An Algebraic Decision Support Model for Inventory Coordination in the Generalized n-Stage Non-Serial Supply Chain with Fixed and Linear Backorders Costs

Abstract

This paper extends and generalizes former inventory models that apply algebraic methods to derive optimal supply chain inventory decisions. In particular this paper considers the problem of coordinating production-inventory decisions in an integrated n-stage supply chain system with linear and fixed backorder costs. This supply chain system assumes information symmetry which implies that all partners share their operational information. First, a mathematical model for the supply chain system total cost is formulated under the integer multipliers coordination mechanism. Then, a recursive algebraic algorithm to derive the optimal inventory replenishment decisions is developed. The applicability of the proposed algorithm is demonstrated using two different numerical examples. Results from the numerical examples indicate that adopting the integer multiplier mechanism will reduce the overall total system cost as compared to using the common cycle time mechanism.

1. Introduction

Background

Formerly, there was no proper coordination of operations decisions among the different members of the supply chain. The absence of collaboration will result in disconnected processes and individual decisions at every partner in the supply chain. According to Mahdavi et al. [1], the absence of coordination also generates a conflict in the objectives of the members sharing the same supply chain. The sustainable supply chain overall optimization can be effectively achieved by fostering the partnership among the supply chain partners and enabling maximum information sharing across the supply chain [2]. In recent times, companies have identified that it is necessary to improve the chain performance and its cost efficiency with deep collaboration between the chain members and with a high level of coordination of different decisions [3]. Effective inventory management system to enhance the operational efficiency and competitiveness of enterprises requires the integration of the entire supply chain system [4]. The advances in information, communication technologies, and supply chain management (SCM) tools have encouraged managers to seek a full integration and close collaboration; see for instance, [5,6,7,8].

The process of coordinating production and inventory operations in the supply chains is a critical planning process. This process covers several operational activities and decisions such as production lot sizing, scheduling, and inventory allocation. It is well known that the decisions can be taken jointly among all the chain members, or independently. In the joint policy for coordinating production and replenishment decisions, the overall cost for the whole supply chain system is optimized jointly. Conversely, in a decentralized policy, each supply chain member optimizes its own total cost individually. According to Kim et al. [9], a centralized inventory control policy generates a better service-level equilibrium than a decentralized policy. There exist in the literature numerous other advantages of joint inventory replenishment decisions and integrated operations in the supply chain systems; for example [10,11,12,13,14,15,16,17,18], among others.

2. Literature Review

In recent times, several papers related to supply chain modeling and optimization have investigated the problem of inventory-distribution synchronization. For example, Ben-Daya et al. [19] proposed integrated supplier-retailer inventory models and Khouja and Goyal [20] presented a review on the joint optimization models proposed for the economic lot-sizing problem. Abdul-Jalbar et al. [21] studied the one-warehouse multi-retailer inventory-distribution systems considering situations under both joint and independent replenishment schemes. They proposed a near-optimal solution for the decentralization case. Moreover, they developed a branch and bound algorithm that gives a near-optimal policy for the centralized case. In the same year, Gnoni et al. [22] hybridized mixed-integer linear programming and simulation to solve the lot sizing and scheduling problem in a multi-site manufacturing supply chain. This system is constrained to capacity and the demand of the multi-product in the multi-period is stochastic. They applied the hybrid model to an industrial case study related to a supply chain of braking components for the automotive industry taking into account centralized and decentralized decision policies. Later, Chen and Chen [23] addressed the multi-item two-echelon supply chain inventory system with deterministic demand. They considered centralized and decentralized decision policies and derived mathematical models to minimize the total supply chain costs. The numerical results have shown that implementing a policy of jointly coordinated replenishment decisions is better in terms of cost reduction and achieving savings than allowing independent replenishment decisions made by each member.

Regularly, in operations research, the differential calculus is used to model and optimize the integrated production inventory systems. On the other hand, some researchers and academicians have been proposing easy to apply and understand solution methods to optimize the production inventory integrated models. In this direction there exist the works of [24,25,26,27,28]. Grubbström [24] applied an algebraic o approach to the EOQ without backlogging. Later, Cárdenas-Barrón [25] also employed the algebraic method to derive and solve the EPQ model with back-ordering assuming the situation of only one type of backorder: the cost per unit and time unit. In a subsequent paper, Cárdenas-Barrón [26] developed an optimization model to solve the inventory decision making problem for a supply chain system involving a firm which supplies its produced goods to multiple buyers. The model was developed based on deterministic rates of production and demand. It is significant to see that this model formulation followed the simplest scheme for inventory coordination. This scheme assumes that all the firms at all tiers in the supply chain will share the same production-replenishment cycle time. He concluded that it is feasible to apply the algebraic optimization approach to derive the solution for the supply chain model without the need of using the traditional derivatives method. Wee and Chung [28] applied the algebraic method to solve the economic lot size of the integrated buyer inventory problem. They concluded that the algebraic method is very helpful for those who lack the required calculus knowledge and skills to understand the solution procedure easily. Afterward, Chung and Wee [27] proposed replenishment policies for an integrated three-stage inventory system with backorders. They were able to algebraically derive the optimal solution for this inventory problem in the supply chain of three echelons. Later, this model was extended by Seliaman [29] to model a four-stage inventory system. Chi [30] developed a simplified algebraic procedure to show that it is possible to find the optimal solution for the economic manufacturing quantity (EMQ) model with imperfect products, without using traditional calculus methods. In the same year, Cárdenas-Barrón [31] dealt with the problem of finding the optimal manufacturing quantity in a single manufacturing facility assuming a rework process. The optimal solutions under two different inventory policies were obtained. Besides, the author derived the conditions for the existence of an optimal policy, the exact solution for the joint total inventory cost under the two inventory schemes, and the formulated structures that can identify the other additional costs resulting from using a suboptimal solution. Afterward, Ben-Daya et al. [32] solved the joint economic quantity sizing problem for a supply chain with three layers to find the optimal order quantities for all the members in the chain to minimize the total costs comprising of production setup costs, costs of holding raw material, and costs of holding finished product inventory. In their model, the replenishment period in each echelon is established to be a whole multiple of the replenishment period for the contiguous direct following stage. To improve the performance in the shipment delivery, lots from a specific batch are permitted to occur during the time of producing batch and not after completing the production of the whole lot. They applied an optimization method without using the concepts of derivatives to obtain the optimal solution in its closed form for their proposed inventory model. Cárdenas-Barrón et al. [33] revised the Ben-Daya et al. [32] model improving their algorithm and obtaining less supply chain total cost and less CPU time. According to Cárdenas-Barrón [34], the algebraic optimization approach for the modeling and optimization of production inventory systems has had remarkable interest from researchers in the domains of operational research and operational management. Also, Cárdenas-Barrón [34] states that the increasing interest in using the algebraic optimization methods for solving these types of operational management problems and models of inventory is because these methods do not require any knowledge of differential calculus or the derivatives methods. A complete and comprehensive survey of literature on the use of these emerging algebraic techniques for the optimization of supply inventory models is done in Cárdenas-Barrón [34]. Those who are interested in this line of research may refer to the relevant inventory models given in [35,36,37,38,39,40].

