Optimal Dynamic Production Planning for Supply Network with Random External and Internal Demands

Abstract

This paper focuses on joint production/inventory optimization in single and multiple horizons, respectively, within a complicated supply network (CSN) consisting of firm nodes with coupled demands and firm nodes with coupled demands. We first formulate the single-epoch joint optimal output model by allowing the production of extra quantity for stock underage, considering the fixed costs incurred by having inventory over demand and shortfalls. Then, the multi-temporal dynamic joint production model is further investigated to deal with stochastic demand fluctuations among CSN nodes by constructing a dynamic input–output model. The K-convexity defined in Rn space is proved to obtain the optimal control strategy. According to physical flow links, all demands associated to the nodes of CSN are categorized into the inter-node demand inside CSN (intermediate demand) and external demand outside CSN (final demand). We exploit the meliorated input–output matrix to describe demand relations, building dynamic input–output models where demand fluctuates randomly in single-cycle CSN and finite multi-cycle CSN. The novel monocyclic and multicyclic dynamic models have been developed to minimize system-wide operational costs. Unlike existent literature, we consider fixed costs incurred by overdemand and underdemand inventory into system operational cost functions and then demonstrate the convexity of objective functions. The cost function with two fixed penalty costs due to excess and shortage of inventory is developed in a multicycle model, and the K-convexity defined in Rn is proved to find out the optimal strategy for joint dynamic production of CSNs in the case of multi-products and multicycles.

1. Introduction

Currently, firms increasingly need to improve their supply chain network to gain competitive advantage and profitability. In the network, nodes typically represent entities (such as people, objects, events, etc.), and the connecting edges between nodes represent the relationships between entities [1]. In a supply chain network, a product is generally fabricated jointly by multiple firm nodes (the nodes may be different organizations of firms or independent firms) within the network, with each node responsible for one component or several components of the final product. The physical configuration of a product determines the demand correlation among nodes along with complex mutual transaction relations. Firm nodes of the network have their decision objectives; that is, they seek individually optimal interests. Although they have the aligned goal of final product fabrication, this typical decentralized decision-making scenario incurs various profit targets; that is, the optimal policy of each node might be a suboptimal result either for its upstream or downstream firms, further leading to the non-optimal decision of the whole network. Therefore, this paper focuses on figuring out network-wide coordination and optimization in some special decentralized decision settings.

With the development of information and communication technology and the industrial revolution, the supply chain has increasingly become a network consisting of multiple relevant organizations in which firms as elements rely on each other to acquire competitive advantages by integrating materials, information, capital, and knowledge flows. Accordingly, SCM is a managerial process within which the firm designs and maintains a suitable mechanism or mode and utilizes its core advantages to make itself and the whole supply network to attain competitive advantages through integrated resources and the support offered by cooperative firms [2]. Inventory optimization and production optimization have become two indispensable pillars in supply chain management. Inventory optimization is generally achieved through scientific methods and advanced technological means, such as demand forecasting [3], inventory control strategies [4], supply chain collaborative inventory management [5], etc., to optimize inventory levels, reduce the risk of inventory backlog and shortage, lower inventory costs, and enhance the flexibility and resilience of the supply chain. In the supply network of joint production, inventory optimization also needs to consider inventory sharing between different nodes, coordination of replenishment strategies, and inventory allocation in emergency situations to ensure the smooth operation and efficient response of the overall supply chain. Production optimization, which goes beyond the traditional scope of optimizing inventory for a single product or multiple products, focuses on how to achieve optimal configuration of production processes in the context of multiple enterprises jointly developing or producing a certain product to meet diversified market demands, becoming a new research focus. Qu Li et al. [6] constructed a joint optimization model for production batch size and preventive maintenance threshold based on nonlinear degradation of production systems, with the goal of maximizing unit time profit. Zhu Kai et al. [7] proposed an integrated data-driven model that coordinates maintenance planning decisions with production planning decisions to solve the scheduling and maintenance planning problems of parallel series production lines. Distinct from the existing literature, which usually considers inventory optimization for a single product or multiple products without demand correlation [8], this paper investigates the joint production optimization in a supply network comprising many demand-dependent nodes. Several firms co-develop or manufacture a particular product to meet one or more kinds of market demands. Their coordination and cooperation in the development, manufacturing, and transaction processes comprise the diverse coupled relationships in the supply network. In this firm, nodes belong to different profit centers. Thus, there is a series of problems in realizing network-wide coordination [8], cooperation, and optimization in cases where the goals of manufacturing operations are congruent [9].

In this study, the manufacturers fabricating a particular product make their components with intertwined supply and demand relationships based on physical configuration among firm nodes. The demand is usually uncertain and may have unpredictability such as seasonality and globality, so suppliers need an efficient and flexible planning system [10]. The demand correlation circumscribes boundaries for the network, separating nodes from the outer supply network. As a supplier, each node needs to make a production plan to satisfy demands from the inside and outside network simultaneously. The former kind of demand is assumed deterministic, and the latter random. This demand within and outside the network is similar to push and pull demand. Peters [11] examined the impact of technology-push and demand-pull policies on technological change. Technological push is the assumption that the transition from research to development and, ultimately, to diffusion is a largely linear process driven by the supply side. Demand pull, on the other hand, assumes that anticipated market demand is the key determinant of technological change. Ref. [12] studied the upstream policy push and downstream demand pull to form a long-term mechanism for green technology innovation in Upstream Policy Promotion and Downstream Demand Pull to Form the Long-term Mechanism of Green Technology Innovation. It can be seen that pull is more likely related to the final demand outside the network in this study, and push is relevant to the dependent demand inside the network.

For instance, a specific auto manufacturing company is a supply network comprising an engine manufacturer, a tire supplier, assembling providers, etc. Among them, this specific tire supplier should also consider demand from other auto manufacturers outside the network boundary besides that from the inside when he makes decision-making on capacity and volumes spanning selling periods. Another case is the supply network of Hewlett–Packard and Compaq (HP) laptops, which includes mainboard suppliers, Intel chips, and monitor manufacturers. As an indispensable portion of this featured network, Intel needs to consider both endogenous and exogenous demands in and out of this computer supply network. Categorizing requirements internally and externally helps business nodes more accurately assess the priority and urgency of different requirements. Internal demand often directly affects the continuity and efficiency of the production process, while external demand determines the market demand for the final product. Timely fulfillment of internal demand helps reduce waiting time and inventory backlogs in the production process, thus improving the responsiveness of the entire supply chain. At the same time, accurate forecasting and a quick response to external demand can enhance the competitiveness of an enterprise in the market. By distinguishing between these two types of demand, an enterprise can allocate resources more efficiently and ensure that critical production processes are not affected.

In the context of a globalized economy, optimizing complex supply chain networks (CSN) is an internal national issue and an international challenge. Cross-border supply chains face a more complex and changing market demand, and economic fluctuations, changes in consumer preferences, and policy adjustments in different countries and regions will directly affect the accuracy of demand forecasts. Cross-border logistics involves the transportation system of many countries and regions, including sea, air, land, and other modes, and its complexity and cost are much higher than that of domestic logistics. Globally, there have been many attempts to improve the efficiency of supply chain networks through joint production optimization. For example, multinational enterprises in the automobile manufacturing industry, such as Toyota and Volkswagen, have achieved standardization and sharing of parts and components by establishing a global production network, reducing production costs, and improving market response speed. At the same time, these enterprises also realize transparent and collaborative supply chain management using information technology, further improving overall operational efficiency. Therefore, discussing the joint optimal dynamic production planning problem in this paper is necessary. Yi Cong-qin et al. [13] reveal the sources of supply chain risk through the evolution of complex supply chain network modeling, identify the critical nodes in the supply chain network, and improve the supply chain network’s ability to withstand risk by closely cooperating with the critical nodes and enhancing the network resilience. Mahendra Piraveenan et al. [14] conducted an extensive structural and topological analysis of supply chain networks constructed from the FactSet Revere dataset to capture global supply chain relationships between firms. The results show that large multinationals predictably dominate the centrality analysis. Robert Wiedmer and Stanley E. Griffis [15] argue that the network structure of a supply chain affects the global firm and supply chain levels, which is addressed by examining the structural topology in the supply chain network. The results show that identifying the network topology is relevant and provides insight into the behavior and performance of the network. Danijela Doncic et al. [16] consider that complex supply chains are evolving into multilevel international supply networks, incorporating the relationships between logistics service providers in the automotive industry into the analysis.

Being different from the existing literature studying perishable products, we investigated the optimal policies for producing functional products with a long life cycle in a multiple-period horizon. In the decentralized setting, every firm node within the network usually has the decision-making of maximizing its own profit individually. Each node producer within the network has to make a production plan for satisfying demands both from the network and the external market. Essentially, the market demand variation will disperse among the nodes through the demand correlation. In the multi-period scenario, the preceding net inventory status should be considered in the succeeding period. We mainly focused on the joint production/inventory control problems in a complex structured supply network (CSN) with special demand correlation under two production policies. One of them is a static single-period issue, and we will study the optimal control strategy that allows for additional production. The other is a dynamic multi-period issue, and we will provide the optimal joint production strategy by means of $K$-convexity in ${ℝ}^n$. This paper provides an in-depth analysis of the core issue of joint production/inventory optimization in a complex supply chain network composed of nodes with coupled demand characteristics. By constructing and modifying the input–output matrix model, we accurately portray the intricate dependencies among nodes and then propose a set of strategy frameworks to promote network-wide coordination and optimization. In addition, the study is not only limited to the single dimension of production/inventory optimization but also novelly introduces a multi-perspective joint optimization analysis, which significantly enhances the overall understanding of the complex decision-making mechanisms in supply chain networks. Against the backdrop of the increasing trend of outsourcing and offshoring in the global manufacturing industry [17], the results of this paper provide valuable theoretical guidelines and practical references for the joint optimization of production (and inventory) planning among node firms in a decentralized decision-making environment in supply networks.

The rest of the paper is organized as follows: Section 2 gives a review on the related literature. Section 3 introduces the structure of a complex supply chain network, categorizing two kinds of demands with its principles. We also describe the input–output method that reflects demand links amongst adjacent node firms with decision variables and parameters determined. And then, considering the scenario with a short-life-cycle product, we formulate it as a static single-period problem and provide the analysis and calculation method. In Section 4, changing to look at the scenario of a relatively long-life cycle product made across multiple periods, we further compare the settings with and without taking the fixed set-up costs into account, respectively. We then devise dynamic models to demonstrate the optimal policy and the related managerial insights. Computational study and relevant implications are provided in Section 5. As a closing, in Section 6, we expose the contribution of this paper and highlight directions for future research.

2. Literature Review

The literature related to this study includes four streams of research, such as single-product production/inventory optimization, multi-product production planning, input–output model-based production planning, and production network design.

Our study is related to the literature on the single-product multi-period production or inventory decision-making problems subject to stochastic demand. Following the pioneering work [18,19,20,21], Gallego and Sethi [22] defined the generalized concept of $K^n$-convexity and demonstrated its related properties, while some other research (like [23,24,25,26,27]) discusses system design or logistics issues in various scenarios of supply chain networks. The literature above just studied either single-product or multi-product issues without considering the demands coupling among each product parts. Although Gallego and Sethi [22] and Pathak, Day [23] study supply chain structures similar to ours, they only considered a scenario of a static single period. Furthermore, we utilize ${ℝ}^n$$K$-convexity to study the optimal policy of joint production in the CSN with a multi-product situation.

More closely related to our study is the literature on multi-product dynamic production planning. The classic literature in this area can be found in Kalin [20], Johnson [28], Lenard and Roy [29,30,31], Hausman, Lee [32], Ohno and Ishigaki [33], and Liu and Esogbue [21], which are followed by some typical research, including Bhattacharya [34], Hill and Pakkala [35], Panda and Maiti [36], Feng, Liu [37], and Topan, BayIndIr [38]. Recently, there has been an increasing body of literature on this field. Vahdani, Behzadi [39] studied optimal multi-product dynamic pricing and inventory coordination strategy based on fuzzy expansion method. Nishi and Sakurai [40] analyzed the factors causing demand uncertainty of multi-cycle production and discuss the concept and strategy of the multi-cycle production plan reconfiguration. Ben-Ammar, Bettayeb [41] developed the optimization method of multi-cycle supply planning under random lead time and dynamic demand. Liu, Zhang [42] showed the optimal production and inventory strategy of the multi-cycle fixed-proportion cogeneration system. Xiu, Duan [43] studied how to construct and optimize the multi-cycle closed-loop supply chain network equilibrium model under the consideration of consumer product preferences. Duan, Yao [44] analyzed the closed-loop supply chain network equilibrium from the perspective of marketing and corporate social responsibility. Hajiaghaei–Keshteli, Rahmanifar [45] designed a multi-cycle dynamic EV supply chain production and routing problem considering energy consumption. Ref. [46] studied multi-product production planning, demand influenced by financial assets, combined optimization of production volumes, and real-time hedging strategies. Ref. [47] studied the multi-phase supplier selection and inventory-volume problems in multi-product serial supply chains. Ref. [48] proposed a dynamic virtual unit reconstruction framework for multi-cycle programming in a dynamic environment. Ref. [49] used approximate dynamic programming methods to solve capacity-planning problems in multi-plant and multi-product supply chains. Ref. [50] explored different batch strategies for supplier selection and order allocation in a two-stage supply chain. Ref. [51] studied the humanitarian supply chain network design problem considering product differences and demand uncertainties. Ref. [52] adopted cross-block convenient scheduling technology to verify the flexibility and economic benefits of the production system. To sum up, current research on multi-cycle production focuses on strategies and optimization methods for the coordination between multi-cycle production planning and inventory under uncertain (such as demand uncertainty and lead time randomness) and fuzzy environments. However, as supply chain management becomes increasingly complex, multiple objectives (such as cost, time, quality, and sustainability) should be considered simultaneously. This requires more advanced optimization algorithms and decision-support systems. There should also be a focus on dynamic tuning and real-time optimization, where real-time data and advanced analytics are applied.

