1.INTRODUCTION AND RESEARCH LITERATURE
The intensification of global competition and the demand for better customer services have considerably increased the need for supply chain integration (SCI) between companies. Integration is defined as “the extent to which all activities within an organization, and the activities of its suppliers, customers, and other supply chain members, are integrated together” (Narasimhan and Jayaram, 1998). SCI aiming at coordinating processes alongside the supply chain (SC) is considered to be an important determinant to maintain a competitive advantage over competitors (Christopher et al., 2006; Lambert, 2008; Pamela and Pietro, 2011). The assessment shows where an organization is and indicates the areas in which the organization needs to improve to fulfill the vision of integration (Campbell and Sankaran, 2005). In this regard, many organizations decide to adopt a lower level of integration or as tapered integration.
The basis of integration can be characterized by cooperation, collaboration, information sharing, trust, partnerships, shared technology and a fundamental shift away from managing individual functional processes to manage integrated chains of processes (Kahn, 1998; Pagell, 2004; Vickery et al., 2003). Of among an extensive series of researches conducted on the SCI, the following recent works can be mentioned based on ISC optimization, modeling, design, develop and examination fields.
In the field of integrated supply chain (ISC) optimization, Zhou et al. (2003) proposed a hierarchical architecture for modeling ISC optimization system which is consisted of such three levels as integrated longrange planning and decision making, sustainable SC planning and scheduling, and unit process simulation and optimization. At the level of longrange planning, a goalprogramming model was utilized to debottleneck the SC . Ayoub et al. (2009) offered up a paper in which an optimization model for bioenergy production ISC s was presented with a solution approach for designing and evaluating the integrated system of bioenergy production SC s at the local level. Khalifehzadeh et al. (2015) studied the design and optimization of an integrated fourechelon SC structure including multiple suppliers, producers, distributors and customers. The paper mathematically formulated the problem as Mixed Integer Linear Programming (MILP) model. In order to solve largesized instances of the problem, the paper proposed a novel heuristic algorithm called Comparative Particle Swarm Optimization. The results of the different numerical experiments endorsed the effectiveness of the proposed heuristics. Kim (2005) designed the ISC system for optimizing inventory control and reduction of material handling costs of pharmaceutical products in the healthcare sector. This optimized SC system, enabled hospitals to improve the procurement processes and inventory control of pharmaceutical products resulting in decreasing total inventory by more than 30%. Elimam and Dodin (2013) formulated and optimized an ISC as a project network (PN) with activities covering sending and receiving orders, processing these orders as well as sending and receiving shipments and their precedence. Najmi et al. (2016) addressed a largescale multiperiod complementary model for an advanced hydrocarbon bio fuel supply chain integrated with existing petroleum refineries. This model simultaneously optimized the SCI and found the equilibrium quantity of feed stocks, crude oil and final products in the integrated supply chain.
In the field of ISC modeling, Stringer and Hall (2007) presented a novel generic model of the integrated food SC for part of the UK Food Standards Agency (FSA). The key motivation of this paper was a project initiated by the UK FSA to create the “Database of Food Safety Breakdowns”. Yao and Chiou (2004) modeled an ISC system in which one vendor supplied items for the demand of many buyers. This SC model was originally presented in Lu’s (1995) paper. Pitty et al. (2008) demonstrated that a dynamic model of an ISC could serve as a valuable quantitative tool that aided in such decision making. They presented dynamicity in the model of an integrated refinery SC . Sarkar and Majumder (2013) presented an integrated vendorbuyer SC model. The aim of their model was to reduce the total system cost by considering the setup cost reduction by the vendor. BenDaya et al. (2013) presented a threelayer SC model in which the SC included a supplier, a manufacturer and multiple retailers. The objective was to specify the timings and quantities of inbound and outbound logistics for all parties involved in such a way that the chainwide total ordering, setup, raw material and finished product inventory holding costs are minimized. Tiwari et al. (2010) mathematically modeled an ISC design. To ensure high customer service levels, they proposed the inclusion of multiple shipping/transportation options and distributed customer demands with fixed lead times into the SC distribution framework and formulated an integerprogramming model for the fivetier SC design problem. The problem was further complex by including realistic assumptions of nonlinear transportation and inventory holding costs and the presence of economies of scale.
In the field of ISC design, Kostin et al. (2012) addressed the multiobjective design of integrated sugar/bioethanol supply chains considering several life cycle assessment (LCA) impacts. Saleh and Roslin (2015) designed a framework for investigating the influence of supply chain relational capital (SCRC) on the execution of SCI by adopting a relational capital theory. Arora et al. (2010) designed the four policies for delivering products to distributors and retailers on the grounds of the demand of the previous periods occurring in the ISC , and at the current inventory level of retailers and distributors. Zhao et al. (2016) designed and considered an ISC with threelevel which faced by a multinational corporation managing a multistage supply chain over an infinite time horizon. Lemmens et al. (2016) reviewed the literature on modelbased ISC network design in order to identify the applicability of these models to the key issues of the structure of a vaccine supply chain.