In this research paper, a generalized model is developed for the multi-tier multi-level supply chain using the mechanism of integer multipliers to coordinate the production and inventory operations among the different partners in the supply chain integrated system. The model assumes two well-known backorder costs. Furthermore, this model extends the generalized model of Seliaman and Ahmad [41]. To the best of the authors’ knowledge, the research work presented in this paper is the first to develop an algebraic algorithm to derive the optimal solution for the generalized n-stage serial or non-serial supply chain inventory problem considering linear and fixed backorder costs. The algebraic algorithm presented in this paper obtains the solution for a supply chain having the general n number of stages (n = 2, 3, 4, 5, 6, …). The remainder of this paper is structured in the following manner. Section 3 describes the model notation, assumptions the model development details. Section 4 shows the development of the algebraic optimization process and the proposed solution algorithm for the developed model and illustrates the use of the solution algorithm through two numerical examples. Section 5 presents a brief discussion of the research results. Finally, Section 6 provides some general conclusions.

3. Materials and Methods

The non-serial supply chain system modeled in this research paper is similar to the complex supply chain configuration described by Cárdenas-Barrón [26]. This configuration can be viewed as a network of multi-vertical layers and multi-horizontal stages. There could be any general number n of stages in this supply chain. At each stage, there are multiple firms. Each firm can process and ship its products to one or more firms in the following stage. This supply chain system assumes information symmetry which implies that all partners share their operational information. While Cárdenas-Barrón [26] considered only the equal cycle coordination mechanism without allowing any backorders, this research paper used both the equal cycle and integer multipliers coordination mechanisms in addition to integrating two types of backorder costs.

3.1. Notation and Assumptions

For the purpose of developing the extended mathematical model of this research, the following notations and assumptions are adapted from the work of Seliaman and Ahmad [41]:

• T = Basic inventory replenishment time duration for the retailers at the last chain tier.
• T_i = Time duration for production/inventory replenishment time at the i-th chain tier.
• S_ij = Cost of setup for the j-th company at the i-th chain tier.
• S_i = The sum of setup total costs for all of the companies at the i-th chain tier.
• K_i = A whole number representing the cycle coordination multiplier at the i-th chain tier.
• h_i = Cost of holding inventory at the i-th chain tier.
• n_i = Total number of companies at the i-th chain tier.
• D_ij = Rate of demand for the j-th company at the i-th chain tier.
• D = Rate of demand for the whole supply chain
• P_ij = Rate of production at the j-th company at the i-th chain tier.
• T_S = the stock-out time at the last chain tier.
• = Linear cost of backorder cost at the last chain tier.
• = Backorder cost per unit (fixed) for each retailer.
• A = The product of rates of production for all supply chain firms.
• B_ij = A values obtained by multiplying all the rates of production for all companies in the supply chain, excluding the j-th firm at the i-th chain tier.

For the development of the extended generalized multi-stage supply model, the following assumptions are made:

(a)

In this non-serial supply chain, a single product is produced.

(b)

Production rates for all firms at all stages are deterministic and uniform.

(c)

End consumers demand rate is deterministic and uniform.

(d)

Holding costs are equal for the partners sharing the same tiers.

(e)

Shipments are delivered in equal sizes between each two chain tiers.

(f)

Shortage is allowed at the last chain tier.

(g)

Replenishment time duration at each tier in the chain is an integer multiple of the time duration at the next chain tier.

3.2. Formulation of the Supply Chain Model

According to the assumptions specified in Section 3, the whole cost for any firm at the last tier is comprised of the cost of inventory holding, the stock out cost, and the cost of orders setup. The inventory behavior for an end retailer is illustrated in Figure 1. From this graphical representation of the inventory and shortage levels, then the complete cost for the jth company is obtained by:

$$T{C}_{n,j}=h_n\frac{T{D}_{n,j}}{2}-T_S{D}_{n,j}{h}_n+h_n\frac{T_S^2{D}_{n,j}}{2T}+\widehat{\pi }\frac{T_S^2{D}_{n,j}}{2T}+\pi \frac{T_{S{D}_{n,j}}}{T}+\frac{S_{n,j}}{T}$$

(1)