The literature focusing on the input–output model-based multi-product inventory optimization in the supply chain is also highly related to our study. Ge [53] combined the input–output model with the production function to estimate the direct-consumption coefficient. The early-stage literature of this kind includes Ren and Liu [54], Xiong and Tang [55], and Zhang, Jun [56]. There has also been a lot of research of this kind in recent years. Ahmadi, Valmorbida [57] expounded that in distributed parameter systems, the convex optimization method can be used to solve the optimization problems of multiple objective functions involved in input and output analysis, such as maximizing system performance, minimizing energy consumption, etc., so as to obtain the optimal input and output strategy of the system. Song and Ieee [58] analyzed that uncertainty quantification is a key step in evaluating the uncertainty of input parameters and output response. Commonly used uncertainty quantification methods include probability distribution fitting, variance analysis, and sensitivity analysis, etc., which can help to understand the uncertainty distribution of input and output and its impact on the optimization process. Ling, Song [59] studied an improved evolutionary stochastic neural network approach based on particle swarm optimization and input–output sensitivity, which performs well when dealing with complex and dynamically changing datasets by combining the global search capabilities of PSO and the adaptability of ERNN. Pashko [60] studied capital investment allocation optimization in an open economy based on the input–output model to improve the accuracy and effectiveness of capital investment allocation optimization. Koch, Montenbruck [61] introduced the basic concepts of input–output systems and the challenges of data-driven inference, discuss different sampling strategies, including random sampling, systematic sampling, model-based sampling, and adaptive sampling, and evaluated their performance in various applications. Ozer, Iftar [62] discussed that eigenvalue optimization ensures stability by modifying the eigenvalues of the system, while delayed output feedback increases the complexity of the problem but, at the same time, provides opportunities for stabilization. For more related literature in this field, please refer to [63,64,65,66].

Our study is also highly related to the literature on manufacturing supply chain network design. Yang, Li [67] introduced the new design of knowledge-driven products based on complex networks by integrating and utilizing cross-domain and multi-level knowledge resources. Lian, Yang [68] discussed the propagation of design changes among different components and subsystems in the process of complex product design and explained the propagation mechanism by introducing complex network theory and methods. Lin, Shieh [69] proposed a product-design process of integrated quality engineering and evolutionary neural networks to improve the efficiency and quality of product design by combining advanced quality-engineering methods and evolutionary neural network models. Guo [70] outlined the causes, effects, and methods of predicting the risk of design change propagation in product-design development networks, emphasizing that predicting and managing the propagation of design changes is critical to ensuring product quality, reducing costs, and maintaining development schedules. Yang, Tian [71] introduced the application of evaluation methods based on consensus measurement, network analysis, and analytic Hierarchy Process (AHP) in product-design evaluation. Tahereh Mohammadi et al. [72] proposed an optimal design of an intelligent supply chain for multiple perishable products using IoT-related techniques. They developed a bi-objective nonlinear integer mathematical planning model. The results show that adopting these techniques in the supply chain can reduce the delivery time and total supply chain cost. Peng, Hu [73] discussed the stability of complex product collaborative-design networks from the perspective of ecology. Guo [74] proposed a product configuration evaluation method based on the importance of network nodes to identify the key nodes that have the greatest impact on product configuration so as to optimize product configuration schemes, reduce unnecessary variant generation, reduce development costs, and improve product market competitiveness to improve the generation efficiency of variant design. Jiao, Wu [75] found that crowdsourcing can improve the quality of product design, and this effect is moderated by network connectivity. When the network connectivity is high, crowdsourcing has a more significant effect on the improvement of product-design quality. Jialiang Pan et al. [76] considered different combinations of carbon abatement policies and various carbon abatement regulations that high-carbon-emitting firms may face to optimize production inventory in a multinational supply chain system. The results show that manufacturers and retailers adjust their decisions to reduce carbon emissions regardless of the increase in carbon price or carbon tax. Wei Dai et al. [77] extended the SCN to a hybrid-parallel version where two SCNs are constructed simultaneously. He proposed an adaptive hyper-parameter tuning method so that the hyper-parameters in the supervisory mechanism are automatically adjusted with the learning process. Bojing Liu et al. [78] addressed the limitations of traditional time series-forecasting methods through an innovative deep-learning framework by proposing a Prophet model that combines deep learning with LSTNet and statistics, and the hybrid model was used with the Particle Swarm Optimization algorithm (PSO) to tuning. More recent research like [79,80,81] also conducted similar investigations.

As can be seen from the above literature, the research on product network design involves many fields such as quality engineering, neural networks, complex networks, and knowledge-driven networks. Theoretical tools such as complex networks are introduced to describe and analyze the dynamics and complexity of design. However, the current methods have challenges such as data sparsity, knowledge heterogeneity, and network complexity. Future research needs to focus on improving the usefulness of these methods so that they can be applied more widely in different areas of product design.

Generally, the focused setting and model in this paper are both obviously different from the existing literature. Even though some of the literature above has similar considerations about the dependent demand among products, their target functions and models are different from ours; some other studies have also examined the input–output method, but most of them are rooted in the macro-economy background. Few studies focus on the firm or supply chain level to model the joint optimal output of the multi-item supply network. Effective methods and theoretical studies on this issue are inadequate. This is exactly what this paper tries to resolve.

3. Preliminaries and Base Model

3.1. Preliminaries and Problem Formulation

We consider a supply network consisting of different participant firms that gather to finish one kind of product assembled by a core firm, and the relation of process and material flow circumscribe the boundary of this network, and all the firms outside the boundary are termed as externalities of the network [82]. Suppose this supply network produces a kind of product that is very complex and needs different processes. This product is composed of $n$ components, and each node firm of the supply network is responsible for only one. Every node should not only provide the internal network with a component or service, but it should also meet the demand from the outside network. The demand of each node for components produced by other nodes is generally decided by the composition and technical characteristic of the product. According to the sources of demand, we categorize internal demand inside the network into intermediate demand and external demand from the outside network into final demand. Pathak, Day [23] define two categories of demand: the final or external independent demand refers to demand from the outside of a specific network, and the dependent demand refers to demand among members of the supply network. Obviously, some components are required to meet both external independent demand and intermediate dependent demand simultaneously.

For the intermediate dependent demand, the technical relation of the product structure among member firms in a supply network determines the certain quantity relationship among each component; that is, one specific component has a certain demand dependence with some other components in the network. Simultaneously, this specific component should also try to meet the external random independent demand, then fluctuation brought about by this uncertainty will increase the difficulty of decision-making in the supply chain [83] and be transmitted to relevant components through the dependent relations among intermediate dependent demands. Zacharias Bragoudakis et al. [84] used an input–output model to consider the interdependence between technology and production sectors, productivity, economic efficiency, and living standards. There are direct and indirect quantity dependencies between various external independent demand and intermediate dependent demand, and we can describe those two relations by the direct consumption coefficient matrix and the complete consumption coefficient matrix [85] in the input–output model. We establish a set as $N=\{1,2,\cdots ,n\}$ to describe all demand items of a virtual manufacturing network when organizing production and supply. The illustration of symbols is as follows: $D_i$ denotes Intermediate Demand Item i inside the network, $I_j$—Final Demand Item $j$ inside the network, $Q_I$—planned output of final demand, $Q_D$—planned output of intermediate demand, and $D_i$ and $I_i$ indicates the intermediate demand item $i$ and the final demand item $j$ inside the network, respectively, for all $i,j\in N$. Vectors $Q_I$ and $Q_D$ are the output planning of final demand and intermediate demand, respectively; $a_{ij}(i,j\in N)$ is the quantity of intermediate demand items of the lower level, which the final demand or intermediate demand items demand for, called the direct consumption coefficient $A={\left\{a_{ij}\right\}}_{n\times n}.$ The intermediate demand $D_i$ produced by manufacturing network firms serves as raw materials or intermediate products to meet the intermediate demand $D_j$ and final demand $I_j$; that is

$$Q_D=A{Q}_D+Q_I \mathrm{or} Q_I=(I-A)Q_D=B\cdot Q_D$$

(1)

where $B=I-A={\left\{b_{ij}\right\}}_{n\times n}$. It can be proved that Matrix $(I-A)$ is a nonsingular matrix. Formula (1) can be reformulated as

$$Q_D={(I-A)}^{-1}{Q}_I=H\cdot Q_I$$

(2)

Here, $H={(I-A)}^{-1}={\left\{h_{ij}\right\}}_{n\times n}$ is the complete consumption coefficient matrix, or Leontief inverse matrix. When the planned vector of final demand $Q_I$ is known, we can use Formula (2) to figure out the demand vector $Q_D$ of every intermediate item easily. Define $X$ to be the market demand for final demand items, and the stochastic market demand for the corresponding intermediate demand items is $Z$ (the stochastic market demand for intermediate items is transited by the final items). From Formula (2), we know $Z=H\cdot X$; that is

$$z_i={\displaystyle {\sum }_{j=1}^n{h}_{ij}{x}_j,i\in N}$$

(3)

A supply network should first decide the planned output $Q_I$ subject to stochastic market demand to obtain maximum profit when organizing production. Meanwhile, the planned output of every intermediate demand item must meet the demand from the final demand; that is, when the planned output of the final demand is $Q_I$, the planned output of the intermediate demand should be

$$Q_p=Q_D+U=H\cdot Q_I+U,U\ge 0$$

(4)

where $q_i^P={\displaystyle {\sum }_{j=1}^n{h}_{ij}{q}_j^I+u_i,} u_i\ge 0,i\in N$, and $u_i$, here, is a control variable, the supply surplus of the intermediate demand. Since $Q_D$ can be calculated directly in Formula (2) by using the complete consumption coefficient and the final demand, and the complete consumption coefficient can be obtained by the BOM of every demand item, so the decision vectors are $Q_I$ and $U.$

So, this paper uses a modified input–output matrix to describe these demand relationships based on the above two types of demand (internal inter-node demand and external demand). Firstly, a single-cycle static model is built, and an novel model is developed to minimize the system operating costs and consider the fixed costs in the case of inventory over- and under-expenditure. Next, the article extends the single-cycle model to a multi-cycle model. It deals with stochastic fluctuations in demand among nodes in a CSN by constructing a dynamic input–output model, thus identifying an optimal strategy for the joint dynamic production of CSNs in the case of multi-products and multi-cycles.

3.2. The Basic Static Model

Shortages and subsequent drops in demand are important factors in improving the production system. To cope with shortages and a subsequent drop in demand, Yue Tan et al. [5] utilizes a machine-learning approach for inventory allocation to achieve the dual goals of cost reduction and customer satisfaction. Bo He and Lvjiang Yin [86] analyze the demand forecasting problem that exists in the context of the Chinese market for cold-chain logistics based on the neural network algorithm, and the mathematical calculations based on the neural network algorithm and grey prediction were conducted. The accurate prediction of cold chain logistics demand is beneficial to optimize the management process. So, facing the internal demand from the network and external market demand outside the network, every member of a CSN may decide whether to add production when confronting the cost incurred by excess and shortage of inventory, and minimizing the total cost usually becomes their goal [87]. Moreover, its decision depends on comparing the optimal profits when the goal is to obtain the maximum profit. On the one hand, the models and solution methods are different under these two situations; on the other hand, because of the variation of parameters, the comparison between the optimal profits will vary correspondingly. The model in this section analyzes the optimal joint production problem when additional production is allowed.

3.2.1. Problem Description

In the single-period situation, if the output of the final demand item $i$ is surplus, let the net residual value of the unit product be ${\kappa }_i^D$ after accounting for inventory cost and future value; additional production is allowed when a shortage happens, and let the penalty cost of each unit for late delivery be $c_p^i$. From common sense, we know ${\kappa }_i^D<c_p^i$. When the final demand output of the supply network is $Q_I$, the difference cost vector between final market demand and planned output of part $i$ is $\Delta C(Q_I)=[\cdots ,\Delta C_i,\cdots ]$, $i\in N$, where $\Delta C_i=c_h^i\cdot {(x_i-q_i^I)}^{-}+c_p^i\cdot {(x_i-q_i^I)}^{+}$, and define $y^{+}=\max \{y,0\}$, $y^{-}=\max \{-y,0\}$. In the process of manufacturing, production cost is composed of fixed costs and variable costs. The fixed cost has nothing to do with the quantity, while the variable cost has a linear relation with it. In general, the fixed cost and the unit variable cost will not change in the range of planned output [88], but they will become higher when the intermediate demand is beyond our plan. Here, we define the intermediate demand vector that is out of the planned range as $Z^{\prime }=[{z_1}^{\prime },{z_2}^{\prime },\cdots ,{z_n}^{\prime }]$ and ${z_i}^{\prime }={(z_i-q_i^p)}^{+}={[z_i-({\displaystyle {\sum }_{j=1}^n{h}_{ij}{q}_j^I+u_i})]}^{+},i\in N$.

