Production Planning Optimization in a Two-Echelon Multi-Product Supply Chain with Discrete Delivery and Storage at Manufacturer’s Warehouse

Abstract

In today’s competitive world, customers expect their demands to be met at the shortest possible time, while manufacturers aspire to deliver the orders within a convenient time and at a minimum cost. Thus, manufacturers are compelled to seek ways of lowering the costs of their services in order to satisfy customers and survive the competition in their respective industries. This research paper investigates a multi-product problem in a two-echelon supply chain consisting of a single manufacturer and several retailers. The main objective of this research is to develop and present a multi-product optimization model in which retailers receive their orders through discrete delivery and surplus manufactured goods are stored in the manufacturer’s warehouse. The objective function of the mathematical model in the economic dimension includes the minimization of the total supply chain costs and the maximization of profit. The retailers in this model place new orders when their inventory level drops to zero, and the manufacturer responds to the retailers’ orders at the same time as it begins processing each product. After delivering the last set of orders, the manufacturer stores surplus items in its warehouse in case the retailers place new orders. This optimization problem is modeled using mixed integer nonlinear programming, while numerical scenarios are coded using the MATLAB software which helps estimate the total cost within a short time. Finally, a sensitivity analysis is performed to determine the effects of a number of factors on the total cost, including problem parameters, demand and production rates, the production quantity, and the number of times the manufacturing machines are operated at each production cycle.

1. Introduction

A multi-echelon production system can only achieve success if, in addition to satisfying the customers, it manages to overcome the time- and cost-related challenges coming its way. One such challenge is meeting the customers’ needs at every inventory level in the shortest possible time while, at the same time, keeping production costs at a minimum. Inventory management in the supply chain represents a fundamental factor in achieving economic balance, requiring optimal negotiation, integration, and coordination at the chain’s levels [1]. An integrated approach becomes imperative in the current competitive business environment as it helps to save costs and to achieve on-time deliveries [2]. Learning the amount of demand and choosing the right times for production and rest may considerably help lower the costs. Integrated supply chain (SC) scheduling is a fundamental topic in SC management, seeing as the simultaneous consideration of due dates, production times, and distribution could lead to reduced costs and, as a result, increased profitability [3]. On the other hand, while innovation is increasingly growing in stature, the average lifespan of products is becoming shorter. In such conditions, developing new multi-objective mathematical models for SC configuration and new product design is an advantage that makes it possible for businesses to survive competitive markets [4]. Production planning and scheduling as well as machine reliability are some of the prominent issues in flexible production systems, which lead to reduced production costs and increased system efficiency [5]. In this study, a multi-product model is proposed in a two-echelon SC which is optimized using the MATLAB programming software and whose chief aim is to reduce production costs. A brief literature review is presented so as to trace the evolution of relevant ideas and methods over time where the research works are reviewed in order of their publishing date.

With the globalization of commerce in recent decades, companies have been producing and distributing raw materials, parts, and final products all over the world. Customers want to purchase products quickly and from trusted sources, even if that means ordering from another continent. Moreover, if the purchased product is delivered with any delay, customers are quick to look elsewhere for other suppliers. As a result, production planning, inventory management, and distribution scheduling are considered extremely important these days [6]. One of the most significant problems faced by companies which purchase raw materials, parts, and final products from suppliers is determining the quantity of ordered goods and licensing. To address this problem, two inventory models known as economic order quantity (EOQ) and economic production quantity (EPQ) were proposed by Harris (1913) and Taft (1918), respectively, which are still considered the simplest and most basic models available. Since their introduction, researchers have continuously been modifying and expanding the two models to adapt them to the latest real-world conditions and factors such as multi-product systems, imperfect item systems, discrete delivery, etc. [7].

Environmentally sustainable supply chain management (ESSCM) is an approach to carrying out supply chain management (SCM) in an eco-friendly manner and according to environmental requirements [8]. Optimizing supply chain management (SCM) can be described as a strategic management decision considering other decisions in the supply chain. In today’s economy, sustainability is one of the crucial issues. Many techniques have been developed to solve complex issues of SCM problems. Traditional optimization methods inherently suffer from high computation complexity and local optimal stagnation. These methods mainly rely on fitness function’s derivatives to determine the direction to search in to achieve the final solution. Moreover, they are very sensitive to the initial estimate, which may lead to converge into a locally optimal solution. As a result, these techniques are not effective enough to efficiently solve SCM problems. Over the past few years, many metaheuristics have been introduced to solve real-world SCM optimization problems. However, their performance is not guaranteed for various SCM problems as they are for random search methods. Therefore, there is a need to utilize practical algorithms utilizing the heuristic information available in the problem [9,10].

Transportation and inventory management are two of the main components of a supply chain that should ideally be optimized jointly. The inventory routing problem (IRP) is a complex logistics problem in which inventory management and transportation decisions are made simultaneously [11]. Production scheduling and the reliability of machinery are prominent issues in flexible manufacturing systems that lead to a decrease in production costs and increase in system efficiency [5]. Scheduling an integrated, detailed supply chain (SC) at the operational level is a crucial aspect of supply chain management (SCM). The simultaneous consideration of scheduling the production and distribution and assigning delivery times could lead to reduced costs and thus more profit. Fuel costs constitute a significant portion of total transportation costs; therefore, reducing fuel consumption could improve transportation efficiency at the operational level [12]. Viswanathan (1998) proposed an integrated vendor–buyer inventory model which employs the following two strategies to address the order quantity problem: (1) an identical delivery quantity at each replenishment and (2) all the vendor’s available inventory is delivered to the buyer [13]. Given the assumptions and mentioned strategies of the research, constraints always seem to lower the quality of studies.