The total cost for all of the retailers together is expressed as follows:

$$\begin{array}{cc}T{C}_n\hfill & ={{\displaystyle \sum }}_{j=1}^{m}T{C}_{n,j}=\left(h_n\frac{T}{2}-T_S{h}_n+h_n\frac{T_S^2}{2T}+\widehat{\pi }\frac{T_S^2}{2T}+\pi \frac{T_S}{T}\right){{\displaystyle \sum }}_{j=1}^m\left(D_{n,j}\right)+\frac{1}{T}{{\displaystyle \sum }}_{j=1}^m\left(S_{n,j}\right)\hfill \\ \hspace{1em}\hfill & =h_n\frac{TD}{2}-T_{S}D{h}_n+h_n\frac{T_S^{2}D}{2}+\widehat{\pi }\frac{T_S^{2}D}{2T}+\pi \frac{T_{S}D}{T}+\frac{S_n}{T}+\hfill \\ \hspace{1em}\hfill & =h_n\frac{TD}{2}+T_S\mathrm{D}\left(\frac{\pi }{T}-h_n\right)+\frac{T_S^{2}D}{2T}\left(\widehat{\pi }+h_n\right)+\frac{S_n}{T}\hfill \\ \hspace{1em}\hfill & =h_n\frac{TD}{2}+\frac{T_S^{2}D}{2T}\left(\widehat{\pi }+h_n\right)-\frac{T_{S}D}{T}\left(h_{n}T-\pi \right)+\frac{S_n}{T}\hfill \\ \hspace{1em}\hfill & =h_n\frac{TD}{2}+\frac{D\left(\widehat{\pi }+h_n\right)}{2T}\left(T_S^2-\frac{2T_S\left(T{h}_n-\pi \right)}{\left(\widehat{\pi }+h_n\right)}\right)+\frac{S_n}{T}\hfill \end{array}$$

(2)

Now, if the method of completing the square for the term $\left(T_S^2-\frac{2T_S\left(T{h}_n-\pi \right)}{\left(\widehat{\pi }+h_n\right)}\right)$ in (2) is applied, then:

$$\begin{array}{cc}T{C}_n\hfill & =h_n\frac{TD}{2}+\frac{D\left(\widehat{\pi }+h_n\right)}{2T}\left(T_S^2-\frac{2T_S\left(T{h}_n-\pi \right)}{\left(\widehat{\pi }+h_n\right)}+{\left(\frac{T{h}_n-\pi }{\widehat{\pi }+h_n}\right)}^2-{\left(\frac{T{h}_n-\pi }{\widehat{\pi }+h_n}\right)}^2\right)+\frac{S_n}{T}\hfill \\ \hspace{1em}\hfill & =h_n\frac{TD}{2}+\frac{D\left(\widehat{\pi }+h_n\right)}{2T}\left(T_S^2-\frac{2T_S\left(T{h}_n-\pi \right)}{\left(\widehat{\pi }+h_n\right)}+{\left(\frac{T{h}_n-\pi }{\widehat{\pi }+h_n}\right)}^2\right)-\frac{D\left(\widehat{\pi }+h_n\right)}{2T}{\left(\frac{T{h}_n-\pi }{\widehat{\pi }+h_n}\right)}^2+\frac{S_n}{T}\hfill \\ \hspace{1em}\hfill & =h_n\frac{TD}{2}+\frac{D\left(\widehat{\pi }+h_n\right)}{2T}{\left(T_S^2-\left(\frac{T{h}_n-\pi }{\widehat{\pi }+h_n}\right)\right)}^2-\frac{D\left(\widehat{\pi }+h_n\right)}{2T}\left(\frac{T^2{h}_n^2}{{\left(\widehat{\pi }+h_n\right)}^2}-\frac{2T{h}_n\pi }{{\left(\widehat{\pi }+h_n\right)}^2}+\frac{{\pi }^2}{{\left(\widehat{\pi }+h_n\right)}^2}\right)+\frac{S_n}{T}\hfill \\ \hspace{1em}\hfill & =h_n\frac{TD}{2}+\frac{D\left(\widehat{\pi }+h_n\right)}{2T}{\left(T_S-\left(\frac{T{h}_n-\pi }{\widehat{\pi }+h_n}\right)\right)}^2-\frac{TD{h}_n^2}{2\left(\widehat{\pi }+h_n\right)}+\frac{D{h}_n\pi }{\left(\widehat{\pi }+h_n\right)}-\frac{D{\pi }^2}{2T\left(\widehat{\pi }+h_n\right)}+\frac{S_n}{T}\hfill \\ \hspace{1em}\hfill & =h_n\frac{TD}{2}-\frac{TD{h}_n^2}{2\left(\widehat{\pi }+h_n\right)}+\frac{D{h}_n\pi }{\left(\widehat{\pi }+h_n\right)}+\frac{D\left(\widehat{\pi }+h_n\right)}{2T}{\left(T_S-\left(\frac{T{h}_n-\pi }{\widehat{\pi }+h_n}\right)\right)}^2-\frac{D{\pi }^2}{2T\left(\widehat{\pi }+h_n\right)}+\frac{S_n}{T}\hfill \end{array}$$

(3)

The cost of holding inventory at any tier in the supply chain, apart from the last tier (the downstream end tier n), is comprised of the holding cost of the raw materials while being transformed into final items through the manufacturing period in the cycle time; and the cost of keeping processed items during the non-production period of the full the cycle. During the production time of the cycle, the rate of holding inventory of raw material and produced items is equal to ${{\displaystyle \prod }}_{s=i}^n\frac{K_{S}T{D}_{i,j}^2}{2P_{i,j}}$. In the period of non-manufacturing, the holding inventory depletes with a time-frequency of T_i+1 units by the amount of T_i+1D_ij beginning with the amount of (T_i − T_i+1)D_ij as it is illustrated in Figure 2 (see also [40,41]). Consequently, the total cost for each company i at any stage j, excluding the final stage, is expressed by:

$$T{C}_{i,j}={{\displaystyle \prod }}_{s=i}^n\frac{K_{S}T{D}_{i,j}^2{h}_{i-1}}{2P_{i,j}}+{{\displaystyle \prod }}_{s=i+1}^n\frac{K_{S}T{D}_{i,j}}{2}\left(K_i\left(1+\frac{D_{i,j}}{P_{i,j}}\right)-1\right)h_i+\frac{S_{i,j}}{{{\displaystyle \prod }}_{S=j}^n{K}_{S}T}$$

(4)

It is assumed that $K_n=1$.