Defining $K^p$ and $K^a$ as the fixed-cost vectors under planning quantity and beyond planning, respectively, $V^p$ and $V^a$ are the variable-cost vectors for unit planning quantity and the counterpart for unit quantity beyond planning. Drawing on the analysis above, one can obtain $K^p\ge V^p,K^a\ge V^a$. Assuming that the stochastic market demand $x_i(i\in N)$ for the final demand item is independent, its probability-density function is $f_i(x_i)$, the cumulative distribution function is $F_i(x_i),i\in N$, and the mean value is ${\mu }_i$. Assuming that the price vector of the final demand of the CSN is an exogenous constant, the expected value of the stochastic market demand vector is also a constant. Thus, the expected value of the total revenue of the whole network earned from these final demands is constant. Therefore, maximizing the profit of the supply network equals to minimizing the operational cost of the whole network. For the sake of convenience, we define the fixed cost function of unplanned intermediate demand as $K^a(Z)=[K_1^a(z_1),K_2^a(z_2),\cdots ,K_n^a(z_n)]$, where $K_i^a(z_i)=K_i^a\cdot 1_{B_i}(x_i),i\in N$, $1_{B_i}(.)$ is an indicator function, $B_i=\left\{x_i|x_i=z_i-q_i^p>0\right\}$, and $1_{B_i}(x_i)=1,\mathrm{if} x_i\in B_i;$ else $1_{B_i}(x_i)=0$. Then the expected value of the total operational cost within a supply network is provided by

$$\tilde{J}(Q^I,U)=E\left[\Delta C(Q_I)\cdot E+K^p\cdot E+V^p\cdot (H\cdot Q^I+U)+K^a(Z)\cdot E+V^a\cdot Z^{\prime }\right]$$

(5)

where $E={(1,1,\cdots ,1)}^T$.

The expected value of the final sales is $E(\min (q_i,X_i))=q_i^I+u_i-{\displaystyle {\int }_0^{q_i^I+u_i}{F}_i(x)dx}$; the expected value of the surplus is ${\Delta }^I(Q_I)={[\cdots ,{\Delta }_i(Q_I),\cdots ]}^T$, where it yields

$${\Delta }_i^I(Q_I)=q_i^I+u_i-E(\min (q_i^I,X_i))={\displaystyle {\int }_0^{q_i^I+u_i}{F}_i(x)dx}$$

(6)

The expected value of the loss of sales is $L^I(Q_I)={[\cdots ,L_i(Q_I),\cdots ]}^T$, where

$$L_i^I(Q_I)={\mu }_i-E(\min (q_i^I,X_i))={\mu }_i-q_i^I-u_i+{\displaystyle {\int }_0^{q_i^I+u_i}{F}_i(x)dx}$$

(7)

Note that we use the term ${\displaystyle {\int }_0^{q_i^I+u_i}{x}_i{f}_i(x_i)d{x}_i}$ here instead of the erroneous term ${\displaystyle {\int }_0^{q_i^I}{x}_i{f}_i(x_i)d{x}_i}$ in Ref. [9] for calculating the expected sales and loss throughout this problem. This emendation significantly avoids ignoring the supply surplus $[u_1,u_2,\cdots ,u_n]$ because it is an indispensable part of overall planning supply to respond to the random market demand. Hence, in this sense, our calculation and deployment in this study substantially amends that of the prior research, which provides bran-new optimal joint production schemes.

The objective function of the whole system is to minimize total network-wide cost as follows

$$\begin{array}{c}\begin{array}{cc}\hfill \min \tilde{J}(Q^I,U)=& {\displaystyle {\sum }_{i\in N}[-{\kappa }_i^D{\displaystyle {\int }_0^{q_i^I+u_i}{F}_i(x)dx}+c_p^i({\mu }_i-q_i^I-u_i+{\displaystyle {\int }_0^{q_i^I+u_i}{F}_i(x)dx})}\hfill \\ & +(K_i^p+v_i^p({\displaystyle {\sum }_{j=1}^n{h}_{ij}{q}_j^I+u_i}))+K_i^a[1-G_i({\displaystyle {\sum }_{j=1}^n{h}_{ij}{q}_j^I+u_i})]\\ & +v_i^a({\displaystyle {\sum }_{j\in N}{h}_{ij}{\mu }_j}-{\displaystyle {\sum }_{j\in N}{h}_{ij}{q}_j^I}-u_i+{\displaystyle {\int }_0^{{\displaystyle {\sum }_{j\in N}{h}_{ij}{q}_j^I}+u_i}{G}_i(z)dz})]\end{array}\\ s.t.q_i^I\ge 0, u_i\ge 0 \mathrm{for} \mathrm{all} i\in N.\end{array}$$

(8)

The Hessian matrix of profit target function $\tilde{J}(Q_I,U)$ is shown as $H_{\tilde{J}}=\left[\begin{array}{cc}{\partial }^2\tilde{J}(Q_I,U)/\partial U^2& {\partial }^2\tilde{J}(Q_I,U)/\partial U\partial Q_I\\ {\partial }^2\tilde{J}(Q_I,U)/\partial Q_I\partial U& {\partial }^2\tilde{J}(Q_I,U)/\partial Q_I^2\end{array}\right]$, and the elements in its upper-left sub-matrix ${\partial }^2\tilde{J}(Q_I,U)/\partial U^2$ are

$${\displaystyle \frac{{\partial }^2\tilde{J}(Q_I,U)}{\partial u_k\partial u_l}} = \left\{\begin{array}{cc}\left[\begin{array}{l}-K_k^a{g^{\prime }}_k({\displaystyle {\sum }_{j\in N}{h}_{kj}{q}_j^I}+u_k)\\ +v_k^a{g}_k({\displaystyle {\sum }_{j\in N}{h}_{kj}{q}_j^I}+u_k)+(c_p^k-{\kappa }_k^D)f_k(q_k^I+u_k)\end{array}\right],& k=l;\\ 0,& k\ne l ; k,l\in N.\end{array}\right.$$

(9)

The elements in its sub-matrix ${\partial }^2\tilde{J}(Q_I,U)/\partial Q_I^2$ in the lower right direction are

$${\displaystyle \frac{{\partial }^{2}J}{\partial q_k^I\partial q_l^I}} = \left\{\begin{array}{cc}\left[\begin{array}{l}(c_p^k-{\kappa }_k^D)f_k(q_k^I+u_k)-\\ {\displaystyle {\sum }_{i\in N}{h}_{ik}^2[K_i^a{g_i}^{\prime }({\displaystyle {\sum }_{i=1}^n{h}_{ij}{q}_j^I+u_i})-v_i^a{g}_i({\displaystyle {\sum }_{j\in N}{h}_{ij}{q}_j^I}+u_i)]}\end{array}\right],& k=l,\\ -{\displaystyle {\sum }_{i\in N}{h}_{ik}{b}_{il}[K_i^a{g_i}^{\prime }({\displaystyle {\sum }_{i=1}^n{h}_{ij}{q}_j^I+u_i})-v_i^a{g}_i({\displaystyle {\sum }_{j\in N}{h}_{ij}{q}_j^I}+u_i)]},& k\ne l, k,l\in N.\end{array}\right.$$

(10)

From (8), we can find the Hessian matrix $H_{\tilde{J}}$’s lower right sub-matrix is a symmetric matrix, and the elements of the lower left and upper right matrixes are

$${\displaystyle \frac{{\partial }^{2}J}{\partial u_k\partial q_l^I}} = {\displaystyle \frac{{\partial }^{2}J}{\partial q_l^I\partial u_k}} = \left\{\begin{array}{cc}\left[\begin{array}{l}{v}_k^a{h}_{kl}{g}_k({\displaystyle \sum _{j=1}^n{h}_{kj}{q}_j^I}+u_k)-K_k^a{h}_{kl}g{\prime }_k({\displaystyle \sum _{j=1}^n{h}_{kj}{q}_j^I+u_k})\\ +(c_p^k-{\kappa }_k^D)f_k(q_k^I+u_k)\end{array}\right],& k=l;\\ v_k^a{h}_{kl}{g}_k({\displaystyle \sum _{j=1}^n{h}_{kj}{q}_j^I}+u_k)-K_k^a{h}_{kl}g{\prime }_k({\displaystyle \sum _{j=1}^n{h}_{kj}{q}_j^I+u_k}), & k\ne l; k,l\in N.\end{array}\right.$$

(11)

3.2.2. Model Analysis

First, assume the expected profit function $\tilde{J}(Q_I,U)$ is a second-order continuous function. When $\tilde{J}(Q_I,U)$ is convex and the first-order derivative of $\tilde{J}(Q_I,U)$ to every final demand $q_k^I(k\in N)$, and the intermediate demand surplus $u_k(k\in N)$ equals zero, we can obtain the sole optimal solution (maximum value) in (9). Work out the partial derivative of $q_k^I$ and $u_k$ in (9), respectively, and let them be zero as follows:

$$\begin{array}{ll}{\displaystyle \frac{\partial \tilde{J}(Q^I,U)}{\partial u_k}}=& v_k^a{G}_k({\displaystyle {\sum }_{j\in N}{h}_{kj}{q}_j^I}+u_k)-({\kappa }_k^D-w_k)F_k(q_j^I+u_k)\\ & -K_k^a{g}_k({\displaystyle {\sum }_{i=1}^n{h}_{kj}{q}_j^I+u_k})-v_k^a+v_k^p-w_k=0\end{array}$$

(12)

$$\begin{array}{l}{\displaystyle \frac{\partial \tilde{J}(Q^I,U)}{\partial q_k^I}}=(w_k-{\kappa }_k^D)F_k(q_k^I+u_k)-w_k-\\ {\displaystyle {\sum }_{i\in N}{h}_{ik}[-v_i^a{G}_i({\displaystyle {\sum }_{j\in N}{h}_{ij}{q}_j^I}+u_i)+K_i^a{g}_i({\displaystyle {\sum }_{i=1}^n{h}_{ij}{q}_j^I+u_i})+v_i^a-v_i^p]}=0\end{array}$$

(13)

Considering (12), then (13) is changed to

$$({\kappa }_k^D-w_k)F_k(q_k^I+u_k)+w_k-{\displaystyle {\sum }_{i\in N}{h}_{ik}[({\kappa }_i^D-w_i)F_i(q_i^I+u_i)+w_i]=0}$$

(14)

where $g_i(z_i)$ and $G_i(z_i)$ are $z_i$’s probability density function and cumulative distribution function, respectively, where $k\in N$.

Furthermore, if the expected function of operational cost $J(Q^I,U)$ is a convex function, then by Formulas (10) and (12), the optimal solutions of $q_k^I$ and $u_k$ can be easily obtained, considering that the probability density of the stochastic market demand $X_i$ of the final demand item $i$ is decreasing with the increase in demand. Based on the economic phenomenon that the probability density of stochastic demand $z_j$ of intermediate demand item $j$ of market demand is also decreasing with increasing $z_j$, we assume the density function $g_j(z_j)(j\in N)$ of stochastic demand is monotonely decreasing.

Theorem 1.

The total profit’s expected-value function $\tilde{J}(Q^I,U)$ is convex when the density function $g_j(z_j)(j\in N)$ of intermediate demand is monotone decreasing; that is, ${g_j}^{\prime }(z_j)<0,$ $z_j\in R^{+},j\in N$.

The Proof of Theorem 1 is shown in the Appendix A. $Q^{\ast }$ and $u^{\ast }$ solved by formulas (12) and (14) are the joint optimal output of final products’ manufacturing sector M and raw materials’ supply sector S. The theorem above largely relaxes the relevant constraints of Ref. [9] and, accordingly substantially generalizes the associated research outcomes in the literature. We can prove the theorem only by the constraint of ${g_j}^{\prime }(y_j)<0,j\in N$. This implies that a globally optimal solution leads to an optimal joint-output strategy. So, firms faced with monotonically decreasing market demand can use the optimization model to find an explicit production and inventory strategy that maximizes profits.

If the condition of Theorem 1 does not hold, then the solution process of (8) will not be a convex programming problem anymore, and we cannot figure out the global optimal solution directly. In this situation, we can employ the direct optimization algorithm to seek the approximate optimal solution.

