is assumed that there are always imperfect quality items, the retailer will undergo a sorting process when a lot arrives at the retailer's location during each shipment. This sorting process incurs a sorting cost per unit product s_i. The products at the retailer's location will be stored before they can be sold during the sales period. The storage process incurs a holding cost h_i^b. Then, to avoid shortage conditions, each retailer agrees to use a partial backorder strategy with a reorder cost per unit of products per unit of time b_i and with a proportion of shortage that will be ordered set at β. It is then agreed that all carbon management costs will be charged as part of the product fulfillment cost by the manufacturer to the retailer. First is the carbon emission cost from loading equipment EG₂ in the process of loading products into the transport vehicle. The transport equipment uses fuel that generates carbon emissions. Next is the carbon emission cost EG₂ from the type of transport vehicle used to deliver products from the manufacturer's location to each retailer's location. It is assumed that the same type of vehicle is used for each retailer, so EG₃ has the same value for each retailer i. Finally, the carbon emission cost component from unloading equipment used to unload products from the transport vehicle that arrives at the retailer's location is considered, assuming that all retailers use the same type of unloading equipment and fuel that can generate carbon emissions.

Firstly, the component of product holding costs will be determined. It is assumed that the sorting process is entirely 100% completed when the products reach the retailer’s location. Imperfect-quality products discovered during this process are promptly separated and returned to the manufacturer during the subsequent product shipment. It is further assumed that no holding expenses are incurred for the identified imperfect-quality products. The assessment of holding costs is derived from the inventory levels are presented in the following diagram.

According to the inventory process illustrated in Figure 1, the cost for holding component in the cost function for each retailer can be formulated as:

$h_i^b\left(\frac{n}{2}\left(\frac{q_i-\gamma_i-\beta q_i}{D}\right)^2+\frac{q_i^2 \gamma_i\left(1-\gamma_i\right)}{D}\right)$

Meanwhile, the shortage cost component is formulated as $\frac{1}{2} C_b \beta n q_i / D$. The emission handling cost is allocated to each retailer i. The type of handling equipment and vehicles are determined by the manufacturer. The emission handling cost element is computed by adding up the costs incurred from the loading and unloading equipment as well as transportation equipment emissions, formulated as:

$J_i E G_1 C_j q_i+C_l E G_2 m_{p p} q_i+C_{u l} E G_3 m_{p p} q_i$.

Consequently, the function of total cost for each retailer is denoted as $\mathbb{R}_i(.):\mathbb{R}^2 \rightarrow \mathbb{R}$ with:

$\begin{gathered}R_{i\left(q_i, n\right)}=C_i^b+n F_i+s_i n q_i+h_i^b \frac{1}{2}+\frac{1}{2} C_b \beta n \frac{q_i}{D}+ \\ J_i E G_1 C_j q_i+C_1 E G_2 m_{p p} q_i+C_{u l} E G_3 m_{p p} q_i\end{gathered}$                   (1)

Now, the manufacturing objective function, denoted as the total cost function, will be detailed. Comprising several components, this manufacturing objective function includes setup costs and compensation expenses incurred due to the identification of imperfect-quality products. Drawing from Figure 1 and building upon the research conducted by Lin [1] and also result by Konstantaras et al. [4], the inventory holding cost for the manufacturer under the integration scheme is established as the product of the cost per unit of product held and the cumulative disparity between the manufacturer's inventory level and retailer i's cumulative inventory level. The formulation of holding costs per cycle is accomplished through the following equation:

$\begin{gathered}h_p\left\lceil n\left(q P^{-1}+(n-1) T\right) q-\frac{n\left(\frac{n}{P}\right)}{2}\right\rceil-h_p[q+2 q+\cdots+ \\ (n-1) q T]=h_p\left(\frac{1}{P} n q^2-\frac{n^2 q^2}{2 P}+\frac{1}{2} D^{-1} n(n-1)(1-\gamma) q^2\right)\end{gathered}$

Hence, the total cost for the manufacturer is formulated by:

$\begin{gathered}M(q, n)=C_p+\omega n q \gamma+h_P \frac{n q^2}{P}-h_P \frac{1}{2 P} n^2 q^2+ h_p\left(\frac{1}{2} D^{-1} n(n-1)(1-\gamma) q^2\right) \\ \text { with } q=\sum_{i=1}^n q_i \text { and } \gamma=\left(\gamma_1, \gamma_2, \ldots, \gamma_n\right) .\end{gathered}$                     (2)

3.2 Optimum analysis

The manufacturer and all retailers are assumed to agree to employ an integration scheme as the foundation for determining the optimal decision variables. The integration scheme in the inventory system is translated into the establishment of the overall cost function for the inventory system which is the combination of the manufacturer's objective function and the retailers' objective function, $J(q, n, B)=M_i\left(q_i, n\right)+\sum_{i=1}^k R_i\left(q_i, n\right)$ as follows:

$\begin{gathered}J(q, n)=h_p\left(\frac{n q^2}{P}-\frac{n^2 q^2}{2 P}+\frac{n(n-1) q^2(1-\gamma)}{2 D}\right) +\sum_{i=1}^k\left(C_i^b+n F_i+s_i n q_i\right) \\ +\sum_{i=1}^k \frac{1}{2} h_i^b n\left(\frac{\left(q_i-\gamma_i-\beta q_i\right)^2}{D} \frac{q_i^2 \gamma_i\left(1-\gamma_i\right)}{D}\right) \\ +\sum_{i=1}^k\left(\frac{1}{2} \frac{c_b \beta n q_i}{D}+J_i E G_1 C_j q_i+C_l E G_2 m_{p p} q_i\right) +\sum_{i=1}^k\left(C_{u l} E G_3 m_{p p} q_i\right)+C_p+\omega n q \gamma .\end{gathered}$                   (3)

where, $q=\sum_{i=1}^n q_i$, and $\gamma=\left(\gamma_1, \ldots, \gamma_n\right)^T$. Since every party in the inventory system also agree to use the synchronization principle, the relationship $q_i=D_i \frac{q}{D}$ hold. Thus, Eq. (3) can be expressed as the following equation:

$\begin{gathered}J(q, n)=\frac{q^2 n}{D^3}\left(\frac{h_p D^3}{P}-\frac{(1-\gamma) D^2 h_p}{2}\right)+ \frac{q^2 n}{D^3} \sum_{i=1}^k h_i^b D_i^2 \gamma_i\left(1-\gamma_i\right)+\frac{q^2 n}{D^3}\left(-\beta \sum_{i=1}^k h_i^b D_i^2+\frac{1}{2} \sum_{i=1}^k \beta^2 D_i^2\right) \\ +\frac{q^2 n}{D^3}\left(\frac{1}{2} \sum_{i=1}^k h_i^b D_i^2\right)+\frac{q^2 n^2}{-\frac{1}{2 P}+\frac{(1-\gamma)}{2 D}}+ q n \frac{1}{D}\left(\omega \gamma+\frac{1}{D} \sum_{i=1}^k s_i D_i-D^{-2} \sum_{i=1}^k h_i^b \gamma_i D_i\right)+ \\ q n \frac{1}{D}\left(\frac{\beta}{D^2} \sum_{i=1}^k h_i^b \gamma_i D_i+\frac{c_b \beta}{2 D}\right)+ \frac{q}{D} \sum_{i=1}^k D_i\left(J_1 E G_1 C_j+C_l E G_2 m_{p p}\right)+\sum_{i=1}^k D_l C_{u l} E G_3 m_{p p}+ \\ \frac{n}{2 D}\left(\sum_{i=1}^k h_i^b \gamma_i^2+\sum_{i=1}^k F_i\right)+\sum_{i=1}^k C_i^b+C_p .\end{gathered}$                    (4)

Due to the length of the product replenishment cycle $\left(T_{t o t}=\frac{n q(1-\gamma)}{D}\right)$ then we have:

$E\left[T_{t o t}\right]=\frac{n q(1-E[\gamma])}{D}$                     (5)

By employing the renewal-reward theorem, the formulated expected average total annual cost per unit of time becomes $E J(q, n, B)=\frac{E[J(q, n, B)]}{E\left[T_{t o t}\right]}$. Then, we have:

$\begin{gathered}E J(q, n)=\frac{q\left(E[\gamma](1-E[\gamma]) \sum_{i=1}^k h_i^b D_i^2\right)}{D^2(1-E[\gamma])}- \frac{q(1-E[\gamma]) D^2 h_p}{2 D^2(1-E[\gamma])}+\frac{q}{D^2(1-E[\gamma])}\left(\frac{h_p D^3}{p}+\frac{1}{2} \sum_{i=1}^k h_i^b D_i^2\right)+ \\ \frac{q}{D^2(1-E[\gamma])}\left(\frac{1}{2} \sum_{i=1}^k \beta^2 D_i^2-\beta \sum_{i=1}^k h_i^b D_i^2\right)+ q\left(E[\gamma](1-E[\gamma]) \sum_{i=1}^k h_i^b D_i^2\right)+\frac{q n}{\frac{1}{2}\left(1-\frac{D}{P}(1-E[\gamma])\right)}+ \\ \frac{1}{q(1-E[\gamma])} \frac{c_b \beta}{2 D}+\frac{1}{q(1-E[\gamma])} \frac{\left(c_b \beta\right)}{2 D} +\frac{1}{q(1-E[\gamma])}\left(\omega E[\gamma]+\frac{1}{D} \sum_{i=1}^k s_i D_i\right) \\ -\frac{\left(\frac{1}{D^2} \sum_{i=1}^k h_i^b E[\gamma] D_i+\frac{\beta}{D^2} \sum_{i=1}^k h_i^b E[\gamma] D_i\right)}{q(1-E[\gamma])} +\frac{1}{n(1-E[\gamma])} \sum_{i=1}^k D_i\left(J_i E G_1 C_j+c_l E G_2 m_{p p}\right) \\ +\sum_{i=1}^k C_{u l} E G_3 m_{p p}+\frac{n}{2}\left(\sum_{i=1}^k h_i^b \gamma_i^2+\sum_{i=1}^k F_i\right) +D \frac{1}{n g(1-E[\gamma])} \sum_{i=1}^k\left(C_i^b+C_p\right)\end{gathered}$                      (6)