Now, the complete cost of production, inventory holding, and stock-out cost in the whole system is:

$$TC={{\displaystyle \sum }}_{i,j}T{C}_{i,j}$$

(5)

Equation (5) is rewritten as:

$$\begin{array}{c}TC=\frac{T}{2}\left(D{h}_n-\frac{D{h}_n^2}{\left(\widehat{\pi }+h_n\right)}+{{\displaystyle \sum }}_{i=1}^{n-1}{{\displaystyle \sum }}_{j}I{C}_{i,j}\right)+\frac{D\left(\widehat{\pi }+h_n\right)}{2T}{\left(T_S-\left(\frac{T{h}_n-\pi }{\widehat{\pi }+h_n}\right)\right)}^2\\ +\frac{1}{T}{{\displaystyle \sum }}_{i=1}^n\frac{S_i}{{{\displaystyle \prod }}_{S=i}^n{K}_S}-\frac{D{\pi }^2}{2T\left(\widehat{\pi }+h_n\right)}+\frac{D{h}_n\pi }{\left(\widehat{\pi }+h_n\right)}\end{array}$$

(6)

where $I{C}_{i,j}=\frac{{{\displaystyle \prod }}_{S=i}^n{K}_{S}T{D}_{i,j}^2{h}_{i-1}}{2P_{i,j}}+\frac{{{\displaystyle \prod }}_{S=i+1}^n{K}_{S}T{D}_{i,j}}{2}\left(K_i\left(1+\frac{D_{i,j}}{P_{i,j}}\right)-1\right)h_i$.

Now, Theorem 1 represents the supply chain total cost of (6) into a compact recursive form.

Theorem 1.

Based on the notation and assumptions stated in Section 3, the total annual cost for the entire supply chain is given by:

$$TC=TY+\frac{W}{T}+\frac{D\left(\widehat{\pi }+h_n\right)}{2T}{\left(T_S-\left(\frac{T{h}_n-\pi }{\widehat{\pi }+h_n}\right)\right)}^2+\frac{D{h}_n\pi }{\left(\widehat{\pi }+h_n\right)}$$

(7)

where:

$$\mathrm{Y}=\frac{K_{n-1}{\mathsf{\Psi }}_{n-i}+{\alpha }_{n-1}}{2A}$$

(8)

$$W=\left(S_n-\frac{D{\pi }^2}{2\left(\widehat{\pi }+h_n\right)}+\frac{{\phi }_{n-1}}{K_{n-1}}\right)$$

(9)

$${\alpha }_{n-1}=AD{h}_n-\frac{AD{h}_n^2}{\left(\widehat{\pi }+h_n\right)}-A{h}_{n-1}{{\displaystyle \sum }}_j{D}_{n-1,j}$$

(10)

$${\alpha }_0=\left(h_0+h_1\right){{\displaystyle \sum }}_j{D}_{1,j}^2{B}_{1,j}+A{{\displaystyle \sum }}_j{D}_{1,j}{h}_1,{\mathsf{\Psi }}_0=0,{\phi }_0=0$$

(11)

For all values of i until (n − 2) and all values of j:

$${\alpha }_i=\left(h_i+h_{i+1}\right){{\displaystyle \sum }}_j{D}_{i+1,j}^2{B}_{i+1,j}+A{{\displaystyle \sum }}_j{D}_{i+1,j}{h}_{i+1}-A{{\displaystyle \sum }}_j{D}_{i,j}{h}_i$$

(12)

For all values of i until (n − 2) and all values of j:

$${\mathsf{\Psi }}_i=K_{i-1}{\mathsf{\Psi }}_{i-1},{\mathsf{\phi }}_i=\left(S_i+\frac{{\phi }_{i-1}}{K_{i-1}}\right)$$

(13)

The proof of Theorem 1 is detailed in Appendix A.

3.3. Development of the Algebraic Optimization Algorithm

Now, following the algebraic approach developed by Cárdenas-Barrón [25] and extended by Seliaman and Ahmad [40], the complete cost for the whole supply chain system as shown by Equation (7) is re-expressed by factorization of 1/T and the perfect square completion. Thus, one obtains the following total cost:

$$TC=\frac{1}{T}\left(T^{2}Y-2T\sqrt{YW}+W+2T\sqrt{YW}\right)+\frac{D\left(\widehat{\pi }+h_n\right)}{2T}{\left(T_S-\left(\frac{T{h}_n-\pi }{\widehat{\pi }+h_n}\right)\right)}^2+\frac{D{h}_n\pi }{\left(\widehat{\pi }+h_n\right)}$$

(14)

Factorizing the perfect squared trinomial in a squared binomial, the following is obtained:

$$TC=\frac{1}{T}{\left(T\sqrt{Y}-\sqrt{W}\right)}^2+2\sqrt{YW}+\frac{D\left(\widehat{\pi }+h_n\right)}{2T}{\left(T_S-\left(\frac{T{h}_n-\pi }{\widehat{\pi }+h_n}\right)\right)}^2+\frac{D{h}_n\pi }{\left(\widehat{\pi }+h_n\right)}$$

(15)

It is clear that Equation (15) will reach the minimum value with regard to the values of the basic cycle time duration T and the stock out time duration T_S after allowing ${\left(T\sqrt{Y}-\sqrt{W}\right)}^2$ and allowing:

$${\left(T_S-\left(\frac{T{h}_n-\pi }{\widehat{\pi }+h_n}\right)\right)}^2=0$$

In this case, the optimal cycle time duration T* will be:

$$T^{*}=\sqrt{\frac{W}{Y}}$$

(16)

and the corresponding time duration of stocking out $T_S^{*}$ is:

$$T_S^{*}=\frac{T^{*}{h}_n-\pi }{\widehat{\pi }+h_n}$$

(17)