A production design in a manufacturing system is to put the machinery in a serial position in a way that each one operates a single task in certain duration, so that the product exits from the system when it passes a series of freed machines after probable waiting times. This is called a flow-shop system [14]. In addition to manufacturing systems, the healthcare system is no stranger to resource challenges in the face of unlimited demand for healthcare delivery. And, the goals of this system are to satisfy patients, maintain service quality, and maximize profits. Healthcare decision makers are responsible for devising effective methods to fairly and equitably allocate scarce resources, which maximizes benefits at the social level [15]. Efforts to bring models closer to the actual challenges faced by industries have often been observed as a recurring theme across research works published in recent years. Pasandideh and Akhavan Niaki (2008) expanded the EPQ model based on three key assumptions: multiple products, constrained warehouse space, and discrete delivery order in different pallets. The researchers formulate the problem as a nonlinear integer programming (NLIP) model and solve it using a genetic algorithm (GA) [16]. A manufacturing system of machineries is exposed to breakdown due to some reasons and this decreases the productive capacity. The breakdown of machineries in a manufacturing system can have dangerous consequences, and thus, a regular maintenance program to enhance the reliability of machineries seems necessary [17]. Constrained warehouse space is usually not employed as a factor in studies, and warehouses are most often considered as having unlimited capacity. Realistic studies which recognize all constraints accurately could make significant contributions to the quality of research on the subjects of production and inventory control. Numerous types of algorithms may be used to solve such models. The above-mentioned study, for instance, employs a GA with evolutionary biology techniques. Widyadanaa and Wee (2009) developed an integrated multi-product vendor–buyer EPQ inventory model with budget constraints and a just-in-time (JIT) production approach. In this model, the buyer asks the vendor to deliver the products in small batches via a discrete delivery order. The problem is modeled as a mixed-integer nonlinear programming (MINLP) model and solved using a Lagrangian method and a branch and bound (B&B) algorithm [18]. In JIT production systems, there are numerous challenges when considering various constraints, such as those placed on the budget, which the above-mentioned study does not address.

In JIT production problems, the manufacturer’s storage costs are considered as zero, but assuming there is unlimited space to store the manufactured items is also unimaginable. Therefore, such models are hardly helpful if, at any point during the production process, a buyer asks that a product be delivered or that the manufacturer keeps the product in its warehouse. Pasandideh et al. (2009) investigated a multi-product EPA inventory model with warehouse space constraints with discrete delivery order in numerous pallets and in which shortages were turned into backorders. The researchers formulate the problem as an NLIP model and propose a GA to solve it [19]. Pasandideh and Akhavan Niaki (2010) presented another expansion of the EPQ inventory model using assumptions that constrained the model based on real-world conditions. One of the assumptions is that orders are constantly being delivered to buyers. The orders in this model are delivered discretely and as multiple pallets. The researchers propose an optimized model and algorithm to determine the optimal quantity of economic orders and the reorder point (ROP). Taleizadeh et al. (2011) developed a multi-product EPA inventory model considering that all products are processed by a single machine with constrained capacity and a discrete delivery order. The problem is formulated as an MINLP model and assesses production, operation, storage, and transportation costs. The researchers propose metaheuristics such as the harmony search algorithm (HSA) and particle swarm optimization (PSO) to solve the model [20].

Designing manufacturing systems has been a great concern in the past, and due to the increase in global competition and quick technological advancements, this issue has found greater significance and has become one of the managers’ greatest concerns and challenges [21]. There are many different studies in the scheduling field, mostly considering uncertain conditions in single-machine environment. A two-stage genetic algorithm is used to generate the predictive schedule. The first stage optimizes the primary objective, which minimizes the makespan, where all data are considered to be deterministic with no expected disruptions. The second stage optimizes two objectives, makespan and stability, that function in the presence of random machine breakdowns. In the second stage, two different versions of multi-objective genetic algorithms, non-dominated sorting genetic algorithm II and the non-dominated ranking genetic algorithm, a used [22]. Production systems which manufacture multiple products and impose constraints on the manufacturing process have also been examined by researchers. Cárdenas-Barrón et al. (2012) propose a heuristic to solve an EOQ-based inventory control system with multiple products, multiple constraints, and backorders, which is managed by the vendor and consists of linear and fixed backorder costs. The researchers argue that the proposed heuristic is superior to GAs in three respects: total cost, the number of evaluations of the total cost function, and computation time [23]. Based on their production quantity and break-even point, each production system has a fixed and a variable set of costs which have drawn the interest of researchers. In addition to production costs, the total costs incurred by vendor–buyer systems are also important when it comes to determining the optimal production–inventory policy. Jha and Shanker (2013), for instance, developed an integrated production–inventory policy in which a single vendor manufactured an item in a production system with serial layout configurations and presented it to a set of buyers. In this model, the buyers constantly examine their inventory and their demand is assumed to be independently and normally distributed. The model is formulated in such a way as to minimize the total vendor–buyer system costs and determine the optimal production–inventory policy. A Lagrangian algorithm assesses the buyers’ lead time in order to identify the optimal lead time, order quantity, buyer safety factor, and the number of delivery tours between the vendor and buyers in each production cycle [24]. Cárdenas-Barrón et al. (2014) proposed a heuristic algorithm to solve an integrated vendor–buyer model with JIT production and a constrained budget [25]. The performance improvement of a manufacturing system is a major component of maintaining the position of a company in the present competitive world. In the present dynamic environments and complex markets and based on the responding time significance, the optimization of performances has become a vital factor for companies and their competition. In the current competitive world, the on-time and quick supply of products to satisfy customers is a big necessity to maintain an organizational competitive position [26]. The growing concerns of companies regarding economic savings, the optimal utilization of resources, and increased environmental protection regulations prompt manufacturers to be more focused on the recycling of products that are at the end of their useful life. One should consider a job shop scheduling problem with reverse flows under uncertainty. Since the main parameter of the model (i.e., the processing time of operations) is tainted with a great degree of uncertainty in real-world applications, a robust programming approach is utilized. Due to the complexity and difficulty of solving the presented model, an exact solution method for small-sized instances and simulated annealing (SA) and discrete harmony search (DHS) algorithms for medium- and large-sized instances are proposed [27]. In order to accurately evaluate the production system and the costs of buyers or retailers, it is important to measure the number of goods shipped from the manufacturer to the retailer. Glock and Kim (2011) studied an integrated inventory model in which a single buyer sources one type of product from multiple vendors. To reduce the number of deliveries to the buyer, large shipments should be delivered earlier and small shipments late in the cycle [28].