3.2.3. Solving Method and Optimal Solution

To figure out the relationship between two arbitrary control variables $u_i$ and $u_j$, $i,j\in N$, simple derivative calculation shows the independence of optimal solution $u_k^{\ast }$ and $u_j^{\ast } (k,j\in N)$ as the planned surplus of intermediate demand. Furthermore, the first-order optimal condition yields the following equations:

$$\left[\begin{array}{c}{F}_1(q_1^I+u_1)\\ F_2(q_2^I+u_2)\\ ⋮\\ F_n(q_n^I+u_n)\end{array}\right] = {\left[\begin{array}{cccc}(h_{11}-1)(w_1-{\kappa }_1^D)\hfill & h_{21}(w_2-{\kappa }_2^D)& \cdots & h_{n1}(w_n-{\kappa }_n^D)\\ h_{12}(w_1-{\kappa }_1^D)\hfill & (h_{22}-1)(w_2-{\kappa }_2^D)& \cdots & h_{n1}(w_n-{\kappa }_n^D)\\ ⋮& ⋮& \ddots & ⋮\\ h_{1n}(w_1-{\kappa }_1^D)\hfill & h_{2n}(w_2-{\kappa }_2^D)& & (h_{nn}-1)(w_n-{\kappa }_n^D)\end{array}\right]}^{-1} \left[\begin{array}{c}{\displaystyle {\sum }_{i\in N}{h}_{i1}{w}_i}-w_1\\ {\displaystyle {\sum }_{i\in N}{h}_{i2}{w}_i}-w_2\\ ⋮\\ {\displaystyle {\sum }_{i\in N}{h}_{in}{w}_i}-w_n\end{array}\right] = \left[\begin{array}{c}{\tilde{\theta }}_1\\ {\tilde{\theta }}_2\\ ⋮\\ {\tilde{\theta }}_n\end{array}\right]$$

(15)

The solution is obtained as

$$q_k^{I^{\ast }}+u_k^{\ast }=F_k^{-1}({\tilde{\theta }}_k), k\in N$$

(16)

Substitute (16) into (12) and the original function is upgraded as

$$G_k({\displaystyle {\sum }_{j\in N}{h}_{kj}{q}_j^I}+u_k)-{\displaystyle \frac{K_k^a}{v_k^a}}{g}_k({\displaystyle {\sum }_{j=1}^n{h}_{kj}{q}_j^I+u_k})-{\displaystyle \frac{v_k^a-v_k^p+w_k+({\kappa }_k^D-w_k){\tilde{\theta }}_k}{v_k^a}}=0$$

(17)

Generally speaking, (17) is a nonlinear function, and it is intractable to figure out the analytical solution. The common method is to employ binary search or other numerical analysis solutions to find out the approximate solution. We, therefore, use ${\mu }_k$ to approximate ${\displaystyle {\sum }_{i=1}^n{h}_{kj}{q}_j^{I^{\ast }}+u_k^{\ast }}$ for $\forall k\in N$. The solutions of the equations set formed by the 2n equations jointly being established from Formulas (16) and (15) are as follows

$$\left[\begin{array}{c}{q}_1^{I^{\ast }}\\ q_2^{I^{\ast }}\\ ⋮\\ q_n^{I^{\ast }}\end{array}\right]={\left[\begin{array}{cccc}{h}_{11}-1\hfill & h_{12}& \cdots & h_{1n}\\ h_{21}\hfill & h_{22}-1& \cdots & h_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ h_{n1}\hfill & h_{n2}& \cdots & h_{nn}\end{array}\right]}^{-1}\left[\begin{array}{c}{\mu }_1-F_1^{-1}({\tilde{\theta }}_1)\\ {\mu }_2-F_2^{-1}({\tilde{\theta }}_2)\\ ⋮\\ {\mu }_n-F_n^{-1}({\tilde{\theta }}_n)\end{array}\right]$$

(18)

$$\left[\begin{array}{c}{u}_1^{\ast }\\ u_2^{\ast }\\ ⋮\\ u_n^{\ast }\end{array}\right]=\left[\begin{array}{c}{F}_1^{-1}({\tilde{\theta }}_1)\\ F_2^{-1}({\tilde{\theta }}_2)\\ ⋮\\ F_n^{-1}({\tilde{\theta }}_n)\end{array}\right]-{\left[\begin{array}{cccc}{h}_{11}-1\hfill & h_{12}& \cdots & h_{1n}\\ h_{21}\hfill & h_{22}-1& \cdots & h_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ h_{n1}\hfill & h_{n2}& \cdots & h_{nn}\end{array}\right]}^{-1}\left[\begin{array}{c}{\mu }_1-F_1^{-1}({\tilde{\theta }}_1)\\ {\mu }_2-F_2^{-1}({\tilde{\theta }}_2)\\ ⋮\\ {\mu }_n-F_n^{-1}({\tilde{\theta }}_n)\end{array}\right]$$

(19)

where $X^{-1}$ means the inverse matrix of $X$ and $F_i^{-1}(.)$ represents the inverse function of $F(.).$

Here, we analyze the uncertainty in the input parameters and output response and consider the effects of three typical demand-uncertainty expressions, i.e., uniform-, normal-, and exponential-probability distributions. We examine the three distributions and rewrite Equation (18) as follows:

$$\left[\begin{array}{c}{q}_1^{I^{\ast }}\\ q_2^{I^{\ast }}\\ ⋮\\ q_n^{I^{\ast }}\end{array}\right]=O*\left[\begin{array}{c}{\mu }_1-F_1^{-1}({\tilde{\theta }}_1)\\ {\mu }_2-F_2^{-1}({\tilde{\theta }}_2)\\ ⋮\\ {\mu }_n-F_n^{-1}({\tilde{\theta }}_n)\end{array}\right] \mathrm{i}.\mathrm{e}., q_i^{I^{\ast }}={\displaystyle {\sum }_{j=1}^n{o}_{ij}[{\mu }_j-F_j^{-1}({\tilde{\theta }}_2)]}$$

First, consider that the demand obeys a uniform distribution $X\sim U(c,d)$, and let the random variables obey a uniform distribution on the interval $[c,d]$, so $d_j-c_j$ is the fluctuation of the demand, so the optimal output can be deduced as follows:

$$\begin{array}{l}{\displaystyle \frac{x_j-c_j}{d_j-c_j}}={\tilde{\theta }}_j\Rightarrow x_j=F_j^{-1}({\tilde{\theta }}_j)={\tilde{\theta }}_j(d_j-c_j)+c_j\\ \Rightarrow q_i^{I^{\ast }}={\displaystyle {\sum }_{j=1}^n{o}_{ij}[{\mu }_j-{\tilde{\theta }}_j(d_j-c_j)-c_j]}\end{array}$$

With ${\displaystyle \frac{\partial q_i^{I^{\ast }}}{\partial (d_j-c_j)}}=-{\displaystyle {\sum }_{j=1}^n{o}_{ij}{\tilde{\theta }}_j}$, we find that fluctuations in demand affect optimal output. If both $o_{ij}$ and ${\tilde{\theta }}_j$ are positive, the optimal production decreases as demand volatility increases. This may be because when the uncertainty of demand increases, firms may prefer a conservative production strategy to minimize the risk of being unable to meet demand or having excess inventory.

Next, consider that the demand obeys the normal distribution $X\sim N(\omega ,{\rho }^2)$. Therefore, the following optimal output can be derived:

$$\begin{array}{l}\Phi \left({\displaystyle \frac{x-\omega }{\rho }}\right)={\tilde{\theta }}_j\Rightarrow x_j=F_j^{-1}({\tilde{\theta }}_j)=\omega +\rho *{\Phi }^{-1}({\tilde{\theta }}_j)\\ \Rightarrow q_i^{I^{\ast }}={\displaystyle {\sum }_{j=1}^n{o}_{ij}[{\mu }_j-\omega -\rho *{\Phi }^{-1}({\tilde{\theta }}_j)]}\end{array}$$

With ${\displaystyle \frac{\partial q_i^{I^{\ast }}}{\partial \rho }}=-{\displaystyle {\sum }_{j=1}^n{o}_{ij}{\Phi }^{-1}({\tilde{\theta }}_j)};{\displaystyle \frac{\partial q_i^{I^{\ast }}}{\partial \omega }}=-{\displaystyle {\sum }_{j=1}^n{o}_{ij}}$, we find that fluctuations in demand affect the optimal yield. If the standard deviation is significant, i.e., the fluctuation range of demand is extensive, and if $o_{ij}$ and ${\Phi }^{-1}({\tilde{\theta }}_j)$ at this time are both positive, then the more significant the fluctuation range of demand, the smaller the optimal output. This is because, in the case of large fluctuations in demand, firms may use more flexible production strategies, such as on-demand production or delayed production, to more accurately match market demand. Instead, the optimal output increases if the mean is lower. This is because firms believe that increasing production when demand is low can help them gain a larger market share or reduce unit costs by increasing production during market downturns.

Consider that the demand follows an exponential distribution $X\sim EXP(\vartheta )$, so the following optimal output can be derived:

$$\begin{array}{l}1-e^{-\vartheta x}={\tilde{\theta }}_j\Rightarrow x_j=F_j^{-1}({\tilde{\theta }}_j)=-{\displaystyle \frac{1}{\vartheta }}\ln (1-{\tilde{\theta }}_j)\\ \Rightarrow q_i^{I^{\ast }}={\displaystyle {\sum }_{j=1}^n{o}_{ij}[{\mu }_j+\frac{1}{\vartheta }\ln (1-{\tilde{\theta }}_j)]}\end{array}$$

With ${\displaystyle \frac{\partial q_i^{I^{\ast }}}{\partial \vartheta }}=-{\displaystyle \frac{1}{{\vartheta }^2}}{\displaystyle {\sum }_{j=1}^n{o}_{ij}}\ln (1-{\tilde{\theta }}_j)$,we find that fluctuations in demand affect the optimal yield. If both $o_{ij}$ and $\ln (1-{\tilde{\theta }}_j)$ are positive, the optimal production decreases if there are more times of the same level of demand in the same period. This means that the demand in the same period may be more stable with little change, so the firm should wait to expand production, and the overexpansion of production may lead to higher costs and lower productivity. Generally, the above analysis shows that the type of probability distribution of demand can have a significant impact on the operations equilibriums of the system.

4. The Dynamic Multi-Period Model of Joint Production Planning

This section proceeds to focus on the dynamic multi-period model of optimal joint production planning in the same supply chain network of manufacturing. The structure of the supply network is identical with the one we discussed in the last section. The relation among each node is decided by the physical flow as well; that is, we can still use the input–output model to describe the demand-coupling relation among nodes. The main difference is that this section studies the control problem of joint optimal output in a repeated multi-period situation. As for functional products, which have relatively long life cycles generally, they tend to need the organization of multi-period production. In this situation, the products unsold in the previous period will be automatically transferred into the next period as inventory, generating corresponding inventory holding costs. However, different from the traditional classical model in the past, what this section considers is the penalty cost resulting from the excess or shortage of inventory. From these, we established an improved single-period inventory control model of CSN and extended this static model to a dynamic model to seek the control strategy of joint optimal output in multi-period situations.

4.1. Notation System and Assumptions in Dynamic Setting

Because of the complication of the dynamic problem, we re-established part of the marks system in this section and provided some corresponding assumptions. We define the following parameters:

• $N=\{1,2,\cdots ,n\}:$ Products set of network nodes;
• $x={[x_1,x_2,\cdots ,x_n]}^{\mathrm{T}}:$ Initial inventory vector of every node;
• $Z={({\xi }_i)}_{n\times 1}:$ Single-period stochastic market demand vector (where some elements could be zero);
• $Z_{\tau }={({\xi }_i^{\tau })}_{n\times 1}:$ Stochastic market demand vector in $\tau$-th period (where some elements could be zero);
• $A={(a_{ij})}_{n\times n} :$ Incidence matrix of every node’s demand in the network (determined by the constituent relationship of parts, some items could be zero);
• $c_m={[c_m^1,c_m^2,\cdots ,c_m^n]}^{\mathrm{T}}:$ Unit manufacturing cost vector in every node;
• $c_h={[c_h^1,c_h^2,\cdots ,c_h^n]}^{\mathrm{T}}:$ Unit inventory holding variable cost vector in every node
• $c_p={[c_p^1,c_p^2,\cdots ,c_p^n]}^{\mathrm{T}}:$ Unit variable cost vector of the loss of shortage in every node;
• $I-A=B={(b_{ij})}_{n\times n}:$ Leontief matrix; ${(I-A)}^{-1}=D={(d_{ij})}_{n\times n}:$ Leontief inverse matrix;
• ${\alpha }^i:$ Fixed penalty cost of excess inventory of product $i$;
• ${\beta }^i:$ Fixed penalty cost of shortage inventory of product $i$;
• $f_i(.) :$ The probability density function of the stochastic variable ${\xi }_i$ of the single-period demand for product $i$;
• $F_i(.):$ The probability distribution function of the stochastic variable ${\xi }_i$ of the single-period demand for product $i$;
• $f_i^{\tau }(.):$ The probability density function of the stochastic variable ${\xi }_i$ of $\tau$-th period demand for product $i$;
• $F_i^{\tau }(.):$ The probabilistic cumulative distribution function of the stochastic variable ${\xi }_i$ of $\tau$-th period demand for product $i$;

Furthermore, the related decision variables are shown as follows:

• $Q=(q_i{)}_{n\times 1}:$ Inventory level vector of intermediate demand;
• $\stackrel{⌢}{Z}={(z_i)}_{n\times 1}:$ Planned output vector of final market demand;
• $J_x(Q) :$ The total cost function corresponding to $Q$;
• ${\Psi }_{\tau }(x):$ The optimal cost function in $\tau$-th period;

Assume each node only produces (or processes) one certain part of a product. Moreover, the nodes belong to a functional product [89] with a relatively long life cycle, thus requiring repeated production and multi-period production decisions. We also need to assume that the structure relation of assembling a product determines the demand correlation among each node, and the product’s structure is stable and will not change in the finite periods we studied. Therefore, the incidence matrix A in the parameters above does not change with time. Assume the production time is very short, and the demand of each node can be met very quickly. As a result, there is no need to consider the lead time. Meanwhile, suppose all the parameters above have already been known. Each node’s initial inventory level vector $x$ only refers to the holding inventory level at the beginning of the studied $t$ period. The production decision and internal and stochastic external demand in the previous period decide the initial inventory level of subsequent periods. The inventory surplus at the end of each period is directly transferred into the next period as initial inventory, but the shortage of inventory will not accumulate into the next period.