Substituting Equations (16) and (17) into Equation (15), we get the minimum possible total cost for the whole system, as:

$$TC=2\sqrt{YW}+\frac{D{h}_n\pi }{\left(\widehat{\pi }+h_n\right)}$$

(18)

The optimal basic cycle time duration for inventory replenishment T* depends on the defined integer multipliers. Therefore, the algebraic methods can be used to derive the optimal values of these integer multipliers using recursion. Substitution of $Y$ and $W$ from previous Equation (8) along with Equation (9) into Equation (18), one obtains:

$$\begin{array}{cc}TC\hfill & =\sqrt{\frac{2}{A}} {\left\{\left(K_{n-1}{\mathsf{\Psi }}_{n-1}+{\alpha }_{n-1}\right)\left(S_n+\frac{{\phi }_{n-1}}{K_{n-1}}\right)\right\}}^{\frac{1}{2}}+\frac{D{h}_n\pi }{\left(\widehat{\pi }+h_n\right)}\hfill \\ \hspace{1em}\hfill & =\sqrt{\frac{2}{A}}{\left\{\frac{1}{K_{n-1}}{\left(K_{n-1}\sqrt{{\mathsf{\Psi }}_{n-1}{S}_n}-\sqrt{{\alpha }_{n-1}{\phi }_{n-1}}\right)}^2+{\left(\sqrt{{\mathsf{\Psi }}_{n-1}{\phi }_{n-1}}+\sqrt{{\alpha }_{n-1}{S}_n}\right)}^2\right\}}^{\frac{1}{2}}+\frac{D{h}_n\pi }{\left(\widehat{\pi }+h_n\right)}\hfill \end{array}$$

(19)

Using Equation (19) and letting ${\left(K_{n-1}\sqrt{{\mathsf{\Psi }}_{n-1}{S}_n}-\sqrt{{\alpha }_{n-1}{\phi }_{n-1}}\right)}^2=0$, will directly drive the optimal integer value of multiplier $K_{n-1}^{*}$ as:

$$K_{n-1}^{*}=\sqrt{\frac{{\alpha }_{n-1}{\phi }_{n-1}}{{\mathsf{\Psi }}_{n-1}{S}_n}}$$

(20)

Because the value of $K_{n-1}^{*}$ is a positive integer, as in Chung and Wee [27] it must satisfy the following condition:

$$\left(K_{n-1}^{*}\right)\left(K_{n-1}^{*}-1\right)\le {\left(K_{n-1}^{*}\right)}^2\le \left(K_{n-1}^{*}\right)\left(K_{n-1}^{*}+1\right)$$

(21)

To obtain the optimal integer value of the next multiplier $K_{n-2}^{*}$, the quantity $\sqrt{{\mathsf{\Psi }}_{n-1}{\phi }_{n-1}}$ in Equation (19) is rewritten as follows:

$$\begin{array}{ll}\sqrt{{\mathsf{\Psi }}_{n-1}{\phi }_{n-1}}& ={\left\{\left(K_{n-2}{\mathsf{\Psi }}_{n-2}+{\alpha }_{n-2}\right)\left(S_{n-1}+\frac{{\phi }_{n-2}}{K_{n-2}}\right)\right\}}^{\frac{1}{2}}\\ & ={\left\{\frac{1}{K_{n-2}}{\left[K_{n-2}\sqrt{{\mathsf{\Psi }}_{n-2}{S}_{n-1}}-\sqrt{{\alpha }_{n-2}{\phi }_{n-2}}\right]}^2+{\left[\sqrt{{\mathsf{\Psi }}_{n-2}{\phi }_{n-2}}-\sqrt{{\alpha }_{n-2}{S}_{n-1}}\right]}^2\right\}}^{\frac{1}{2}}\end{array}$$

(22)

From Equation (22), making ${\left[K_{n-2}\sqrt{{\mathsf{\Psi }}_{n-2}{S}_{n-1}}-\sqrt{{\alpha }_{n-2}{\phi }_{n-2}}\right]}^2=0$, then the optimal integer value of the next multiplier $K_{n-2}^{*}$ is obtained by:

$$K_{n-2}^{*}=\sqrt{\frac{{\alpha }_{n-2}{\phi }_{n-2}}{{\mathsf{\Psi }}_{n-2}{S}_{n-1}}}$$

(23)

Since the value of $K_{n-2}^{*}$ is a positive integer, then the integer value conditions must be met:

$$\left(K_{n-2}^{*}\right)\left(K_{n-2}^{*}-1\right)\le {\left(K_{n-2}^{*}\right)}^2\le \left(K_{n-2}^{*}\right)\left(K_{n-2}^{*}+1\right)$$

(24)

This recursion procedure is followed until obtaining the optimal integer value of the last multiplier ($K_1^{*}$) as:

$$K_1^{*}=\sqrt{\frac{{\alpha }_1{\phi }_1}{{\mathsf{\Psi }}_1{S}_1}}$$

(25)