Sales and operations planning (S&OP) remains one of the most critical challenges of supply chain management in terms of improving customers’ experience level and managing supply chain costs [29]. Parametric uncertainties such as demand and supply are aggravated by disruptions, with devastating effects on strategic, tactical, and operational decisions [30]. There are various models in production and inventory systems in terms of logistics, the most widely used of which are the single-buyer single-vendor and the single-buyer multi-vendor models. In this regard, Nobil et al. (2016) assess two inventory models: a single-buyer single-vendor one with lost sales and discrete delivery and a single-buyer multi-vendor model where the vendors supply the same products to the buyer. The two models are developed in the form of a three-echelon multi-product SC wherein all products are processed by a single machine with multiple suppliers, a single manufacturer and a single distributor. The basic assumption in the model is that some of the packages delivered to the distributor contain imperfect items. The researchers also developed two heuristic models based on a genetic and biogeography-based optimization (BBO) algorithm [31].

Decisions as to the reworking of imperfect products is another crucial topic in the field of production systems. Reworking helps restore such products back into the production cycle and prevent the wastage of raw materials, ultimately minimizing the total loss incurred by the manufacturing process. Shafiee-Gol et al. (2016) developed an EPQ inventory model with the assumption that some of the manufactured products may be imperfect and should be reworked to reduce the total cost. In this multi-product single-machine production system, pricing and other decisions are made based on discrete delivery orders and product rework. Moreover, a Lagrangian method is used to determine the optimal production times and quantity [32].

Because of stochastic factors such as the failure of machines regarding the competitive power of manufacturing organizations in the competitive global marketplace, the importance of production planning is doubled. To deal with uncertainty conditions in manufacturing systems, failure-prone manufacturing systems have arisen [33]. The problem of controlling the production rates of failure-prone manufacturing systems has stochastic features that make it more complex and challenging. Given uncertainty and complexity in the production planning of a network failure-prone manufacturing system (NFPMS), this study presents a simulation–optimization approach to estimate the control policy parameters. The system in question is a network of dissimilar manufacturing machines with perishable final goods. Under constant demand, the production system is subject to stochastic failures of machines, and corrective repairs are carried out at the time of failure [34]. Failure-prone manufacturing systems (FPMSs) are a subset of flexible systems in which the stochastic breakdown and maintenance of machines is assumed. In these systems, the purpose is to determine the production rate so that holding, shortage, and maintenance costs are minimized in a long-term planning horizon. Failure-prone manufacturing systems are one of the models of exploring manufacturing systems with due attention to system uncertainties [35].

There have been many studies on multi-echelon production systems which are often distinguished by their basic assumptions. Afzalabadi et al. (2016) presented a model for inventory control decisions in a two-echelon SC with a single vendor and multiple retailers. The vendor orders a product from the supplier, receives it instantaneously (the transportation time from supplier to vendor is considered zero), and delivers it to several retailers. The order quantities and cycles are different for each retailer, while the supplier and vendor do not experience any shortages. The researchers manage to introduce an efficient algorithm to minimize operational costs in an infinite time horizon [36].

Machines are the main sources of production scheduling in manufacturing industries. Two services are directly in contact with machines, i.e., production and preventive maintenance. Scheduling comprises allocating determinate sources to a set of jobs to optimize a definite objective function such as minimizing the tardiness/earliness of jobs, minimizing job completion times, and minimizing job processing times [37]. A system consisting of network machines with random breakdowns and repair times is considered. The machines in this system can be in one of four states: operational, in repair, starved, and blocked. The failure and repair times of the machines are exponentially distributed [38]. In the development of multi-echelon models, assumptions such as the production rate, number of shipments delivered by the manufacturer, and storage conditions by the manufacturer and retailer are significant in order to have a realistic and applicable model. Thus, researchers have always sought to bring theoretical models closer to being completely usable by real-world production systems. As a scenario, Seifbarghy et al. (2016) formulated a two-echelon, single-manufacturer multi-buyer SC model in which the manufacturer adopted a constant production rate and EPQ policy. Each buyer’s operational parameters include sales quantity, sales price, and production rate. The model is categorized as a nonlinear integer program and is solved using the discrete particle swarm optimization algorithm (DPSO) [39]. Nobil et al. (2017b) developed a multi-product EPQ inventory model for a multi-product, single-machine vendor–buyer system. The number of orders in this system is a discrete value and the buyer decides the size of the batches in which shipments are delivered. The purpose of the study is to determine the optimal cycle length and number of batches for each product to minimize the total inventory cost. The problem in this study is modeled using MINLP based on the number of orders, capacity, and budget constraints. Three different solution approaches were developed for this problem by the researchers, namely (1) an exact method, (2) a heuristic algorithm, and (3) a hybrid GA. The research was concluded with a sensitivity analysis [40].

None of the studies reviewed thus far deal with the inner workings of a production facility and the way manufacturing machines operate. Nobil et al. (2017c) present two inventory control models, (1) for a single product and (2) for multiple products, processed by a single machine. The researchers propose a single-machine economic batch quantity (EBQ) inventory model for an imperfect production system manufacturing perfect and reworked items. The required time to set up the machines for rework is considered as zero in this model [41]. Jonrinaldi et al. (2018) presented a single-vendor multi-buyer mathematical (EPQ)/(EOQ) inventory model in which discrete and continuous demand are simultaneously considered. The model takes into account the storage costs of multiple buyers as well as transportation costs in order to minimize the total costs of production and inventory. The model is solved using MINLP and algorithmic methods [42].