In the field of ISC development, Mohammadi Bidhandi and Yusuff (2011) developed an ISC model in the design of multicommodity, singleperiod SC network problems under uncertainty. Susarla and Karimi (2011) modified and enhanced the model set forth by Sundaramoorthy et al. (2006) to model disparate entities and their activities in a global ISC in a seamless fashion with a granularity at the level of production lines. Diabat and Deskoores (2016) developed a capacitated multiechelon joint locationinventory ISC model, according to which a single product is distributed from a manufacturer to retailers through a set of warehouses whose locations were to be determined by the model. They developed a GAbased heuristic to solve the problem. Diabat and AlSalem (2014) developed a nonlinear mixed integer program that minimized the cost of a stochastic twoechelon SC . They developed a GA to solve the model and compared it to optimal solutions obtained by using GAMS BARON solver, whenever obtained.
In the past, experience alone was an adequate guide when changes could be made in small increments (Sobhanallahi et al., 2016a; Sobhanallahi et al., 2016b). In the field of ISC model examinations and investigation, Kim (2006) examined the causal relationship between ISC practice, competition capability, the level of SCI and firm performance. It must be further mentioned that such effort should also enable us to derive a set of recommended strategies from SCM practices for SCI . Matsui (2012) examined the economic role of transfer pricing for a vertically ISC when a riskaverse productiondivision manager faced uncertainty on the outcomes from research and development (R&D) investment. Zhang et al. (2014b) investigated an ISC network design problem that involved the determination of the locations for DCs and the assignment of customers and suppliers to the corresponding ones . The problem simultaneously engaged the distribution of products from the manufacturer to the customers and the collection of components from the suppliers to the manufacturer via cross docking at DCs . The colocation of different types of DCs and coordinated transportation were introduced to achieve cost savings. An LR based algorithm was then developed, and in their paper, a spectrum of extensive computational experiments displayed that the proposed algorithm has stable performance and outperforms CPLEX in largescale problems. Zhang et al. (2014a) investigated the optimal ISC design for commodity chemicals via wood biomass fast pyrolysis and hydroprocessing pathway. This work reflected the economic feasibility and the optimal production planning and facility locations for commodity chemicals via wood biomass fast pyrolysis. AlfallaLuque et al. (2015) examined the relationship between employee commitment and SCI dimensions to explain several performance measures such as flexibility, delivery, quality, inventory, and customer satisfaction. The findings of their model suggested that the relationship between employee commitment and operational performance was fully mediated by SCI . Klein (2007) examined eBusiness ISC relationships between service providers and clients, focusing on the performance impacts of the level of customization implemented by clients using vendorprovided eBusiness solutions and the subsequent realtime access achieved by operational information provided by vendors. Trkman and Groznik (2006) carried out a study whose core idea reiterated that the successful implementation of SCI projects is not as much a technological problem and that a thorough study of the current and desired states of business processes in all companies involved was required. Jaber and Goyal (2008) investigated the coordination of order quantities amongst the players in a threelevel supply chain. The first level of the supply chain included multiple buyers, the second level of a vendor, and the third level of multiple suppliers. They developed a mathematical model in their paper which guaranteed the costs for the levels that had either remained the same as before coordination or decreased as a result of it. The current work is a fourlevel integrated supply chain with stochastic constraints as integrated. Mentioned chain levels consist of single supplier, single producer, single wholesaler and multiple retailers. Also, there are several singlestage products between levels. The supplier at the first level of the chain supplies required items to the producer at the second level to produce required products. Then, the producer ships the products to the wholesaler at the third level of the SC. Next, the wholesaler sales demanded products to multiple retailers. The levels involved in the SC cooperate with each other in order to minimize the total inventory cost including ordering and holding, and so, they agree on having the same stockpile and period length for their products to make an integrated chain. The aim of the paper is to determine the agreed optimum stockpiles and the agreed optimum period length for each product on every level. In fact, optimum stockpile is lot quantity to make the total cost minimum by considering the balance between ordering cost and inventory carrying cost. Also, optimum period length is the replenishment cycle time while total inventory cost is minimized.
It is noted, every product is made up l items. In other words, finished product consists of l separate items with different consumption rate. Products are considered in shortage condition and shortage is allowed for all products in the mentioned model. Additionally, to make the model more applicable to realworld supply chains, some stochastic constraints on the costs of procurement or production, space and order quantity are considered. As the mathematical formulation of the problem becomes nonlinear with a rather large size, the developed SQP method as an effective algorithm with superlinear convergence rate and less iteration for solving the nonlinear and convex models is used. Another realistic assumption, commonly ignored, is stochastic constraint on the total number of all products on each level can move in order. Based on the abovementioned topics, the main contributions and novelties in theory and experimental techniques in this research are as follows:

Design and optimization of an integrated inventory model in a fourechelon supply chain;

Adoption of new variables in making decisions which contain the number of agreed optimum stockpile and period length for products on each level;

Number of the levels in a supply chain, multiproduct and use of real stochastic constraints in model.

Use of a developed algorithm with less iteration and super linear convergence rate for solving the nonlinear and large models such as the current research model.
The outline of the rest of the paper is as follows: Section 2 casts light on the problem definition and assumptions. Notations and mathematical formulation of a fourlevel integrated supply chain will be described respectively in Sections 3 and 4. Section 5 deals with the developed SQP as solution method and in section 6, a numerical example is presented. At last, sensitivity analysis and conclusion are given respectively in sections 7 and 8.
2.PROBLEM DEFINITION AND ASSUMPTIONS
The fourechelon supply chain under stochastic condition involves several singlestage products while resources follow normal distributions with known means and variances on each level. Thus, the product stockpiles, total procurement and production costs, space costs and number of orders are all assumed to be stochastic. The goal hereby is to minimize the total cost of the integrated inventory model in supply chain aiding the cooperation between levels while the stochastic constraints are fulfilled. So, the main questions of the research remain as follows:

What is the number of agreed optimum stockpile for each level?

What is the agreed optimum period length for each level?
As stated earlier, we further actualize the problem by assuming constraints under probability condition as stochastic. The following assumptions are used for modeling the fourlevel supply chain in shortage condition:

There are “l” items for each product on the level of supplier and there are “n” products respectively on the levels of producer, wholesaler and retailers.

Demand rate for products on each level is known, deterministic and dependent.

Shortage is allowed for all products in the foresaid model (${\widehat{\pi}}_{i}\ne 0$ and π_{i} ≠ 0). So, the aforementioned the fourlevel integrated SC is considered to be in shortage condition.

Discount does not exist.

The products on different levels of the chain share unique period length and stockpile.

Holding costs of the products and items per unit and per time unit vary on different levels of the chain.

The planning horizon in this fourlevel supply chain is infinite.

The total procurement cost of the products and items on each level is constrained with a probability greater than a specified value.

Number of orders for products and items on each level is limited with a probability greater than a certain value.

The space cost required for the products and items on each level is limited with a probability greater than a specified value.

The total number of products ordered by each level is limited with a probability greater than a certain value.

There is no limitation for production time capacity.

There is no setup cost for supplier and producer levels.

The replenishment policy in our research will be fallen in fixed order size category.

The order size is determined by solving model and we will order when inventory reaches to zero.

The resources of the problem follow normal distributions with known means and variances.
3.NOTATIONS
As clarified earlier, the final objectives of this research are to find the number of agreed optimum stockpile and agreed optimum period length for products to minimize the total inventory cost of each level while the stochastic constraints are met. The following notations are used for modeling:
3.1.Set of Indices

k : An index for retailers (k =1, 2,…, m).

j : An index for items which make upproducts (j =1, 2,…, l).

i : An index for products (i =1, 2,…, n).
3.2.Level Parameters
3.2.1.Supplier Parameters

μ_{s}: The mean of the resources in the supplier level.