4.2. Model Setup

The product structure determines the quantity relation of the components that make it up; that is, the demands for components are complementary. A BMW car needs four hubs, four outer rubber tires, four inner tiers, one engine, one pair of windshield wipers, four seats, etc. It is easy to find that, in the BMW car supply network, the demands among each node have direct or indirect relevance and coupling relations. Some node firms in this network, like the tier firm, will meet the demands from the BMW supply network, as well as those outside the network from other car manufacturing firms. The production of each product $i$ should not only meet the demand from related nodes inside the network as raw materials or intermediate products but also should meet the final demand ${\xi }_i$ from external markets. This kind of demand correlation can be described by the model of input-output matrix in Zhong and Chen [90]. Matrix $A={(a_{ij})}_{n\times n}$ records and shows the demand correlation among $n$ parts, and the $i-j$ item $a_{ij}$ represents the required quantity of part $i$ when producing one unit part $j$ in this supply network. This paper focuses on functional products that have stable structures in their life cycle, and every item in the demand relationship matrix $A={(a_{ij})}_{n\times n}$ is constant.

To meet stochastic market demand, each node firm in the network needs to make corresponding planned external demand output matrix $\stackrel{⌢}{Z}={[z_1,z_2,\cdots ,z_n]}^{\mathrm{T}}$. Meanwhile, at the beginning of every period, each firm should decide the output of this period and the inventory level $Q={[q_1,,q_2,\cdots ,q_n]}^{\mathrm{T}}$ after finishing the production. The input–output relation between the planned outputs of intermediate demand and external demand is $Q=AQ+\stackrel{⌢}{Z}$; this is

$${\displaystyle {\sum }_{j\in N}{b}_{ij}{q}_j}=z_i,\forall i\in N.$$

(20)

where $(I-A)$ is the Leontief matrix [91]. We further assume that the stochastic demand outside the network of each product is independent and an exogenous variable, and that it will not be influenced by the system’s intermediate demand. At the beginning of every period, we assume that there exists a central decision-maker in the system. Thus, the target problem is how to design the output of each node firm in this period and how to determine the inventory level matrix $Q={[q_1,q_2,\cdots ,q_n]}^{\mathrm{T}}$ based on the initial inventory level matrix of each node product $x={[x_1,x_2,\cdots ,x_n]}^{\mathrm{T}}$. Then, the decision variable of the node firm $i$ is $q_i,\forall i\in N.$

This section discussed the features of the production decision of every node firm in the CSN with an input–output demand relation in a single-period situation. Then, we extended the issue to a multi-period situation. According to the classical single-product model and the $(s,S)$ strategy, we know the set-up cost is the cause for the occurrence of the bottom and top control points $s$ and $S$ of inventory level. Inspired by this, in the ${ℝ}^n$ space, we considered the existence and inexistence of the set-up cost, respectively, and then compared the optimal control strategies under these two situations.

The cost function in this paper is distinct from the classical single-period model frequently discussed by the traditional literature. The traditional classical single-period inventory model only examined the production-variable cost, inventory holding-variable cost, and loss-variable cost of shortage. In addition to those variable costs above, this paper also added the fixed penalty costs caused by excess and shortage of inventory that sometimes must be considered in practical decisions but is seldom seen in the existing literature. To clarify, the production set-up cost referred to above is totally different from the two fixed penalty costs caused by the excess and shortage of inventory. We can easily distinguish them: the former is caused by the decision of whether to start the production line during this period, and the latter is led by the difference between the inventory level and actual demand (from inside and outside the network). In industrial practice, there indeed exists fixed penalty costs caused by excess and shortage of inventory. That can be taken as the sunk cost and compensation for the shortage or excess of inventory. If service is included when providing the product, those fixed penalty costs should be cared for particularly.

We first study the single-period decision problem without considering the fixed set-up cost and then add it in. In these two situations, the cost composition contains the production-variable cost, inventory holding-variable cost, loss-variable cost of shortage, and fixed penalty costs ${\alpha }^i$ and ${\beta }^i$ cased by excess and shorting of inventory.

4.3. The Improved Monocyclic Model

4.3.1. The Monocyclic Model without Fixed Production Set-Up Cost

Without considering the fixed production set-up cost, the system cost function of the supply network is

$$\begin{array}{ll}{J}_x(Q)& ={\displaystyle {\sum }_{i\in N}{E}_{x_i}[C_i]}\\ & ={\displaystyle {\sum }_{i\in N}E[c_m^i(q_i-x_i)+({\alpha }^i-{\beta }^i)\delta (z_i-{\xi }_i)+c_h^i{(z_i-{\xi }_i)}^{+}+c_p^i{(z_i-{\xi }_i)}^{-}]}\\ & ={\displaystyle {\sum }_{i\in N}\{c_m^i(q_i-x_i)+{\displaystyle {\int }_0^{{\displaystyle {\sum }_{j\in N}{b}_{ij}{q}_j}}[{\alpha }^i+c_h^i({\displaystyle {\sum }_{j\in N}{b}_{ij}{q}_j}-\epsilon )]f_i(\epsilon )d\epsilon }+}\\ & {\displaystyle {\int }_{{\displaystyle {\sum }_{j\in N}{b}_{ij}{q}_j}}^{\infty }[{\beta }^i+c_p^i(\epsilon -{\displaystyle {\sum }_{j\in N}{b}_{ij}{q}_j})]f_i(\epsilon )d\epsilon }\}\end{array}$$

(21)

where function $\delta (.):ℝ\to \{0,1\}$ satisfies $\delta (y)=1 \mathrm{if} y>0,$ otherwise it is 0 and the first item in $E[.]$, which represents the production variable cost. The second and the fourth item, respectively, indicate the fixed penalty costs of excess and shortage of inventory, the third item indicates the inventory holding-variable cost, and the fifth item indicates the loss-variable cost of shortage. Thus, the target function of the optimal model of this problem is ${\min }_{Q\ge x}{J}_x(Q)$.

Theorem 2.

$J_x(Q)={\displaystyle {\sum }_{i\in N}[c_m^i(q_i-x_i)+{\alpha }^i\delta (z_i-{\xi }_i)+c_h^i{(z_i-{\xi }_i)}^{+}+{\beta }^i\delta ({\xi }_i-z_i)+c_p^i{(z_i-{\xi }_i)}^{-}]}$ is a convex function in the space of ${ℝ}_{+}^n$.

The Proof of the above theorem is shown in Appendix A. The theorem above denotes the existence of a unique global optimal solution of the function $J_x(Q)$ in the domain ${ℝ}_{+}^n$. Therefore, when enterprises make production decisions, they should consider the convexity of the function or other related functions to ensure that the decision-making process is globally optimal and avoid local optimal solutions. By the first-order optimality condition $\partial J_x(Q)/\partial q_k=0$, $\forall k\in N$, we have

$$\begin{array}{ll}{\displaystyle \frac{\partial J_x(Q)}{\partial q_k}}& ={\displaystyle {\sum }_{i\in N}[c_m^k+({\alpha }^i-{\beta }^i)b_{ik}{f}_i({\displaystyle {\sum }_{j\in N}{b}_{ij}{q}_j})+{\displaystyle {\int }_0^{{\displaystyle {\sum }_{j\in N}{b}_{ij}{q}_j}}{c}_h^i{b}_{ik}}{f}_i(\epsilon )d\epsilon -{\displaystyle {\int }_{{\displaystyle {\sum }_{j\in N}{b}_{ij}{q}_j}}^{\infty }{c}_p^i{b}_{ik}}{f}_i(\epsilon )d\epsilon }]\\ & ={\displaystyle {\sum }_{i\in N}[(c_m^k-b_{ik}{c}_p^i)+({\alpha }^i-{\beta }^i)b_{ik}{f}_i({\displaystyle {\sum }_{j\in N}{b}_{ij}{q}_j})+(c_h^i+c_p^i)b_{ik}{F}_i({\displaystyle {\sum }_{j\in N}{b}_{ij}{q}_j})]}=0, \forall k\in N.\end{array}$$

from which we can obtain the equation set, $B(f+F)=C_{\Delta },$ where $f={[{\gamma }^i{f}_i]}_{n\times 1},$$F={[(c_h^i+c_p^i)F_i]}_{n\times 1},$ $C_{\Delta }={({\displaystyle {\sum }_{i\in N}{b}_{il}{c}_p^i}-n{c}_m^l)}_{n\times 1}$ are all column vectors. Furthermore, we can obtain $f+F=B^{-1}{C}_{\Delta },$ let ${\gamma }^k={\alpha }^k-{\beta }^k,$ $\forall k\in N$. After simplification, we have:

$${\gamma }^k{f}_k({\displaystyle {\sum }_{j\in N}{b}_{ij}{q}_j})+(c_h^k+c_p^k)F_k({\displaystyle {\sum }_{j\in N}{b}_{kj}{q}_j})=(d_{1k},d_{2k},\cdots ,d_{nk}){({\displaystyle {\sum }_{i\in N}{b}_{il}{c}_p^i}-n{c}_m^l)}_{n\times 1}$$

(22)

Let $Q^{\ast }={({q_1}^{\ast },{q_2}^{\ast },\cdots ,{q_n}^{\ast })}^{\mathrm{T}}$ be the solution of Function (22). This is the system’s optimal joint output. According to Theorem 1, we can easily prove the following Theorem 3.

Theorem 3.

When the network systemic cost function $J_x(Q)$ is given, then $Q^{\ast }={({q_1}^{\ast },{q_2}^{\ast },\cdots ,{q_n}^{\ast })}^{\mathrm{T}}$ is the global optimal solution.

The Proof of Theorem 3 is shown in the Appendix A. Theorem 3 states that a globally optimal solution exists under the conditions of a particular network system cost function, which helps firms find the optimal production and inventory strategies through mathematical planning methods when facing complex network systems.

Remark 2.

The method-solving function (22) and the explanation of its economic meaning are as follows. When ${\gamma }^k\ne 0$ holds, (22) is probably an implicit function with a close-form solution, while we can obtain the numerical solution with the binary search method; when ${\gamma }^k=0$ holds, it changes to an explicit function with a close-form solution. Now, we will discuss it in three situations as follows:

$(i) \mathrm{if} \forall k\in N,{\gamma }^k=0, \mathrm{then} F_k({\displaystyle {\sum }_{j\in N}{b}_{kj}{q}_j})=(d_{1k},d_{2k},\cdots ,d_{nk}){({\displaystyle {\sum }_{i\in N}{b}_{il}{c}_p^i}-n{c}_m^l)}_{n\times 1}/(c_h^k+c_p^k)$ Firm $k$’s planned output of the demand from the outside network is ${z_{kc}}^{\ast }={{\displaystyle {\sum }_{j\in N}{b}_{kj}{q}_j}}^{\ast }=F_k^{-1}[(d_{1k},d_{2k},\cdots ,d_{nk}){({\displaystyle {\sum }_{i\in N}{b}_{il}{c}_p^i}-n{c}_m^l)}_{n\times 1}/(c_h^k+c_p^k)]$, corresponding to the optimal output of this period $({q_k}_c^{\ast }-x_k)$, where ${q_k}_c^{\ast }$, the inventory level after finishing production, is provided as

$${q_k}_c^{\ast }={\displaystyle {\sum }_{j\in N}{d}_{kj}{F}_j^{-1}[({\displaystyle {\sum }_{i\in N}{\displaystyle {\sum }_{l\in N}{b}_{il}{d}_{lj}{c}_p^i}-n{\displaystyle {\sum }_{l\in N}{d}_{lj}{c}_m^l}})/(c_h^j+c_p^j)]}, \forall k\in N$$

(23)

where $F_j^{-1}(\cdot )$ is the inverse function of the probability distribution function $F_j(\cdot )$ of external stochastic demand for part $j$ in this single period. Vector $Q_c^{\ast }={({q_1}_c^{\ast },{q_2}_c^{\ast },\cdots ,{q_n}_c^{\ast })}^{\mathrm{T}}$ is the special case of $Q^{\ast }$ when $\forall k\in N,{\gamma }^k=0$ is called as the critical value of function $J_x(Q)$’s optimal solution relative to $\exists k\in N,{\gamma }^k\ne 0$.