In today’s SCs, much emphasis is put upon determining the sales quantity of each product during a specific period and to a specific buyer. Salehi Amiri et al. (2020) proposed a single-vendor multi-buyer model to determine the optimal sales quantity of perishable goods in a two-echelon SC using a vendor-managed inventory (VMI) system [43]. Recently, a new feedback control policy, named the Environmental Hedging Point Policy, to control the inventory and production was introduced. In this policy, inventory, backlog, and emission costs were taken into account. To solve the problem, a new simulation-based optimization method combining opt Quest, experimental design, variance analysis, and response surface methodology was conducted [44]. The innovations in the study considerably distinguish it from prior works, since few researchers in the field of production planning and inventory control systems have put emphasis on multi-product models with storage at the manufacturer’s warehouse. Therefore, neglecting the following factors in prior works compose the research gap with respect to the present study: (a) the simultaneous consideration of discrete delivery in two-echelon inventory systems, (b) assuming multiple products in production system planning, and (c) storage in manufacturer warehouse; these are factors that render the production process more complex in the real world and thereby optimize the model proposed in this study.

Therefore, the present study is an attempt to fill the above-mentioned research gap in the literature, and the proposed mathematical model is meant to be as close to real-world conditions as possible. The objective is to develop, optimize, and present a multi-product model for production planning, with the assumption that the manufacturer begins the production process and delivers the products to retailers simultaneously, the surplus goods are stored in the manufacturer’s warehouse, and the orders are delivered discretely in specific intervals.

After modeling the problem, the production quantity of each product per cycle is calculated as well as the number of products delivered to retailers per cycle. Finally, the model is compared with ones developed by prior research. In brief, the main objectives of this study are as follows:

(1)

Developing a model with discrete delivery to retailers and the storage of surplus production in a manufacturer’s warehouse;

(2)

Optimizing the proposed model.

2. Problem Statement

Today, the increasing popularity of the strategic policies adopted by manufacturing factories and retailers to minimize costs has impelled researchers to develop new models to reduce and control production, storage, and order placement costs, thereby paving the way toward cost-efficient production. In this section, the central problem of research studies is described at length and then the proposed mathematical model is determined.

This study investigates a two-echelon SC in which several retailers buy their required product(s) from a single supplier. Thus, a coalition forms between the supplier and buyers who determine the size of shipments and number of times the production process is initiated per cycle as a collective decision, with the delivery method chosen to be discrete. The production process and delivery of the orders are started by the manufacturer at the same time. The manufacturer stores the surplus items in its own warehouse so that when the production process has finished and the retailers place new orders, the demands can be satisfied through the warehouse inventory. In this arrangement, the buyers have to pay for both the order placement and the storage of their purchased product in the manufacturer’s facilities. The costs of setting up, manufacturing, storing, and delivering the orders are paid by the supplier. The manufacturer can directly deliver the item(s) ordered by the buyer. Figure 1 illustrates the structure of the proposed single-manufacturer multi-retailer SC as follows.

This article deals with the fact that at the same time the manufacturer starts to produce products and sending orders to the retailer, there is no limit for the warehouse space. When the manufacturer starts sending the products at the same time as the production takes place, there is no need to store the products to occupy the warehouse space and store the products. Considering that the budget of the manufacturer is not part of the assumptions of our problem, and for this reason, we did not explain the manufacturer’s budget, in this issue, only costs and cost minimization have been considered.

In this model, the limitation of storage and depot space is not taken into account, and the manufacturer’s budget is not addressed, and only cost measurement and cost minimization are addressed. This article and model can be used in industries such as medicine, retail, food, sanitary detergents and packaged foods and help to solve the problems of these industries.

It is almost impossible to obtain a real case that has all the conditions according to the model. For this reason, a sensitivity analysis was performed in this model to show the generalizability of the model, and a real and accurate model was not presented as a scenario.

The basic assumptions in the model are as follows:

(1)

Each of the retailers’ customers have fixed demands.

(2)

The retailers always have customers and must fulfill their demands.

(3)

Each retailer buys a different type of product from the manufacturer.

(4)

Each of the manufacturer’s products has a different production rate.

(5)

The transportation time from manufacturer to retailer is insignificant and therefore ignored.

(6)

The retailers place new orders when their inventory level drops to zero.

(7)

The manufacturer starts the production process and delivery of orders simultaneously.

(8)

Surplus items are stored in the manufacturer’s warehouse and delivered to retailers at specific intervals.

(9)

The manufacturer has an unlimited budget and warehouse space.

(10)

The idle and setup times to switch the machinery from one product to another are insignificant and considered as zero.

(11)

The time horizon is unlimited.

(12)

The manufacturer’s setup time is fixed.

(13)

The cost of delivering each retailers’ orders is a different but fixed value.

(14)

The warehousing cost is different for each product type in the manufacturer’s central warehouse and the retailers’ warehouses but remains fixed throughout the planning horizon.

The model’s parameters and variables of the proposed model are detailed in

Table 1. Model’s parameters and variables.

Symbol	Description
Indices
i	Number of products $\mathrm{i}=1:\mathrm{I}$
Decision variables
Qi	Quantity of product i delivered to a retailer
ni	Factor number of Qi products per each production cycle
Ni	Number of setup times to manufacture product i
Parameters
$R_i$	Production rate of product i
$D_i$	Demand for product i
$A_i$	Cost of each time product i starts being manufactured
$C_i$	Cost of each time product i is delivered to a retailer
$T{R}_i$	Time it takes to manufacture $n_i{Q}_i$
$T{B}_i$	Time it takes to manufacture a batch of Qi products
$T{S}_i$	Time of rest period after manufacturing $n_i{Q}_i$
$C{T}_i$	Time it takes to consume a batch of Qi products
$T_i$	Total time of a production cycle and a rest period
$H_{mi}$	Cost of storing product i at manufacturer’s warehouse
$H_{ri}$	Cost of storing product i at retailers’ warehouse
B	Number of breaks in the production

.