Substituting:

$${\mathsf{\Psi }}_1={\alpha }_0=\left(h_0+h_1\right){{\displaystyle \sum }}_j{D}_{1,j}^2{B}_{1,j}+A{{\displaystyle \sum }}_j{D}_{1,j}{h}_1,$$

$${\phi }_1=S_1\mathrm{and}$$

$${\alpha }_1=\left(h_1+h_2\right){{\displaystyle \sum }}_j{D}_{2,j}^2{B}_{2,j}+A{{\displaystyle \sum }}_j{D}_{2,j}{h}_2-A{{\displaystyle \sum }}_j{D}_{1,j}{h}_1$$

into Equation (25) one has:

$$K_1^{*}=\sqrt{\frac{\left(\left(h_1+h_2\right){{\displaystyle \sum }}_j{D}_{2,j}^2{B}_{2,j}+A{{\displaystyle \sum }}_j{D}_{2,j}{h}_2-A{{\displaystyle \sum }}_j{D}_{1,j}{h}_1\right)S_1}{\left(\left(h_0+h_1\right){{\displaystyle \sum }}_j{D}_{1,j}^2{B}_{1,j}+A{{\displaystyle \sum }}_j{D}_{1,j}{h}_1\right)S_1}}$$

(26)

The value of $K_1^{*}$ is assumed a positive integer. Hence, it must meet the following condition:

$$\left(K_1^{*}\right)\left(K_1^{*}-1\right)\le {\left(K_1^{*}\right)}^2\le \left(K_1^{*}\right)\left(K_1^{*}+1\right)$$

(27)

When the optimal value of $K_1^{*}$ is derived from Equations (26) and (27), then the optimal value of $K_2^{*}$ can be derived as:

$$K_2^{*}=\sqrt{\frac{{\alpha }_2{\phi }_2}{{\mathsf{\Psi }}_2{S}_3}}$$

(28)

Substituting:

$${\alpha }_2=\left(h_2+h_3\right){{\displaystyle \sum }}_j{D}_{3,j}^2{B}_{3,j}+A{{\displaystyle \sum }}_j{D}_{3,j}{h}_3-A{{\displaystyle \sum }}_j{D}_{2,j}{h}_2\mathrm{and}$$

$${\phi }_2=\left(S_2+\frac{S_1}{K_1}\right),\mathrm{one}\mathrm{gets}\text{:}$$

$$K_2^{*}=\sqrt{\frac{\left(\left(h_2+h_{23}\right){{\displaystyle \sum }}_j{D}_{3,j}^2{B}_{3,j}+A{{\displaystyle \sum }}_j{D}_{3,j}{h}_3-A{{\displaystyle \sum }}_j{D}_{2,j}{h}_2\right)\left(S_2+\raisebox{1ex}{$S_1$}\!\left/ \!\raisebox{-1ex}{$K_1$}\right.\right)}{{\mathsf{\Psi }}_2{S}_3}}$$

(29)

where:

$$\begin{array}{l}{\mathsf{\Psi }}_2=K_1\left[\left(h_0+h_1\right){{\displaystyle \sum }}_j{D}_{1,j}^2{B}_{1,j}+A{{\displaystyle \sum }}_j{D}_{1,j}{h}_1\right]\\ +\left[\left(h_1+h_2\right){{\displaystyle \sum }}_j{D}_{2,j}^2{B}_{2,j}+A{{\displaystyle \sum }}_j{D}_{2,j}{h}_2-A{{\displaystyle \sum }}_j{D}_{1,j}{h}_1\right]\end{array}$$

Since the value of $K_1^{*}$ is a positive integer, the following condition must be satisfied:

$$\left(K_2^{*}\right)\left(K_2^{*}-1\right)\le {\left(K_2^{*}\right)}^2\le \left(K_2^{*}\right)\left(K_2^{*}+1\right)$$

(30)

The above solution procedure is applied every time deriving the optimal integer value of $K_{i+1}$ from the optimal integer value $K_i$ derived in the previous step, until obtaining the optimal integer value of the multiplier $K_{n-1}$. Afterward, the optimal values of the cycle time duration (T*) and the optimal stock-out duration time ($T_S^{*}$) which are both functions of the obtained integer values of the multipliers, are obtained from Equation (13).

4. Results

4.1. Solution Algorithm

Based on the algebraic analysis made in Section 5, an algebraic algorithm optimization is developed. This algorithm has four basic steps. In step zero, the algorithm initializes the predefined recursive structures required by step one for computing the optimal integer multipliers. Then, step one continues iteratively computing the remaining predefined recursive structures required for determining the optimal integer multipliers K₁ thru K_n−1. In step two, the algorithm calculates the values of Y and W that correspond to the optimal values of the basic inventory cycle time T and the optimal stock-out time T_S. Finally, the optimal values of the basic inventory cycle time T, the optimal stock-out time T_S and the total cost of the whole supply chain are computed in step three.

Step 0 (the initialization step)

Set: ${\mathsf{\Psi }}_0=0$, ${\alpha }_0=\left(h_0+h_1\right){{\displaystyle \sum }}_j{D}_{1,j}^2{B}_{1,j}+A{{\displaystyle \sum }}_j{D}_{1,j}{h}_1$, $K_0=1$, ${\phi }_0=0$ and

$${\mathsf{\Psi }}_1={\alpha }_0=\left(h_0+h_1\right){{\displaystyle \sum }}_j{D}_{1,j}^2{B}_{1,j}+A{{\displaystyle \sum }}_j{D}_{1,j}{h}_1,$$

$${\phi }_1=S_1\mathrm{and}$$

$${\alpha }_1=\left(h_1+h_2\right){{\displaystyle \sum }}_j{D}_{2,j}^2{B}_{2,j}+A{{\displaystyle \sum }}_j{D}_{2,j}{h}_2-A{{\displaystyle \sum }}_j{D}_{1,j}{h}_1$$

$${\alpha }_{n-1}=AD{h}_n-\frac{AD{h}_n^2}{\left(\widehat{\pi }+h_n\right)}-A{h}_{n-1}{\displaystyle \sum _j{D}_{n-1.j}}$$

Step 1. Compute the following (For values of I from 1 to n − 2 and for all values of j):

$${\alpha }_i=\left(h_i+h_{i+1}\right){\displaystyle {\sum }_j{D}_{i+1,j}^2}{B}_{i+1,j}+A{\displaystyle {\sum }_j{D}_{i+1,j}}{h}_{i+1}-A{\displaystyle {\sum }_j{D}_{i,j}{h}_i}$$

$${\psi }_i=K_{i-1}{\psi }_{i-1}+{\alpha }_{i-1}$$

$${\phi }_i=\left(S_i+\frac{{\phi }_{i-1}}{K_{i-1}}\right)$$

Calculate $K_i^{*}$ that satisfy integer number condition.