where,

$\begin{aligned} & K_1=\frac{h_p D^3}{P}-\frac{(1-E[\gamma]) D^2 h_p}{2}+E[\gamma](1-E[\gamma]) \sum_{i=1}^k h_i^b D_i^2+ \frac{1}{2} \sum_{i=1}^k h_i^b D_i^2-\beta \sum_{i=1}^k h_i^b D_i^2+\frac{1}{2} \sum_{i=}^k \beta^2 D_i^2, \\ & K_2=\frac{1}{2}\left(1-\frac{D}{P(1-E[\gamma])}\right), \\ & K_3=\omega E[\gamma]+\frac{1}{D} \sum_{i=1}^k s_i D_i-\frac{1}{D^2} \sum_{i=1}^k h_i^b E[\gamma] D_i+ \frac{\beta}{D^2} \sum_{i=1}^k h_i^b E[\gamma] D_i+\frac{c_b \beta}{2 D}, \\ & K_4=\sum_{i=1}^k D_i\left(J_i E G_1 C_j+C_l E G_2 m_{p p}+C_{u l} E G_3 m_{p p}\right)+ \frac{n}{2}\left(\sum_{i=1}^k h_i^b \gamma_i^2+\sum_{i=1}^k F_i\right), \\ & K_5=\left(\sum_{i=1}^k C_i^b+C_p\right) .\end{aligned}$

Next, the optimization process for the formulated expected average total annual cost per unit of time (function (6)) will be carried out using classical optimization principles, specifically partial derivatives. The first step is to determine the extreme values of function (6) by taking the partial derivative with respect to each decision variable, q and n. Thus, we obtain the following equation:

$\begin{gathered}\frac{\partial E J(q, n)}{\partial q}=\frac{K_1}{D^2(1-E[\gamma])}+\frac{n K_2}{2}-\frac{K_3}{q^2(1-E[\gamma])}- \frac{D K_5}{n q^2(1-E[\gamma])}=0\end{gathered}$           (7)

$\frac{\partial E J(q, n)}{\partial n}=\frac{q K_2}{2}-\frac{K_4}{n^2(1-E[\gamma])}-\frac{D K_5}{n^2 q(1-E[\gamma])}=0$                      (8)

To obtain the optimal values of (q and n) (i.e., $q^*$ and $n^*$), we perform algebraic manipulation of Eqs. (7) and (8) and derive the analytical solution as follows:

$q^*=\frac{DK_5}{\sqrt{\frac{\left(n^*\right)^2 K_2(1-E[\gamma])}{2} \sqrt{\frac{K_3+\frac{D K_5}{n}}{\left(\frac{K_1}{D^2}+\frac{\left(n^*\right) K_2(1-E[\gamma])}{2}\right)}}-K_4}}$             (9)

Since function (6) is a convex function, the value of q^* obtained in Eq. (9) represents a minimum extreme point. Furthermore, determining an explicit analytical solution for the optimal solution based on Eq. (9) is not feasible, so the determination of the optimal solution can be achieved using a numerical computational approach, which will be explained in the next section.

1.png

Figure 1. Retailer’s inventory level

A multi-compartment inventory model was developed for a single manufacturer and multiple retailers. The inventory model was designed considering imperfect quality products and carbon emission costs. Due to the involvement of numerous parameters, obtaining an explicit analytical solution was not feasible, and only implicit solutions could be derived analytically. The focus of this study was on utilizing genetic algorithms implemented in Python to find numerical approximations. From the simulation outcomes, it can be deduced that the inventory model created demonstrates resilience in the face of variations in crucial parameters like the rate of imperfect quality and the proportion of ordered shortages. This implies that although variations in these parameters affect the optimal value of q, the resulting changes are within tolerable limits. Furthermore, alterations in these parameter values also impact the computed value of n, but only at the decimal level. Consequently, the optimal value of n, which should be an integer, remains unchanged at n=2.