3. Mathematical Model

Equation (1) is employed to obtain the optimal delivery quantity. The number of times the production process is started, which has a considerable impact on the quantity delivered to the retailer within the range between minimum demand and $D_i$. In each cycle, when the manufacturer makes a delivery to a retailer, the shipment size is represented by ni which, according to Equation (3), ranges from 1 to net demand quantity ($\frac{D_i}{N_i}$).

$$Q_i=\frac{D_i}{N_i\times n_i}$$

(1)

$$N_i=1:D_i$$

(2)

$$n_i=1:\frac{D_i}{N_i}$$

(3)

3.1. Supplier–Retailer Relationship Modes

A major part of each cycle consists of product manufacture, with a smaller period allocated to rest breaks. A rest break refers to the time when, after manufacturing $Q_i$ products, a period of time equal to the difference between demand and production quantity is created.

Full cycle = length of production time + length of rest time

$$T_i=T{R}_i+T{S}_i=n_i\times C{T}_i$$

(4)

If $C{T}_i<\left(T{R}_i-T{B}_i\right)$, then

$$T{B}_i=\frac{Q_i}{R_i}$$

(5)

$$C{T}_i=\frac{Q_i}{D_i}$$

(6)

$$T{R}_i=\frac{n_i\times Q_i}{R_i}$$

(7)

$$T_i=\frac{n_i\times Q_i}{D_i}$$

(8)

$$T{S}_i=\frac{n_i\times Q_i}{D_i}-\frac{n_i\times Q_i}{R_i}$$

(9)

If $C{T}_i<T{R}_i$, then there will be no breaking point. This means

$$\frac{Q_i}{D_i}<\frac{\left(n_i\times Q_i\right)}{R_i}$$

(10)

Consequently,

$$\frac{R_i}{D_i}<n_i$$

(11)

3.1.1. Equal Production and Consumption

Figure 2 illustrates the quantity of shipments delivered by the manufacturer and the retailers’ consumption rate when $n_i$ = 1, which indicates a situation where the production quantity and consumption quantity are equal to $Q_i$.

The blue lines represent factory production and the orange lines represent retailer consumption. Equations (12) and (13) are used to calculate the area under the curve (AUC) and expresses the manufacturer’s production rate and the retailers’ demand. The AUC is also applicable in calculating the manufacturer’s storage costs. Equation (14) shows the total cost, which is the sum of the manufacturer and retailers’ storage and fixed costs.

$$\mathrm{s}\left(1\right)=\frac{1}{2}\times T{B}_i\times Q_i$$

(12)

$$\mathrm{S}(N_i,n_i)=\mathrm{s}(1)$$

(13)

$$\mathrm{T}\mathrm{C}=\left[N_i\times H_{mi}\times \left(\mathrm{S}\left(N_i,n_i\right)\right)\right]+\left({\mathrm{N}}_i\times A_i\right)+\left(\frac{1}{2}\times H_{ri}\times Q_i\right)+\left(\frac{D_i}{Q_i}\times C_i\right)$$

(14)

3.1.2. Two Productions Per One Consumption

In this mode, the rate of consumption is lower than the production quantity; therefore, for every instance of consumption by the retailer, two batches of $Q_i$ products are processed by the manufacturer. Moreover, in this mode, only one shipment is delivered to the retailer during the production cycle. Equations (15)–(17) ensure that in order to prevent shortages, the net production quantity is always larger than the retailers’ demanded quantity.

$$\frac{Q_i}{D_i}>\frac{Q_i}{R_i}\left(n_i-1\right)$$

(15)

$$n_i-1<\frac{R_i}{D_i}$$

(16)

Or

$$n_i<\frac{R_i}{D_i}+1$$

(17)

In Figure 3, the blue lines represent the manufacturer and the orange lines represent the retailers. Equations (18) and (19) are used to calculate the AUC, while Equation (20) is used to calculate the total cost of Equation (14).

$$\mathrm{s}\left(1\right)=\frac{T{B}_i\times Q_i}{2}$$

(18)

$$\mathrm{s}\left(2\right)=C{T}_i+\frac{\left(C{T}_i-T{B}_i\right)\times Q_i}{2}$$

(19)

$$\mathrm{S}\left(N_i,n_i\right)=\mathrm{s}\left(1\right)+\mathrm{s}\left(2\right)$$

(20)

3.1.3. B Deliveries Per n Production Cycles of Q-Product Batches

In the third mode, after the first delivery, B more shipments are delivered before the production cycle ends. The number of deliveries in production cycle T equals n and the first shipment is delivered at the same time as when the production cycle starts.

$$\mathrm{T}{R}_i=\frac{n_i{Q}_i}{R_i}$$

(21)

After the first delivery,

$$\frac{\left(n_i-1\right)\times Q_i}{R_i}$$

(22)

$$\mathrm{B}=\frac{\frac{\left(n_i-1\right)\times Q_i}{R_i}}{C{T}_i}=\frac{\left(n_i-1\right)\times D_i}{R_i}$$

(23)

$$\mathrm{B}=\frac{\frac{\left(n_i-1\right)\times Q_i}{R_i}}{\frac{Q_i}{D_i}}\to \mathrm{B}=\frac{\left(n_i-1\right)\times D_i}{R_i}$$

(24)

In Figure 4, the blue lines represent the manufacturer’s first product and the green lines represent the next product. In S2, up to Q2 products are manufactured and as many as $Q_{i2}-Q_{i3}$ products are delivered to the retailer. The same condition is true for $\sum s_j$ as well. First, the manufacturer’s costs were obtained by calculating the AUC:

$$S_T=S_1+\left(S_2+\cdots +S_B\right)+S_{B+1}+\left(S{S}_{n-B-1}+\cdots +S{S}_1\right)$$