Step 2. Use Equation (8) to calculate Y and apply Equation (9) to get W.

Step 3. Calculate:

$$T^{*}=\sqrt{W/Y},T^{*}{}_S=\frac{T^{*}{h}_n-\pi }{\widehat{\pi }+h_n}$$

and:

$$TC=2\sqrt{YW}+{\frac{D{h}_n\pi }{\left(\widehat{\pi }+h_n\right)}}^{}$$

4.2. Numerical Examples

To illustrate the use of the developed algorithm, two numerical examples are presented and solved in this section. The first example considers the three-stage supply chain system which is adapted from Khouja [42]. The second example represents a four-stage supply chain system.

Example 1.

This example of a three-stage supply chain is taken from Khouja [42] and modified to fit the purpose of this paper by permitting shortages at the end retailers and including the two backordering costs. The data for this example is given in

Table 1. Numerical Example 1 input data.

Stage i	Party Index j	Parent (i,j)	Inventory Cost Incoming hi	Annual Production Pij	Annual Demand Dij	Setup Cost Si	Backordering Costs $\mathit{\pi }$	$\widehat{\mathit{\pi }}$
Supplier	1	_	0.8	399,000	133,000	800	-	-
Manufactures	1	1	2	140,000	70,000	200	-	-
	2	1	2	108,000	36,000	200	-	-
	3	1	2	108,000	27,000	200	-	-
Retailers	1	1	5	-	10,000	50	0.1	9.5
	2	1	5	-	20,000	50	0.1	9.5
	3	1	5	-	40,000	50	0.1	9.5
	4	2	5	-	12,000	50	0.1	9.5
	5	2	5	-	24,000	50	0.1	9.5
	6	3	5	-	9000	50	0.1	9.5
	7	3	5	-	18,000	50	0.1	9.5

.

By applying the developed algorithm, the results for this example are obtained and presented in

Table 2. Results for Example 1.

Stage	Multiplier	Cycle Time	Shortage Duration	Cost
Supplier	2	0.095	-	$12,788.18
Manufactures	1	0.048	-	$16,176.75
Retailers	-	0.048	0.0095	$30,355.05
Entire Supply chain	-	-	-	$59,319.99

. Detailed computational steps of the algorithm are provided within the solution of the second example.

Example 2.

A four-stage supply chain system is assumed. This system consists of a single vendor, two producers, four wholesale distributors, and six retailers selling directly to the end consumers. The input data for this hypothesized example is presented in

Table 3. Example 2 input data.

Tier i	Member j	Index (i,j)	Hi	Pij	Dij	Si	Backordering Costs $\mathit{\pi }$	$\widehat{\mathit{\pi }}$
Supplier	1	_	0.8	396,000	130,000	1000	-	-
Manufactures	1	1	2	180,000	60,000	200	-	-
	2	1	2	200,000	70,000	200	-	-
Distributors	1	1	4	90,000	35,000	50	-	-
	2	1	4	70,000	25,000	50	-	-
	3	2	4	80,000	30,000	50	-	-
	4	2	4	100,000	40,000	50	-	-
Retailers	1	1	7	-	20,000	10	0.1	9.5
	2	1	7	-	15,000	10	0.1	9.5
	3	2	7	-	25,000	10	0.1	9.5
	4	3	7	-	30,000	10	0.1	9.5
	5	4	7	-	15,000	10	0.1	9.5
	6	4	7	-	25,000	10	0.1	9.5

. The inventory holding cost at the supplier of the vendor is h₀ = 0.1.

This problem is solved first assuming the equal cycle time coordination mechanism. The solution of this example under the equal cycle time coordination mechanism is straightforward. The optimal common cycle time can be directly derived from Equation (16) as T* = 0.057. Similarly, the optimal backorder time can be directly derived from Equation (17) as 0.018. Accordingly, the optimal total supply chain wide cost can be obtained from Equation (18) as TC* = 61,346.68.

Second, the same example is solved assuming an integer multiplier mechanism for coordinating the inventory production operational decisions. The solution is obtained by applying the developed algorithm. Results for this example are obtained and presented in

Table 4. Results for Example 2.

Tier	Multiplier K	T	TS	Cost
Supplier	2	0.1092	-	14,093.40
Producers	2	0.0546	-	14,276.55
Wholesale Distributor	1	0.0273	-	12,518.31
Retailers	-	0.0273	0.0055	10,864.68
Entire Supply chain	-	-		51,752.94

. Detailed computational steps of the algorithm are provided within the solution after this example.

Comparing the results from the model when using the equal cycle with results of using the integer multiplier cycle time, it is observed that the latter mechanism can achieve more than a 15% reduction in the total supply chain system costs.

The effects of changing the linear and fixed backorder costs on the percent savings resulting from the integer multiplier cycle time as compared to the equal cycle time are exhibited in Figure 3 and Figure 4 respectively.