${\text{\sigma}}_{s}^{2}$ : The variance of the resources in the supplier level.

α_{s} : The violating probability for each supplier’s stochastic constraint.

Z_{∝s} : The critical point of the standard normal value for α_{s}.

π_{is} : Shortage fixed cost for included items of ith product which is tolerated by supplier.

${\widehat{\pi}}_{is}$ : Shortage cost which is time dependent for included items of ith product, tolerated by supplier.

b_{is} : Total shortage which occurred in period [0, T] for included items of ith product.

b_{is} : The average shortage that occurred in period [0, T] for included items of ith product.

A_{is} : Fixed ordering cost for included items of ith product, tolerated by supplier.

h_{jis} : Holding cost of jth item from ith product, tolerated by supplier.

D_{is} : Demand rate for included items of ith product.

λ_{s} : Maximum number of supplier’s stockpile.

f_{is} : The cost per unit of space for included items of ith product.

S_{s} : Maximum space cost for supplier.

c_{is} : Supply cost for included items of product i.

C_{s} : Maximumsupply cost for supplier.

O_{s} : Maximum number of orders for supplier’s items.
3.2.2.Producer Parameters

μ_{p} : The mean of the resources in the producer level.

${\sigma}_{p}^{2}$ : The variance of the resources in the producer level.

α_{p} : The violating probability for each producer’s stochastic constraint.

Z_{∝ p} : The critical point of the standard normal value for α_{p}.

π_{i}p : Shortage fixed cost for per unit of ith product which is tolerated by producer.

${\widehat{\pi}}_{ip}$ : Shortage cost which is time dependent for ith producer’s product, tolerated by producer.

b_{ip} : Total shortage which occurred in period [0, T] for ith producer’s product.

b_{ip} : The average shortage that occurred in period [0, T] for ith producer’s product.

A_{ip} : Fixed ordering cost per order of ith producer’s product.

h_{jip} : Holding cost of jth item from ith producer’s product.

h_{ip} : Holding cost for ith producer’s product.

D_{ip} : Demand rate for ith producer’s product.

a_{jip} : Fixed ordering cost per order of jth item from ith producer’s product.

λ_{p} : Maximum number of stockpile for producer’s products.

f_{ip} : The cost per unit of space for ith producer’s product

S_{p} : Maximumspace cost for producer.

c_{ip} : Production cost for per unit of ith producer’s product

C_{p} : Maximum production cost for producer.

O_{p} : Maximum number of orders for producer.
3.2.3.Wholesaler Parameters

μ_{w} : The mean of the resources in wholesaler level.

${\sigma}_{w}^{2}$ : The variance of the resources in wholesaler level.

α_{w} : The violating probability for each wholesaler’s stochastic constraint.

Z_{∝w} : The critical point of the standard normal value for α_{w}.

π_{iw} : Shortage fixed cost for per unit of ith product which is tolerated by wholesaler.

${\widehat{\pi}}_{iw}$ : Shortage cost which is time dependent for ith wholesaler’s product, tolerated by wholesaler.

b_{iw} : Total shortage which occurred in period [0, T] for ith wholesaler’s product

b_{iw} : The average shortage that occurred in period [0, T] for ith wholesaler’s product

A_{iw} : Fixed ordering cost per order of ith wholesaler’s product.

h_{iw} : Holding cost for ith wholesaler’s product.

D_{iw} : Demand rate for ith wholesaler’s product.

λ_{w} : Maximum number of stockpile for wholesaler’s products.

f_{iw} : The cost per unit of space for ith wholesaler’s product.

S_{w} : Maximum space cost for wholesaler.

c_{iw} : Procurement cost per unit of ith wholesaler’s product.

C_{w} : Maximum procurement cost for wholesaler.

O_{w} : Maximum number of orders for wholesaler.
3.2.4.Retailer Parameters

μ_{rk} : The mean of the resources in the kth retailer level.

${\sigma}_{rk}^{2}$ : The variance of the resources in the kth retailer level.

α_{rk} : The violating probability for kth retailer’s stochastic constraint.