(25)

The AUC is made up of S1. In this area, the manufacturer can only have simultaneous productions and deliveries. From S2 to SB, the manufacturer simultaneously manufactures products, delivers shipments to retailers, and stores the surplus items in its warehouse. S’B+1 is a situation where the manufacturer first completes the production cycle and then stores the products. From S’’B+1 onward, the manufacturer fulfils the retailers’ demands only using its warehouse supply.

$$s_1=\frac{1}{2}\times T{B}_i\times Q_i=\frac{1}{2}\times \frac{Q_i}{R_i}\times Q_i=\frac{1}{2}\times \frac{{Q_i}^2}{R_i}$$

(26)

$$s_2=\frac{1}{2}\times C{T}_i\times Q_{i2}=\frac{1}{2}\times \frac{Q_i}{D_i}\times Q_{i2}$$

(27)

$$Q_{i2}=R_i\times C{T}_i=R_i\times \frac{Q_i}{D_i}$$

(28)

$$s_2=\frac{1}{2}{R}_i{\left(\frac{Q_i}{D_i}\right)}^2$$

(29)

$$s_3=s_3^{\prime }+s_2$$

(30)

$$s_3^{\prime }=Q_{i4}\times C{T}_i$$

(31)

$$Q_{i3}=Q_{i2}-Q_i$$

(32)

$$Q_{i4}=Q_{i3}-Q_i=\left(R_i-D_i\right)\times C{T}_i=\left(R_i-D_i\right)\times \frac{Q_i}{D_i}$$

(33)

$$s_3^{\prime }=(R_i-D_i)\times {\left(\frac{Q_i}{D_i}\right)}^2$$

(34)

$$s_3={\left(\frac{Q_i}{D_i}\right)}^2\left(\frac{3}{2}{R}_i-D_i\right)$$

(35)

$$s_4=s_4^{\prime }+s_2$$

(36)

$$s_4^{\prime }=Q_{i6}\times C{T}_i$$

(37)

$$Q_{i6}=2\left(R_i-D_i\right)\times C{T}_i=2(R_i-D_i)\times \frac{Q_i}{D_i}$$

(38)

$$s_4^{\prime }=2\left(R_i-D_i\right)\times {\left(\frac{Q_i}{D_i}\right)}^2$$

(39)

$$s_4={\left(\frac{Q_i}{D_i}\right)}^2\times \left(\frac{5}{2}{R}_i-2D_i\right)$$

(40)

$$s_j={\left(\frac{Q_i}{D_i}\right)}^2\times \left(\frac{2j-3}{2}{R}_i-\left(j-2\right)D_i\right)$$

(41)

$$\sum _{j=2}^B{s}_j={\left(\frac{Q_i}{D_i}\right)}^2\times \left(\frac{2j-3}{2}{R}_i-\left(j-2\right)D_i\right)$$

(42)

$$\begin{gathered} s_j={\left(\frac{Q_i}{D_i}\right)}^2\times \left(R_i-D_i\right)\times \left[\left(\left(\frac{\left(B\right)\times \left(B+1\right)}{2}\right)-1\right)-\left(\left(B-1\right)\times \left(\frac{3}{2}{R}_i-2D_i\right)\right)\right] \\ \sum _{j=2}^B{s}_j={\left(\frac{Q_i}{D_i}\right)}^2\times \left(R_i-D_i\right)\times \left(\frac{B\left(B+1\right)}{2}-1\right)-\left[\left(B-1\right)\times \left(1.5R_i-2D_i\right)\right] \end{gathered}$$

(43)

After calculating all the production-related AUCs, two sections are established on the diagram that represent the limited production quantity needed to satisfy the warehouse demand. The two sections, called T₁ and T₂, take the form of a trapezoid and a rectangle, respectively, and their areas are calculated as follows:

$$T_1=\frac{\left(Q_{i2B+1}-Q_{i2B}\right)}{R}$$

(44)

$$T_2=\left(C{T}_i\right)-\left(T_1\right)$$

(45)

As previously mentioned, in ${s\prime }_{B+1}$, the manufacturer begins production with the purpose of preventing shortage, then stores the products in the warehouse, and, within a CT-long period, manufactures niQi products.

$${\mathrm{s}}_{B_{+1}}={\mathrm{s}}_{\mathrm{B}+1}^{\prime }+{\mathrm{s}}_{B+1}^{″}$$

(46)

$${\mathrm{s}}_{\mathrm{B}+1}^{\prime }=\frac{1}{2}\times \left({\mathrm{Q}}_{\mathrm{i}2\mathrm{B}}+{\mathrm{Q}}_{\mathrm{i}2\mathrm{B}+1}\right)\times {\mathrm{T}}_1$$

(47)

$${\mathrm{Q}}_{\mathrm{i}2\mathrm{B}}=\left(\mathrm{B}-1\right)\times \left(R_i-D_i\right)\times C{T}_i=\left(\mathrm{B}-1\right)\times \left({\mathrm{R}}_i-{\mathrm{D}}_i\right)\times \frac{Q_i}{D_i}$$

(48)

$${\mathrm{Q}}_{\mathrm{i}2\mathrm{B}+1}={\mathrm{n}}_i{\mathrm{Q}}_i-\mathrm{B}{\mathrm{Q}}_i=\left({\mathrm{n}}_i-\mathrm{B}\right){\mathrm{Q}}_i$$

(49)

$${\mathrm{T}}_1=\frac{Q_{i2B+1}-Q_{i2B}}{R_i}$$

(50)

$$\mathrm{s}{\prime }_{\mathrm{B}+1}=\frac{1}{2R_i}\left({\mathrm{Q}}_{i2B+1}+Q_{i2B}\right)\left(Q_{i2B+1}-Q_{i2B}\right)$$