$(ii) \mathrm{if} \exists k\in A \subseteq N,{\gamma }^k \ge 0, \mathrm{then} F_k({\displaystyle {\sum }_{j\in N}{b}_{kj}{q}_j}) \le (d_{1k},d_{2k},\cdots ,d_{nk}){({\displaystyle {\sum }_{i\in N}{b}_{il}{c}_p^i}-n{c}_m^l)}_{n\times 1} / (c_h^k+c_p^k)$

By comparing the planned output ${z_k}^{\ast }$ of firm $k$ for demand from the outside network in this situation with that in situation $(i)$, we can see

$${z_k}^{\ast }={\displaystyle {\sum }_{j\in N}{b}_{kj}{q}_j^{\ast }}\le F_k^{-1}[(d_{1k},d_{2k},\cdots ,d_{nk}){({\displaystyle {\sum }_{i\in N}{b}_{il}{c}_p^i}-n{c}_m^l)}_{n\times 1}/(c_h^k+c_p^k)]={\displaystyle {\sum }_{j\in N}{b}_{kj}{q}_{jc}^{\ast }}$$

$(iii) \mathrm{if} \exists k\in N\backslash A,{\gamma }^k\le 0, \mathrm{then} F_k({\displaystyle {\sum }_{j\in N}{b}_{kj}{q}_j})\ge (d_{1k},d_{2k},\cdots ,d_{nk}){({\displaystyle {\sum }_{i\in N}{b}_{il}{c}_p^i}-n{c}_m^l)}_{n\times 1}/(c_h^k+c_p^k)$

With respect to (i) and (ii), the planned output ${z_k}^{\ast }$ of firm $k$ for demand from the outside network is:

$${z_k}^{\ast }={\displaystyle {\sum }_{j\in N}{b}_{kj}{q}_j^{\ast }}\ge F_k^{-1}[(d_{1k},d_{2k},\cdots ,d_{nk}){({\displaystyle {\sum }_{i\in N}{b}_{il}{c}_p^i}-n{c}_m^l)}_{n\times 1}/(c_h^k+c_p^k)]={\displaystyle {\sum }_{j\in N}{b}_{kj}{q}_{jc}^{\ast }}$$

Conditions (ii) and (iii) reveal the fact that when there is a difference between the fixed penalty costs caused by excess and shortage of inventory, the optimal planned output will deviate from the critical optimal output in situation (i). When the fixed penalty cost caused by excess is higher, then the firm should, compared to the critical situation, reduce its output or order inventory level to minimize loss. On the contrary, this employs the opposite strategy. To conclude, the difference between the fixed penalty costs caused by excess and shortage of inventory will directly influence the deviation between the production decision and the critical value.

4.3.2. The Monocyclic Model with Fixed Production Set-Up Cost

Based on the study above, we will take the fixed production set-up cost into consideration. The fixed cost in the traditional single-product single-period model is merely a scalar constant. Literature [9,18,19] examined the $(s,S)$ inventory control strategy in this classical simple model. For facilitating, we quote the generalized $K$-convexity in ${ℝ}^n$ defined by Gallego and Sethi [22] and then apply it to our model as follows.

Definition 1.

Function $K(.):{ℝ}^n \to {ℝ}_{+}$ and Q satisfy $K(Q-x)=K_0\delta (e^{\prime }(Q-x))+{\displaystyle {\sum }_{i\in N}{K}_i\delta (q_i-x_i)},$ where function $\delta (.)$ is similar to prior definition, $e^{\prime }=(1,1,\dots ,1{)}^{\prime }\in R^n,$ the composition of the production set-up fixed cost is $K={[K_0,K_1,K_2,\cdots ,K_n]}^{\mathrm{T}}\in {ℝ}^{n+1}$, $K_0,K_i\in {ℝ}_{+},\forall i\in N$.

For the sake of brevity, denote the representation $\Phi (Q)$ as follows:

$$\begin{array}{ll}\Phi (Q)& ={\displaystyle {\sum }_{i\in N}[c_m^i{q}_i+({\alpha }^i-{\beta }^i)F_i({\displaystyle {\sum }_{j\in N}{b}_{ij}{q}_j})+{\displaystyle {\int }_0^{{\displaystyle {\sum }_{j\in N}{b}_{ij}{q}_j}}{c}_h^i({\displaystyle {\sum }_{j\in N}{b}_{ij}{q}_j}-\epsilon )f_i(\epsilon )d\epsilon }}\\ & \hspace{1em}\hspace{1em}\hspace{1em} +{\displaystyle {\int }_{{\displaystyle {\sum }_{j\in N}{b}_{ij}{q}_j}}^{+\infty }{c}_p^i(\epsilon -{\displaystyle {\sum }_{j\in N}{b}_{ij}{q}_j})f_i(\epsilon )d\epsilon ]}\\ & ={\displaystyle {\sum }_{i\in N}{c}_m^i{q}_i+H(Q)}\end{array}$$

(24)

Let $S=Q^{\ast }={[q_1^{\ast },q_2^{\ast },\cdots ,q_n^{\ast }]}^{\mathrm{T}}$ be the global optimal solution of the production decision of the supply network, which satisfies $\Phi (y)=\Phi (S)+K$. Then, we have:

$$\Phi (y)=\Phi (S)+K_0\delta (e^{\prime }(Q-x))+{\displaystyle {\sum }_{i\in N}{K}_i\delta (q_i-x_i)}$$

(25)

$$J_x(Q)=\Phi (Q)-c_m^{\mathrm{T}}x=H(Q)+c_m^{\mathrm{T}}Q-c_m^{\mathrm{T}}x$$

(26)

From above formulas, we can identify some special points in ${ℝ}_{+}^n$, and some definitions are provided as follows:

Definition 2.

$\partial \sigma =\{y\in {ℝ}^n|\Phi (y)=\Phi (S)+K_0\}$, $\partial {\sigma }^{-}=\{y\in {ℝ}_{+}^n|y\in \partial \sigma \mathrm{and} y<S\}$, when the composition of fixed costs is $K={[K_0,0,0,\cdots ,0]}^{\mathrm{T}}\in {ℝ}^{n+1},K_0\ne 0$, then $K=K_0\delta (e^{\prime }(Q-x))$.

Definition 3.

$\Sigma =\{y\in {ℝ}^n|y=\lambda s+\overline{\lambda }S;\forall s\in \partial \sigma ,\lambda \in [0,1] \mathrm{and} \lambda +\overline{\lambda }=1\},$${\sum }^{-} = \{y\in {ℝ}_{+}^n| y \in \sum \mathrm{and}$$y<S\}$.

Definition 4.

Denote $\sigma ={\sigma }^{+}\cup {\sigma }^{-}\cup {\sigma }_{\prec }$, where we have ${\sigma }^{+}=\{y\in {ℝ}^n|y>S \mathrm{and} y\notin \Sigma \},$${\sigma }^{-}=\{y\in {ℝ}^n|y<S \mathrm{and} y\notin \Sigma \}$ and ${\sigma }_{\prec }=\{y\in {ℝ}^n|y\in {ℝ}_{+}^n\backslash ({\sigma }^{+}\cup {\sigma }^{-}\cup \sum )\}$.

Definition 5.

Function $J_x(Q):{ℝ}_{+}^n\to ℝ$ has $K$-convexity in $R^n$ space if $J_x(Q)$ satisfies $J_x(\lambda Q_1 + \overline{\lambda }{Q}_2) \le \lambda J_x(Q_1) + \overline{\lambda }[J_x(Q_2) + K(Q_2 - Q_1)]$ where for $\forall Q_1\le Q_2$ and $\forall \lambda \in$ [0, 1] $K(Q-x)=K_0\delta (e^{\prime }(Q-x))+{\displaystyle {\sum }_{i\in N}{K}_i\delta ((q_i-x_i))}$, and $\overline{\lambda }=1-\lambda .$ Furthermore, we say $J_x(Q)$ has $K_0$-convexity if the formula above holds when $K_i=0,\forall i\in N$.

Remark 3.

According to the definitions above, we can clearly conclude that $\partial \sigma \subset \sum , \partial {\sigma }^{-}\subset {\sum }^{-}, S\in \sum , \sigma \cap \sum =\varnothing$ and $\sigma \cup \sum ={ℝ}_{+}^n$, $\sigma$ and $\sum$ make up one space partition of ${ℝ}_{+}^n$. Because the existence of fixed joint production set-up cost, as long as one kind of product starts its production, cost $K_0$ will occur. As to $\forall y\in {\sigma }^{+}$, every element in vector $y$ is greater than the corresponding item in vector $S$; to $\forall y\in {\sigma }_{\prec }$, some elements in vector $y$ are greater than the corresponding items in vector $S$; both situations do not allow additional production, which will enable the inventory level to reach the global optimal point $S$. Thus, in the space above, only in ${\sigma }^{-}\cup {\sum }^{-}$ exists the possibility of reaching $S$. Therefore, the following discussion mainly focuses on ${\sigma }^{-}\cup {\sum }^{-}$.

Theorem 4.

Since the function $J_x(Q):{ℝ}_{+}^n\to ℝ$ satisfies $J_x(q)\to \infty$ when $‖q‖\to \infty$ and $J_x(S)+K_0=J_x(s),\forall s\in \partial {\sigma }^{-}$, then we have: (1) $J_x(Q)$ has the property of $K_0$-convexity; (2) $J_x(S)+K_0=J_x(s)\le J_x(Q),\forall s\in \partial {\sigma }^{-},\forall Q\in {\sigma }^{-}$; (3) $J_x(Q)$ is a monotone non-increasing function in ${\sigma }^{-};$ (4) $J_x(w)\le J_x(z)+K_0,\forall z,w\in {ℝ}_{+}^n\backslash {\sigma }^{-}$.

The Proof of Theorem 4 is shown in the Appendix A. The proof of Theorem 4 is in Appendix A. Conclusion (1) shows that the properties of the cost function $J_x(Q)$ in ${ℝ}_{+}^n$ are similar to those in $ℝ$, which can be regarded as the extension of the latter in finite multidimensional space. Conclusion (2) shows that the infimum of the function $J_x(Q)$ in ${\sigma }^{-}$ is $J_x(s)$. Thus, when the initial inventory $x\in {\sigma }^{-}$, the optimal output for minimum cost is the vector $(S-x)$, which will lift the inventory level to $S$. Therefore, when enterprises make production plans, they should consider the convexity of the production cost function and use this property to optimize production lots and reduce costs.

Theorem 5.

Since the function $\Psi (x):{ℝ}^n\to ℝ,$ which satisfies $\Psi (x)={\min }_{q\ge x}\{J_x(Q)+K\}$, here $K=K_0\delta (e^{\prime }(Q-x)),$ then we have (1) Function $\Psi (x)={\min }_{Q\ge x}\{J_x(Q)+K\}$ has the property of $K_0$-convexity; (2) $\Psi (x)=\left\{\begin{array}{l}{K}_0+\Phi (S)-{\displaystyle {\sum }_{i\in N}{c}_m^i{x}_i} x\in {\sigma }^{-},\hfill \\ \Phi (x)-{\displaystyle {\sum }_{i\in N}{c}_m^i{x}_i} x\in {ℝ}_{+}^n\backslash {\sigma }^{-}\hfill \end{array}\right.$.

The Proof of Theorem 5 is shown in the Appendix A. Theorem 5 shows that firms facing a cost function with convexity should use appropriate mathematical tools and optimization techniques to find the optimal solution to maximize cost-effectiveness.

Remark 4.

To prove Theorem 5, we should take two points randomly in the area $x\in {ℝ}_{+}^n$ and examine the correctness of the conclusion above when the values of these two points vary at ${\sigma }^{-},{\sum }^{-},{\sigma }^{+}$, and $\Sigma$ respectively. On the basis of Theorem 4 and the related theorem proved by Gallego and Sethi [22], we can reach the two conclusions in this theorem.