For this four-stage supply chain example, we can substitute n = 4 in Equation (3) to obtain the total cost for all of the six retailers at the fourth stage as:

$$T{C}_4=h_4\frac{TD}{2}-\frac{TD{h}_4^2}{2\left(\widehat{\pi }+h_4\right)}+\frac{D{h}_4\pi }{\left(\widehat{\pi }+h_4\right)}+\frac{D\left(\widehat{\pi }+h_4\right)}{2T}{\left(T_S-\left(\frac{T{h}_4-\pi }{\widehat{\pi }+h_4}\right)\right)}^2-\frac{D{\pi }^2}{2T\left(\widehat{\pi }+h_4\right)}+\frac{S_4}{T}$$

(31)

Substituting n = 4 in the Equation (7) through (13), the total cost for each of the three chain members is:

$$\begin{array}{c}TC=\frac{T}{2A}\left(AD{h}_4-\frac{AD{h}_4^2}{\left(\widehat{\pi }+h_4\right)}-A{h}_3{\displaystyle \sum _j{D}_3}\right)+\hfill \\ \frac{T}{2A}{K}_3\left(\left(h_2+h_3\right){\displaystyle \sum _j{D}_{3.j}^2{B}_{3.j}+A{h}_3{\displaystyle \sum _j{D}_{3.j}-A{h}_2{\displaystyle \sum _j{D}_{2.j}}}}\right)+\hfill \\ \frac{T}{2A}{K}_3{K}_2\left(\left(h_1+h_2\right){\displaystyle \sum _j{D}_{2.j}^2{B}_{2.j}+A{h}_2{\displaystyle \sum _j{D}_{2.j}-A{h}_1{\displaystyle \sum _j{D}_{1.j}}}}\right)+\hfill \\ \frac{T}{2A}{K}_3{K}_2{K}_1\left(\left(h_0+h_1\right){\displaystyle \sum _j{D}_{1.j}^2{B}_{1.j}+A{h}_1{\displaystyle \sum _j{D}_{1.j}}}\right)+\hfill \\ \frac{D\left(\widehat{\pi }+h_4\right)}{2T}{\left(T_S-\left(\frac{T{h}_4-\pi }{\widehat{\pi }+h_4}\right)\right)}^2+\frac{1}{T}\left({\displaystyle \sum _{i=1}^4\frac{S_i}{{\displaystyle \prod _{s=i}^4{K}_s}}-\frac{D{\pi }^2}{2\left(\widehat{\pi }+h_4\right)}}\right)+\frac{D{h}_4\pi }{\left(\widehat{\pi }+h_4\right)}\hfill \end{array}$$

Detailed computational steps

Numerically, the developed algorithm is applied to solve this example as outlined in the following four steps:

Step 0. This step initializes the predefined recursive structures required by step one for computing the optimal integer multipliers.

${\alpha }_0=$	1.02321 × 10+41
${\psi }_0=$	0
${\phi }_0=$	0

Step 1. This step determines the integer multipliers K₁, K₂, and K₃. It first calculates K₁ the integer multiplier at the first stage. Second, it computes K₂ the integer multiplier at the second stage. Finally, the integer multiplier at the third stage K₃ is calculated.

For i = 1:
${\alpha }_1=$	2.01612 × 1041
${\psi }_1=$	1.02321× 1041
${\phi }_1=$	1000
$K_1=$	2
For i = 2:
${\alpha }_2=$	4.01455× 1041
${\psi }_2=$	4.06254× 1041
${\phi }_2=$	900
$K_2=$	2
For i = 3:
${\alpha }_3=$	2.83046× 1039
${\psi }_3=$	1.21396× 1042
${\phi }_3=$	650
$K_3=$	1

Step 2. This step computes the values of Y and W that correspond to the optimal values of the basic inventory cycle time T and the optimal stock-out time T_S as follows:

Y =	846,756.926
W =	631.212

Step 3. This step computes the optimal values of the basic inventory cycle time T, the optimal stock-out time T_S and the total cost as follows:

T* =	0.0273
TS* =	0.0055
TC* =	51752.94

5. Discussion

While Cárdenas-Barrón [26] considered only the equal cycle coordination mechanism without shortages and backorders at the last downstream stage, this research paper developed the most generalized non-serial multi-layer multi-tier supply chain optimization model. The model assumed that backorders are allowed at firms in the last downstream stage. In case of shortages, two types of backordering costs are incurred. The model also assumed two inventory coordination mechanisms. The first mechanism is the equal cycle time mechanism which implies that all the firms operating across the supply chain share a common cycle time. The second mechanism is called the integer multiplier mechanism; under which, the cycle time followed at any stage is an integer multiple of the cycle time at the next stage. Results from the numerical analysis show that using the integer multiplier mechanism can result in lower total system cost as compared to using the equal cycle time mechanism. The percentage of this cost reduction depends on the values of both the linear and fixed backorder costs. Namely, the rate of cost reduction increase as either of these two costs increases.

6. Conclusions

This research paper attempts to contribute to the literature of supply chain management by providing a generalized n-stage, non-serial supply chain production-inventory model with linear and fixed backorder costs. The proposed model uses the integer multipliers mechanism for inventory coordination. The paper also proposes a more generalized solution algorithm and extends research works that use the simple none traditional differential calculus methods to derive optimal supply chain inventory decisions. Most of the previous models consider only one type of backorder costs (see for instance [34]).

Practically, the proposed model contributes by generalizing the well-known and widely applied operational management deterministic models: the economic production quantity (EPQ) and economic order quantity (EOQ). These inventory operational management models are widely adopted for supporting the decision making process in practice. Besides, these models are extensively used in research despite their restrictive suppositions.

From a cost-efficiency point of view, our results provide managerial insights for the supply chain partners about the benefits of coordinated production-inventory decisions. The proposed algebraic algorithm can be easily implemented in a decision support system to enhance managerial decisions related to the coordination of supply chain operations. This coordination requires symmetric information sharing among the supply chain players.

One major limitation of our algebraic algorithm is that it cannot identify the cases in which there is more than one optimal value for each integer multiplier. One direction of future research is to consider using the integral procedures proposed by García-Laguna et al. [43] and Cárdenas-Barrón et al. [33] to overcome this limitation. The extended algebraic algorithm can also be extended to several general supply chain inventory models such as considering the effect of imperfect product quality [44] or considering the rework process in the supply chain as in [17].