(51)

$${\mathrm{s}\prime }_{\mathrm{B}+1}=\frac{{Q_i}^2}{2{D_i}^2{R}_i}\times ({D_i}^2\left({n_i}^2-1\right)+\left(B^2-2B+1\right)\times \left(2D_i{R}_i-{R_i}^2\right))$$

(52)

$$\mathrm{s}{″}_{\mathrm{B}+1}=Q_{i2B+1}{\times T}_2=Q_{i2B+1}\times \left(C{T}_i-{\mathrm{T}}_1\right)$$

(53)

$$s{″}_{B+1}=\left(n_i-B\right)Q_i\times \left(\frac{Q_i}{D_i}-\frac{\left(n_i-B\right)Q_i{R}_i-\left(B-1\right)\left(R_i-D_i\right)Q_i}{D_i{R}_i}\right)$$

(54)

After assessing the production-related ACUs, the costs of storing the finished products, which are delivered to retailers according to the quantity of production, demand, and Qi factor, are examined. The ACUs, which take the shape of a square, are obtained as follows:

$$S{S}_1=Q_i\times C{T}_i=\frac{{Q_i}^2}{D_i}$$

(55)

$$S{S}_2=2Q_i\times C{T}_i=\frac{2{Q_i}^2}{D_i}$$

(56)

$${SS}_j=j{Q}_i\times C{T}_i\frac{j{Q_i}^2}{D_i}$$

(57)

$$\sum _{j=1}^{n-B-1}S{S}_j=\frac{{Q_i}^2}{D_i}\times \frac{\left(n_i-B-1\right)\left(n_i-B\right)}{2}$$

(58)

The total AUC is obtained as follows:

$$S_T={\mathrm{S}}_1+\sum _{j=2}^B{s}_j+{\mathrm{s}}_{B+1}^{\prime }+{\mathrm{s}″}_{\mathrm{B}+1}+\sum _{j=1}^{n-B-1}{\mathrm{S}\mathrm{S}}_j$$

(59)

The manufacturer’s costs in this section consist of fixed and storage costs:

(1) Manufacturer’s fixed cost

$$N_i{A}_i=\left(\frac{D_i}{Q_i{n}_i}\right)\times A_i$$

(60)

(2) Manufacturer’s storage cost

$$N_i\times H_{mi}\times S_T$$

(61)

The retailers’ costs in this section consist of fixed and storage costs:

(1) Retailers’ fixed cost

$$\frac{D_i}{Q_i}\times C_i$$

(62)

(2) Retailers’ storage cost

$$(\frac{1}{2})\times H_{ri}\times Q_i$$

(63)

Total cost = Manufacturer’s costs + retailers’ costs
(fixed + storage) (fixed + storage)

$$TC={N_{i}H}_{mi}{S}_T+N_i{A}_i+\left(\frac{1}{2}\right)H_{ri}{Q}_i+\left(\frac{D_i}{Q_i}\right)C_i$$

(64)

4. Computational Instances and Data Analysis

Now, the equations discussed in the previous section are analyzed as well as the results of solving the numerical scenario.

4.1. Theory Analysis

The analyses presented in this study were all performed using the MATLAB programming software. The analysis was carried out based on the effects of making changes to the problem’s parameters (including the manufacturer’s storage cost, the production setup cost, the cost of each delivery, the production quantity, and the demanded quantity). In the end, the results of our evaluations clarify the lowest possible total cost, the optimal number of production start-ups, and the optimal number of shipments delivered to retailers.

4.2. Numerical Scenario

Based on relevant assessments, the following numbers were selected as numerical scenarios for the problem. These numerical scenarios are given randomly and with the help of information from a fertilizer and agricultural material production plant in

Table 2. Single-product scenario.

Scenario No.	Parameters
1	Hmi = 1	Ai = 20	Hri = 5	Ci = 10	Ri = 4000	Di = 1000
	Min TC = 470		Ni = 6		ni = 2
2	Hmi = 1	Ai = 20	Hri = 5	Ci = 10	Ri = 12,000	Di = 3000
	Min TC = 813		Ni = 10		ni = 2
3	Hmi = 1	Ai = 20	Hri = 5	Ci = 10	Ri = 15,000	Di = 6000
	Min TC = 1140		Ni = $\begin{array}{c}9\\ 10\end{array}$		ni = 4
4	Hmi = 2.5	Ai = 20	Hri = 5	Ci = 10	Ri = 4000	Di = 1000
	Min TC = 502		Ni = 6		ni = 2
5	Hmi = 2.5	Ai = 20	Hri = 5	Ci = 10	Ri = 12,000	Di = 3000
	Min TC = 867		Ni = 11		ni = 2
6	Hmi = 2.5	Ai = 20	Hri = 5	Ci = 10	Ri = 15,000	Di = 6000
	Min TC = 1226		Ni = 15		ni = 2
7	Hmi = 1	Ai = 15	Hri = 5	Ci = 10	Ri = 4000	Di = 1000
	Min TC = 440		Ni = 6		ni = 2
8	Hmi = 1	Ai = 15	Hri = 5	Ci = 10	Ri = 12,000	Di = 3000
	Min TC = 761		Ni = 11		ni = 2
9	Hmi = 1	Ai = 15	Hri = 5	Ci = 10	Ri = 15,000	Di = 6000
	Min TC = 1076		Ni = $\begin{array}{c}15\\ 16\end{array}$		ni = 2
10	Hmi = 1	Ai = 20	Hri = 5	Ci = 8	Ri = 4000	Di = 1000
	Min TC = 442		Ni = 4		ni = 4
11	Hmi = 1	Ai = 20	Hri = 5	Ci = 8	Ri = 12,000	Di = 3000
	Min TC = 766		Ni = 7		ni = 4
12	Hmi = 1	Ai = 20	Hri = 5	Ci = 8	Ri = 15,000	Di = 6000
	Min TC = 1060		Ni = $\begin{array}{c}9\\ 10\end{array}$		ni = $\begin{array}{c}5\\ 4\end{array}$

and

Table 3. Multi-product scenario.