4.4. Dynamic Multi-Cycle Joint Optimal Production Planning of Supply Chain Network

This section extends the inventory control model of the CSN with the input–output relationship discussed above from a single period to finite multiple periods, and it further employs $K$-convexity to demonstrate that the optimal control strategy in the multi-period is also $({\sigma }_{\tau }^{-},S_{\tau })$. Assume that the initial inventory of the supply network is $x$, production and inventory ordering lasts for $t(\ge 2)$ periods, and take ${\Psi }_t(x)$ as the expected value of the sum of optimal total cost during $t$ periods under the optimal strategy. Other parameters are the same as the marks provided in a single period. The optimal value function is provided by

$${\Psi }_t(x)={\inf }_{Q\ge x}\{J_x(Q)+{\displaystyle {\int }_0^{\infty }{\Psi }_{t-1}(Q-\xi )d\xi }\}={\inf }_{Q\ge x}\{{\Phi }_t(Q)-c_m^{\mathrm{T}}x\}$$

(27)

Some of the equations and conclusions in the single-period model can be extended to the multi-period. From (22), we know the first-order optimality condition is:

$${\gamma }^k{f}_k^{\tau }({\displaystyle {\sum }_{j\in N}{b}_{ij}{q}_j})+(c_h^k+c_p^k)F_k^{\tau }({\displaystyle {\sum }_{j\in N}{b}_{kj}{q}_j})=(d_{1k},d_{2k},\cdots ,d_{nk}){({\displaystyle {\sum }_{i\in N}{b}_{il}{c}_p^i}-n{c}_m^l)}_{n\times 1}$$

(28)

by which the critical value of the global optimal point in $\tau$-th cycle can be obtained as $S_{\tau }=Q_{\tau }^{\ast }={(q_1^{\tau \ast },q_2^{\tau \ast },\cdots ,q_n^{\tau \ast })}^{\mathrm{T}}$, where

$${q_{kc}^{\tau }}^{\ast }={\displaystyle {\sum }_{j\in N}{d}_{kj}{F_j^{\tau }}^{-1}[({\displaystyle {\sum }_{i\in N}{\displaystyle {\sum }_{l\in N}{b}_{il}{d}_{lj}{c}_p^i}-n{\displaystyle {\sum }_{l\in N}{d}_{lj}{c}_m^l}})/(c_h^j+c_p^j)]}, \forall k\in N.$$

(29)

Take $F_j^{\tau }(.)$ as the probability-distribution function of the outside demand for parts $j$ in the $\tau$-th period, then ${F_j^{\tau }}^{-1}(\cdot )$ in the formula above is the inverse function of $F_j^{\tau }(\cdot )$. So, we can know in advance the optimal inventory of every node firm and correspondingly obtain the optimal output in this period. To facilitate the discussion, let $\tilde{\theta }=({\displaystyle {\sum }_{i\in N}{\displaystyle {\sum }_{l\in N}{b}_{il}{d}_{lj}{c}_p^i}-n{\displaystyle {\sum }_{l\in N}{d}_{lj}{c}_m^l}})/(c_h^j+c_p^j)$. Assume that the stochastic demand-distribution function of product $j$ in $t$ periods is subject to $F_j^1(q)\ge F_j^2(q)\ge \cdots \ge F_j^{\tau }(q)\ge \cdots \ge F_j^t(q)$, and that $F_j^{\tau }(q)(\forall \tau \in T)$ is a monotone-increasing function. Thus, ${F_j^1}^{-1}(\tilde{\theta })\le {F_j^2}^{-1}(\tilde{\theta })\le \cdots \le {F_j^{\tau }}^{-1}(\tilde{\theta })\le \cdots \le {F_j^t}^{-1}(\tilde{\theta })$. Deducing from Formula (29), we can conclude that the critical values of inventory cost’s global optimal point at each stage in t periods are subject to the relationship of ${q_{kc}^1}^{\ast }\le {q_{kc}^2}^{\ast }\le \cdots \le {q_{kc}^t}^{\ast }$, and then the relationship of each critical value vector satisfies $S_1^c\le S_2^c\le \cdots \le S_{\tau }^c\le \cdots \le S_t^c$.

Then, we will discuss the finite multi-period dynamic production (inventory) control strategy of the CSN with the input–output relation when ${\gamma }^k={\alpha }^k-{\beta }^k=0$. Without special explanation, $S_{\tau }=S_{\tau }^c$. In this section, we still assume the composition of ordering the fixed cost of each product in the supply network to be $K=K_0\delta (e^{\prime }(Q-x))$. Some definitions as follows:

Definition 6.

$\partial {\sigma }_{\tau }=\{y\in {ℝ}^n|{\Phi }_{\tau }(y)={\Phi }_{\tau }(S)+K_0\}$, $\partial {\sigma }_{\tau }^{-}=\{y\in {ℝ}_{+}^n|y\in \partial {\sigma }_{\tau }\mathrm{and}y<S_{\tau }\}$, when the composition of fixed cost is $K={[K_0,0,0,\cdots ,0]}^{\mathrm{T}}\in {ℝ}^{n+1},K_0\ne 0$, then $K=K_0\delta (e^{\prime }(Q-x))$.

Definition 7.

${\Sigma }_{\tau } = \{y \in {ℝ}^n | y = \lambda s_{\tau } + \overline{\lambda }{S}_{\tau };\forall s_{\tau }\in \partial {\sigma }_{\tau },\lambda \in [0,1] \mathrm{and} \lambda +\overline{\lambda }=1\},$${\sum }_{\tau }^{-} = \{y \in {ℝ}_{+}^n | y \in {\sum }_{\tau },y < S_{\tau }\}$.

Definition 8.

${\sigma }_{\tau }={\sigma }_{\tau }^{+}\cup {\sigma }_{\tau }^{-}\cup {\sigma }_{\prec }^{\tau },$ where we define ${\sigma }_{\tau }^{+}=\{y\in {ℝ}_{+}^n|y>S_{\tau } \mathit{and} y\notin {\Sigma }_{\tau }\},$${\sigma }_{\tau }^{-}=\{y\in {ℝ}_{+}^n|y<S_{\tau }$$\mathit{and} y\notin {\Sigma }_{\tau }\}$ and ${\sigma }_{\prec }^{\tau }=\{y\in {ℝ}_{+}^n|y\in {ℝ}_{+}^n\backslash ({\sigma }_{\tau }^{+}\cup {\sigma }_{\tau }^{-}\cup {\sum }_{\tau })\}$.

Denote the periodic ordinal number set as $T=\{1,2,\cdots ,t\}$ to prove the $K^n$-convexity of the optimal value function ${\Psi }_{\tau }(x),\forall \tau \in T$ equals the prove function ${\Phi }_{\tau +1}(Q)=c_m^{T}x+J_x(Q)+$${\displaystyle {\int }_0^{\infty }{\Psi }_{\tau }(Q-\xi )d\xi }$, which satisfies $K^n$-convexity. Literature Scarf, Arrow [18] employed the mathematical induction in ${ℝ}_{+}$ to successfully prove the optimal strategy of single-product inventory $(s,S)$ ($s,S\in {ℝ}_{+}$). We extend a similar method to prove the optimal joint output strategy of multi-product production $({\sigma }_{\tau }^{-},S_{\tau })$ (${\sigma }_{\tau }^{-}\subset {ℝ}_{+}^n,S\in {ℝ}_{+}^n$). Since ${\Phi }_1(Q)=c_m^{\mathrm{T}}x+J_x(Q)$ obviously satisfies $K^n$-convexity, assume ${\Phi }_1(Q),{\Phi }_2(Q),\cdots ,{\Phi }_{{\tau }_0}(Q)$ to be $K^n$ convex. Then, to prove that ${\Phi }_{{\tau }_0+1}(Q)$ also satisfies $K^n$-convexity, we only need to demonstrate the $K^n$-convexity of function ${\Psi }_{{\tau }_0}(x)$.

From the above assumption of the $K^n$-convexity of ${\Phi }_{{\tau }_0}(Q)$, we can know ${\Phi }_{{\tau }_0}(y_{{\tau }_0})={\Phi }_{{\tau }_0}(S_{{\tau }_0})+K_0,\forall y_{{\tau }_0}\in \partial {\sigma }_{{\tau }_0}$, the optimal production inventory control strategy is: lift the inventory level to $S_{{\tau }_0}$ when $x\in \partial {\sigma }_{{\tau }_0}$, or keep the current inventory level when $x\notin \partial {\sigma }_{{\tau }_0}$; that is

$${\Psi }_{{\tau }_0}(x)=\left\{\begin{array}{l}{K}_0+{\Phi }_{{\tau }_0}(S_{{\tau }_0})-{\displaystyle {\sum }_{i\in N}{c}_m^i{x}_i} x\in {\sigma }_{{\tau }_0},\hfill \\ {\Phi }_{{\tau }_0}(x)-{\displaystyle {\sum }_{i\in N}{c}_m^i{x}_i} x\in {\Sigma }_{{\tau }_0}.\hfill \end{array}\right.$$

(30)

Based on the formula above, we will demonstrate the $K^n$-convexity of function ${\Psi }_{{\tau }_0}(x)$. Now, we consider those scenarios and associated calculations and inductions as follows:

$$\begin{array}{l}\mathrm{Scenario} (i):\mathrm{if} x,\tilde{x}\in {\sigma }_{{\tau }_0},\mathrm{obviously}, \lambda x+\overline{\lambda }\tilde{x}\in {\sigma }_{{\tau }_0},\\ \hspace{1em} {\Psi }_{{\tau }_0}(\lambda x+\overline{\lambda }\tilde{x})=K_0+{\Phi }_{{\tau }_0}(S_{{\tau }_0})-{\displaystyle {\sum }_{i\in N}{c}_m^i(\lambda x_i+\overline{\lambda }{\tilde{x}}_i)}=\lambda {\Psi }_{{\tau }_0}(x)+\overline{\lambda }{\Psi }_{{\tau }_0}(\tilde{x})\\ \hspace{1em}\hspace{1em} \hspace{1em} \le \lambda {\Psi }_{{\tau }_0}(x)+\overline{\lambda }[{\Psi }_{{\tau }_0}(\tilde{x})+K_0\delta (e^{\prime }(\tilde{x}-x))].\end{array}$$

$$\begin{array}{l}\mathrm{Scenario} (ii):\mathrm{if} x\in {\sigma }_{{\tau }_0},\tilde{x}\in {\sum }_{{\tau }_0},\mathrm{and} \lambda x+\overline{\lambda }\tilde{x}\in {\sigma }_{{\tau }_0},\\ {\Psi }_{{\tau }_0}(\lambda x+\overline{\lambda }\tilde{x})=K_0+{\Phi }_{{\tau }_0}(S_{{\tau }_0})-{\displaystyle {\sum }_{i\in N}{c}_m^i(\lambda x_i+\overline{\lambda }{\tilde{x}}_i)}\\ \hspace{1em}\hspace{1em} =\lambda [K_0+{\Phi }_{{\tau }_0}(S_{{\tau }_0})-{\displaystyle {\sum }_{i\in N}{c}_m^i{x}_i}]+\overline{\lambda }[K_0+{\Phi }_{{\tau }_0}(S_{{\tau }_0})-{\displaystyle {\sum }_{i\in N}{c}_m^i{\tilde{x}}_i}]\\ \hspace{1em}\hspace{1em} \le \lambda [K_0+{\Phi }_{{\tau }_0}(S_{{\tau }_0})-{\displaystyle {\sum }_{i\in N}{c}_m^i{x}_i}]+\overline{\lambda }[K_0+{\Phi }_{{\tau }_0}(\tilde{x})-{\displaystyle {\sum }_{i\in N}{c}_m^i{\tilde{x}}_i}]\\ \hspace{1em}\hspace{1em} \le \lambda {\Psi }_{{\tau }_0}(x)+\overline{\lambda }[{\Psi }_{{\tau }_0}(\tilde{x})+K_0\delta (e^{\prime }(\tilde{x}-x))].\\ \mathrm{or} \mathrm{if} x\in {\sigma }_{{\tau }_0},\tilde{x}\in {\sum }_{{\tau }_0},\mathrm{and} \lambda x+\overline{\lambda }\tilde{x}\in {\sum }_{{\tau }_0}, \\ {\Psi }_{{\tau }_0}(\lambda x+\overline{\lambda }\tilde{x})={\Phi }_{{\tau }_0}(\lambda x+\overline{\lambda }\tilde{x})-{\displaystyle {\sum }_{i\in N}{c}_m^i(\lambda x_i+\overline{\lambda }{\tilde{x}}_i)}\\ \hspace{1em}\hspace{1em} \le \lambda {\Phi }_{{\tau }_0}(x)+\overline{\lambda }[{\Phi }_{{\tau }_0}(\tilde{x})+K_0\delta (e^{\prime }(\tilde{x}-x))]-{\displaystyle {\sum }_{i\in N}{c}_m^i(\lambda x_i+\overline{\lambda }{\tilde{x}}_i)}\\ \hspace{1em}\hspace{1em} \le \lambda [K_0+{\Phi }_{{\tau }_0}(x)-{\displaystyle {\sum }_{i\in N}{c}_m^i{x}_i}]+\overline{\lambda }[{\Phi }_{{\tau }_0}(\tilde{x})-{\displaystyle {\sum }_{i\in N}{c}_m^i{\tilde{x}}_i+K_0\delta (e^{\prime }(\tilde{x}-x))}]\\ \hspace{1em}\hspace{1em} \le \lambda {\Psi }_{{\tau }_0}(x)+\overline{\lambda }[{\Psi }_{{\tau }_0}(\tilde{x})+K_0\delta (e^{\prime }(\tilde{x}-x))]\end{array}$$

$$\begin{array}{l}\mathrm{Scenario} (iii):\mathrm{if} x,\tilde{x}\in {\sum }_{{\tau }_0},\mathrm{obviously}, \lambda x+\overline{\lambda }\tilde{x}\in {\sum }_{{\tau }_0},\\ \hspace{1em} {\Psi }_{{\tau }_0}(\lambda x+\overline{\lambda }\tilde{x})={\Phi }_{{\tau }_0}(\lambda x+\overline{\lambda }\tilde{x})-{\displaystyle {\sum }_{i\in N}{c}_m^i(\lambda x_i+\overline{\lambda }{\tilde{x}}_i)}\\ \hspace{1em}\hspace{1em} \hspace{1em} \le \lambda {\Phi }_{{\tau }_0}(x)+\overline{\lambda }[{\Phi }_{{\tau }_0}(\tilde{x})+K_0\delta (e^{\prime }(\tilde{x}-x))]-{\displaystyle {\sum }_{i\in N}{c}_m^i(\lambda x_i+\overline{\lambda }{\tilde{x}}_i)}\\ \hspace{1em}\hspace{1em} \hspace{1em} \le \lambda [{\Phi }_{{\tau }_0}(x)-{\displaystyle {\sum }_{i\in N}{c}_m^i{x}_i}]+\overline{\lambda }[{\Phi }_{{\tau }_0}(\tilde{x})-{\displaystyle {\sum }_{i\in N}{c}_m^i{\tilde{x}}_i+K_0\delta (e^{\prime }(\tilde{x}-x))}]\\ \hspace{1em}\hspace{1em} \hspace{1em} \le \lambda {\Psi }_{{\tau }_0}(x)+\overline{\lambda }[{\Psi }_{{\tau }_0}(\tilde{x})+K_0\delta (e^{\prime }(\tilde{x}-x))]\end{array}$$

Thus, ${\Psi }_{{\tau }_0}(\lambda x+\overline{\lambda }\tilde{x})\le \lambda {\Psi }_{{\tau }_0}(x)+\overline{\lambda }[{\Psi }_{{\tau }_0}(\tilde{x})+K_0\delta (e^{\prime }(\tilde{x}-x))],$ that is, ${\Psi }_{{\tau }_0}(x)$ satisfies $K_0$-convexity. Because $J_x(Q)$ satisfies $K^n$-convexity and ${\Phi }_{{\tau }_0+1}(Q)=c_m^{T}x+J_x(Q)+$ ${\displaystyle {\int }_0^{\infty }{\Psi }_{{\tau }_0}(Q-\xi )d\xi }$, then we know ${\Phi }_{{\tau }_0+1}(Q)$ is a $K_0$ convex function, as well and from the mathematical induction above we can make sure that ${\Phi }_{\tau }(Q)$ is $K_0$ convex.

Furthermore, by ${\Psi }_{\tau }(x)={\inf }_{Q\ge x}\{J_x(Q)+{\displaystyle {\int }_0^{\infty }{\Psi }_{\tau -1}(Q-\xi )d\xi }\}={\inf }_{Q\ge x}\{{\Phi }_{\tau }(Q)-c_m^{\mathrm{T}}x\}$, we conclude that $\forall \tau \in T$, ${\Psi }_{\tau }(x)=\left\{\begin{array}{l}{K}_0+{\Phi }_{\tau }(S_{\tau })-{\displaystyle {\sum }_{i\in N}{c}_m^i{x}_i} x\in {\sigma }_{\tau }^{-},\hfill \\ {\Phi }_{\tau }(x)-{\displaystyle {\sum }_{i\in N}{c}_m^i{x}_i} x\in {ℝ}_{+}^n\backslash {\sigma }_{\tau }^{-}.\hfill \end{array}\right.$; that is, the optimal production control strategy in the $\tau$-th period is also a $({\sigma }_{\tau }^{-},S_{\tau })$ problem.

Remark 5.

The discussion above tried to work out the control policy of the CSN based on ${\gamma }^k=0,\forall k\in N$; that is, product $k$’s fixed penalty cost of excess is equal to that of the shortage. And the critical optimal vectors of two adjacent periods must satisfy the relation $S_{\tau -1}^c\le S_{\tau }^c$, $\forall \tau \in T$. Because it ensures that the current surplus value of the production (inventory) after meeting the outside demand will not exceed next period’s critical value and thus ensures the accurate implementation of the optimal production control strategy in the closely sequent periods. The situation is more complicated when ${\gamma }^k\ne 0,\forall k\in N$, and it is not easy to obtain the relation of optimal control values $S_{\tau }$ between adjacent periods from (28) and the distribution feature of product’s periodical demand. If we assume that the production inventory levels of two adjacent periods satisfy $S_{\tau -1}\le S_{\tau }(\forall \tau \in T)$ when ${\gamma }^k\ne 0(\forall k\in N)$, then the demonstration above is also tenable, and the corresponding control strategy is the optimal strategy. However, this assumption has the same deficiency with Literature Johnson [28] when dealing with similar problems. Both of them take this assumption as the prior condition for implementing this strategy, which should be overcome in further research.

As the demonstration above, at the beginning of a certain period, we assume that there exists a central controller in the CSN, and he is responsible for making the optimal output decision for the whole supply network according to the initial inventory of each node firm that he observed and collected. If $x_{\tau }\in {\sigma }_{\tau }^{-}$, then the optimal joint production strategy is that the output vector of each node in this period should satisfy ${\Delta }_{\tau }=S_{\tau }-x_{\tau }$; that is, the output of the $i$-th node is ${\Delta }_{\tau }^i=S_{\tau }^i-x_{\tau }^i,\forall i\in N$; if $x_{\tau }\in {ℝ}_{+}^n\backslash {\sigma }_{\tau }^{-}$. No node firm needs to arrange new production. In the case of this strategy, the whole supply network can coordinate well and minimize its operation cost.

5. Computational Study and Implications

In this section, we will do some numerical experiments to examine the conclusions and results mentioned above. We take a supply chain network with four nodes in the assembly system as an example for our numerical study (see Figure 1).

In this system, node firm A and its associated product A is the final good, which is only subject to external market random demand, while products B, C, and D are all part componets subject to external random market demand, as well as internal intermediate demands incurred by other related node firms. According to those analyses as aforementioned theoretic induction, we set the system’s direct consumption coefficient matrix A = [0 0 0 0; 2 0 0 0; 1 1 0 0; 1 3 1 0], while we assume the random market demand ${\xi }_i^{\tau }$ for node firm i within period $\tau$ follows the exponential distribution with parameter ${\lambda }_i^{\tau }$, mean ${\mu }_i^{\tau }$, and, accordingly, probability-density function $f_i^{\tau }\left({\xi }_i^{\tau }\right)={\lambda }_i^{\tau }{e}^{-{\lambda }_i^{\tau }{\xi }_i^{\tau }}$ and cumulative distribution function $F_i^{\tau }\left({\xi }_i^{\tau }\right)=1-e^{-{\lambda }_i^{\tau }{\xi }_i^{\tau }}.$

We execute several numerical experiments to testify the correctness and reliability of the aforementioned joint production optimal policy $\left({\sigma }_{\tau }^{-},S_{\tau }\right)$ for dynamic multiple periods in ${ℝ}^n$ space. Set the initial inventory level vector X = [10 20 20 30] and assume there are 20 periods for production, stock holding, and replenishment. Obviously, in this study, we only concentrate on the product growth stage of its whole product life cycle. To cater to this focus, we assume all components’ external market demand means increasing over multiple finite horizons. To ensure that the former preconditions hold, we set the mean or expected value of stochastic market demand for each firm i during period j as shown in

Table 1. Expected external market demand for all four node firms.

$\tau$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
${\mu }_1$	110	115	122	127	133	137	147	154	163	169	175	190	192	203	209	238	254	270	275	287
${\mu }_2$	120	124	130	133	141	148	154	163	174	180	186	203	212	223	231	236	263	267	287	294
${\mu }_3$	100	108	112	119	126	135	142	148	152	157	168	179	193	196	241	246	256	260	286	299
${\mu }_4$	140	146	153	158	164	170	177	186	192	201	220	231	241	254	262	275	280	284	290	302

, where $i\in \{1, 2, 3, 4\}$ and $j\in \{1, 2, \cdots , 19, 20\}.$ Upon the calculation using the aforementioned formulae to reflect the relationship between expected production planning and random demand cumulative distribution function, we can obtain the relation $F_j^1(q) \ge F_j^2(q) \ge \cdots \ge F_j^{\tau }(q) \ge \cdots \ge F_j^t(q)$ hold coinciding with former corresponding assumptions for node firm j over 20 periods.

Additionally, set the variable production-cost vector $c_m=[1.6,0.4,0.4,0{.08]}^T$, variable inventory-holding-cost vector $c_h={[0.6,0.4,0.3,0.1]}^T$, and variable shortage-cost vector $c_p={[0.8,0.5,0.4,0.2]}^T$. Using the optimal production/inventory control policy for minimizing expected total supply network-side cost yields critical values of production/inventory for optimization at the end of each period for all four node firms as shown in

Table 2. Critical judgement values for production optimization at end of each period.

Period	Firm A’s	Firm B’s	Firm C’s	Firm D’s
1	17.4	68.4	149.6	407.5
2	18.2	71.1	158.2	426.5
3	19.3	75.0	165.8	448.7
4	20.1	77.5	173.4	465.6
5	21.0	81.6	183.0	490.1
6	21.7	84.8	192.6	511.5
7	23.3	89.7	203.5	540.3
8	24.4	94.4	213.2	567.5
9	25.8	100.4	223.1	598.2
10	26.7	103.9	230.8	619.8
11	27.7	107.5	242.3	647.8
12	30.1	117.0	261.2	700.4
13	30.4	120.2	273.6	725.1
14	32.1	126.7	283.8	760.0
15	33.1	130.9	317.6	809.2
16	37.7	141.5	336.0	867.1
17	40.2	154.1	357.5	930.4
18	42.7	160.3	368.8	963.7
19	43.5	167.5	393.3	1012.1
20	45.4	173.2	409.3	1050.3

and Figure 2, respectively.

Drawing from Figure 2, one can easily observe that for each node firm/component, the critical value for globally optimizing production or inventory amount is increasing in time over all periods, which validates aforementioned conclusions. Subsequently, we can obtain the relative reference value $K_0+{\Phi }_{\tau }(S_{\tau })$, which is compared with the function ${\Phi }_{\tau }(x_{\tau })$ due to the initial stock level vector $x_{\tau }$ at period $\tau$.

6. Concluding Remarks

This paper focuses on the supply chain network with a kind of input–output relation among nodes. The whole network produces a complex structured and multi-process product, while each node (member firm) is responsible for only one part. Every node not only provides the internal network with components but also meets the stochastic demand from the outside network. Thus, we divide demand into intermediate demand and external final demand. We employ dynamic input–output analysis to model this system subject to random demand in single-period and multi-period horizons. We also respectively develop the decision model of static single-period joint optimal output of innovative products that allow additional production and the dynamic multi-period joint optimal output model of functional products. Moreover, the fixed costs incurred by stocking overdemand and stocking underdemand are considered in the system operating cost functions and proven to be convex.

In the static single-period model, we employ the input–output model to establish the relation model among members and nodes in MVN. Further, it provides the joint optimal output model to maximize the profit of the whole network. The model allows for additional quantities to meet demand if there is an inventory shortage. This model provides a foundational framework for understanding and optimizing production decisions in supply chain networks. It also proves the existence of the optimal solution and provides the solution method. Unlike the existing literature, in this paper, we consider the fixed costs incurred by inventory exceeding the actual demand and inventory shortage in the system’s operating cost function and prove its convexity, which helps to find the globally optimal solution. The methods proposed by this paper have operability in industrial practice, and this paper also has a specific theoretical meaning of solving the production coordination problem rooted in the virtual manufacturing network having multiple profit centers and certain practical meaning of directing and coordinating the manufacturing network and making the joint optimal output plan for each member firm.

In the dynamic multi-period model, we establish the cost function that involves two fixed penalty costs caused by excess and inventory shortage and proved the function’s convexity, providing a theoretical basis for finding out the optimal CSN joint dynamic production strategy in a multi-product, multi-cycle scenario.

Based on the analysis of the optimal production inventory strategy in single-period stochastic model, we extended it to finite multiple periods. When the fixed penalty cost of excess inventory of product $k$ is equal to that of shortage, according to the demand features of functional products, we can take advantage of $K -\mathrm{convexity}$ defined in ${ℝ}^n$ to find out the optimal policy for joint production $({\sigma }_{\tau }^{-},S_{\tau })$ $({\sigma }_t\subset {ℝ}^n,$ $S\in {ℝ}^n)$ in the CSN with a multi-product situation. This strategy makes the optimal output decision of the whole supply network on the basis of the initial inventory level vector of each node product. If $x_{\tau }\in {\sigma }_{\tau }^{-}$, then the optimal joint production strategy is that the output vector of each node in this period should satisfy ${\Delta }_{\tau }=S_{\tau }-x_{\tau }$; that is, the output of the $i$-th node is ${\Delta }_{\tau }^i=S_{\tau }^i-x_{\tau }^i,\forall i\in N$; if $x_{\tau }\in {ℝ}_{+}^n\backslash {\sigma }_{\tau }^{-}$, no node firm arranges new production. In the case of this strategy, the whole supply network can coordinate very well and minimize its operation cost. The study becomes more complicated when the fixed penalty costs of excess and shortage are not equal, and the application of that control strategy in this situation deserves further research.

The study discusses the joint production/inventory optimization problem in a complex supply chain network consisting of firm nodes with coupled demand. The demand relationship is described by a modified input–output matrix, which reveals the complex interdependence among nodes and proposes a method to achieve network-wide coordination and optimization. Meanwhile, this paper conducts production/inventory optimization analysis under a single perspective and explores the joint optimization problem from multiple perspectives. This multi-perspective analysis method helps to understand the complex decision-making process in the supply chain network more comprehensively. In the background of global manufacturing with the features of outsourcing and offshore manufacturing, the study in this paper has an instructive sense for the joint optimization practice in production (inventory) planning among firm nodes of the supply network in the decentralized decision scenario.

Although the present research has specific potential inspirations for further investigation of the complex supply chain networks specified here and refined from the setting characterized by global sourcing and manufacturing and offshoring in worldwide virtual networks, some work still can be further investigated. For example, one may consider other scenarios with dependent demands among nodes. The associated complexity will arise due to coupled stochastic disturbs distributed sparsely in different notes and let the production planning twined together and be a challenge to be determined. At the same time, it is close to industry practices and can facilitate joint optimal production decisions in a more complicated decentralized circumstance.