Scenario No.	Parameters
1	Hm1	Hm2	A1	A2	Hr1	Hr2	C1	C2	R1	R2	D1	D2
	1	2.5	20	15	5	5	10	8	4000	12,000	1000	3000
	MinTC1 = 513				N1 = 9				n1 = 2
	MinTC2 = 801				N2 = 9				n2 = 2
2	Hm1	Hm2	A1	A2	Hr1	Hr2	C1	C2	R1	R2	D1	D2
	1	2.5	20	15	5	5	10	8	12,000	15,000	3000	6000
	MinTC1 = 726				N1 = 15				n1 = 2
	MinTC2 = 1091				N2 = 15				n2 = 2
3	Hm1	Hm2	A1	A2	Hr1	Hr2	C1	C2	R1	R2	D1	D2
	1	2.5	20	15	5	5	3	2.5	4000	12,000	1000	3000
	MinTC1 = 412,427				N1 = 11, 12				n1 = 2
	MinTC2 = 647,632				N2 = 11, 12				n2 = 2

Initially, the model is evaluated as a single-product one. To this end, first, the quantities of production and demand are changed. The results are presented in figures listed in scenario 1 in Table 1 including the total cost, the number of times the production operation starts, and the number of shipments delivered to retailers.

As can be seen, increasing the production rate causes a significant increase in the total cost and, to a lesser extent, in the number of production start-ups. If the demand quantity rises, there is no way but to match it by increasing the production quantity. As previously mentioned, this increases the total cost, the number of production start-ups, and the number of deliveries. Scenario 4 indicates the outcome of changing the other variables, such as the manufacturer’s storage costs.

It can be observed that increasing the manufacturer’s storage costs leads to an increase in the total cost and has a slight effect on the number of production start-ups. The reason for this is the manufacturer’s decision to process smaller batches with the purpose of lowering the costs. The effects of reducing the cost of each instance of production are detailed in scenarios 7, 8, and 9.

As expected, reducing the cost of production start-up causes the total cost to decrease. In the last step, we can see the effects of changing the cost of each delivery in the single-product model. The results are listed in scenarios 10 and 11.

In the multi-product model, first, the demand quantity is estimated by increasing the production quantity variable. Then, when the demand quantity is increased, the total cost increases proportionately. Next, keeping the initial demand and production quantities unchanged, the cost of each delivery is reduced. Since this reduces fixed production costs, the total cost decreases as well. In the multi-product tables, we attempted to obtain the two decision variables Ni and ni, which were then used in the sensitivity analysis performed on the total cost. In the sensitivity analysis, The manufacturer’s and retailers’ costs were examined. This was conducted using the numbers obtained by the aforementioned scenarios and executing the scenarios again by MATLAB. The results are detailed in

Table 4. Manufacturer and retailers’ costs.

Scenario No.	Manufacturer Fixed Costs	Manufacturer Storage Costs	Retailer Fixed Costs	Retailer Storage Costs	Total Cost
1	120	21.3333	120	208.3333	470
2	200	38	200	375	813
3	200	165	400	375	1140
4	120	53.3333	120	208.3333	502
5	220	86.4773	220	340.9091	867
6	300	126.25	300	500	1226
7	90	21.3333	120	208.3333	440
8	165	34.5909	220	340.9091	761
9	225	50.5	500	300	1076
10	80	78.125	128	156.25	442
11	140	133.9286	224	267.8571	766
12	180	186.6667	360	333.3333	1060

. and also the calculations associated with Table 4 are shown as a diagram in Figure 5. It should be noted that the currency used in the cost calculations is the USD.

Scenarios 2, 3, and 6 have the highest retailer storage costs. As a result, in case of a rise in the production rate and delivered quantity, more costs are incurred on the retailer for the storage of the received products. Moreover, the highest fixed retailer cost belongs to scenario 9, in which the cost of each production start-up decreases, but the cost of each delivery does not changed. Additionally, because the production quantity in this scenario increases, the retailers’ fixed costs increase as well. The highest total cost is seen in scenarios which have the highest production rate.

5. Conclusions and Discussion

In this study, a two-echelon production planning model is optimized featuring a single manufacturer that manufactures multiple products and delivers them to multiple retailers. In this model, it is assumed that the products are stored in the manufacturer’s warehouse and delivered to buyers via discrete orders. The objectives of the mathematical model are to minimize the total SC cost and maximize profit.

A single-objective, two-echelon, multi-product mathematical model was described and presented. In this model, Qi is a decision variable representing the quantity of the shipment delivered by the manufacturer to retailers. Another decision variable is ni, which represents the quantity at which product i is manufactured. The other variable is Ni, which expresses the number of times the production process is started. Therefore, in this model, the quantity of delivered shipments, number of batches delivered to retailers, and number of times the machines are operated to manufacture each product are obtain. The production plan must be devised in such a way as to supply the products demanded by retailers in the shortest time and lowest total cost possible.

In the model, until a certain point, the manufacturer processes products, delivers orders to retailers, and stores the surplus items in its warehouse all at the same time. Afterward, the manufacturer stops production and fulfills the retailers’ demands using its warehouse supplies. After completing the modeling process, the MATLAB R2016a software was executed to analyze and code the model as well as to solve numerous numerical scenarios. First, a number of single-product scenarios were solved; then, the scenarios were expanded into multi-product ones. By making changes in the production and demand quantities, in addition to the fixed and storage costs of the manufacturer and retailers, the following conclusions were drawn.

Increasing the production rate and demand has a direct impact on increasing the total cost and the number of times manufacturing machines are started (Ni) while reducing delivery costs dramatically decreases the total cost and increases Ni. On the other hand, increasing the cost of each instance of starting the manufacturing machines increases the total cost and decreases Ni. Finally, increasing storage costs causes both the total cost and Ni to increase.