Z_{∝rk} : The critical point of the standard normal value for α_{rk}.

π_{irk} : The cost per unit of lost sales of ith product which is tolerated by kth retailer.

b_{irk} : Total lost sales which occurred in period [0, T] for ith product of kth retailer.

A_{irk} : Fixed ordering cost per order of ith product of kth retailer.

h_{irk} : Holding cost of ith product of kth retailer.

D_{irk} : Demand rate of ith product of kth retailer.

f_{irk} : The cost per unit of space for ith productof kth retailer.

S_{rk} : Maximum space cost for kth retailer.

c_{irk} : Procurement cost of per unit of product i for kth retailer.

C_{rk} : Maximumprocurement cost for kth retailer.

O_{rk} : Maximum number of orders for kth retailer.
3.3.Other Parameters Related to the Levels

u_{ji} : Number of jth item required in one unit of ith product

f_{i} : Space occupied by each unit of product i.

TC_{s} : Total inventory cost of supplier.

TC_{p} : Total inventory cost of producer.

TC_{r} : Total inventory cost of retailers.

TC_{w} : Total inventory cost of wholesaler.

TC_{T} : Total inventory cost of a fourlevel integrated supply chain.
3.4.Variables

λ_{i} : Number of agreement optimum stockpile of ith product which supplier, producer and wholesaler have agreed to minimize total inventory cost of SC.

T_{i} : The agreement optimum period length of ith product which levels have agreed to minimize total inventory cost of SC.
4.MATHEMATICAL FORMULATION
When retailers place Q_{i} or order quantity of product i(Q_{i} = T_{i} ×D_{i} ) with the wholesaler every T_{i}, wholesaler places an order of size Q_{i} with the producer and then producer determines its order quantity λ_{ip}Q_{i} for supplier and places orders for items of products which are supplied by supplier. For the producer to fulfill the order for wholesaler, will request u_{ji}λ_{ip}Q_{i} items of product i. So, the annual total inventory cost for supplier is written as Eq. (1). Where the term $\frac{{A}_{is}}{{\lambda}_{is}{\lambda}_{ip}{\lambda}_{iw}{T}_{is}}$ is the order cost. To adjust the order quantity of the supplier to producer, producer to wholesaler and wholesaler to retailers; product’s stockpiles of each level is considered as multiple of product’s stockpiles of lower levels. So, there are three Lambda values in denominator; One for supplier level (λ_{is} ) and the other two for lower levels (producer level λ_{ip} and wholesaler level λ_{iw} ). Term $\frac{1}{2}{\displaystyle {\sum}_{i=1}^{n}(({\displaystyle {\sum}_{j=1}^{l}({h}_{jis}{u}_{ji})}))}{D}_{is}{T}_{is}({\lambda}_{is}1){\lambda}_{ip}{\lambda}_{iw}$ is holding cost for supplier to meet the demand for items requested by the producer. It is clear that first stockpile hasn’t any holding cost, so it comes (λ_{is} −1) and product’s stockpiles of lower levels (λ_{ip}, λ_{iw}) is multiplied on.
Supplier level is single and exclusive. If the supplier didn’t meet the producer demand, it would be tolerated shortage cost as backlogged. So, if shortage is occurred in supplier level, producer’s orders will be deferred by supplier, so supplier has to pay shortage cost or fine. So, the term ${\text{\pi}}_{is}{b}_{is}+{\widehat{\pi}}_{is}{\overline{b}}_{is}{T}_{is}$ is shortage cost which supplier must be tolerated.
It is noted, in Eq. (1), the ordering cost depends on the number of orders and number of optimum stockpiles. Also, holding cost depends on the items which made up products and consumption rate of the foresaid items in production.
The annual total inventory cost for producer, including the ordering cost in the first term, holding cost of produced products in the second term, holding cost of items (which have been delivered from supplier) in the third term and shortage cost in the fourth term is as Eq. (2):
Same as supplier level, producer level is single and exclusive. So as usual, shortage at producer level is backlogged type. In other words, in shortage condition, wholesaler’s orders will be deferred by producer, so producer has to pay shortage cost or fine. So, the term ${\text{\pi}}_{ip}{b}_{ip}+{\widehat{\pi}}_{ip}{\overline{b}}_{ip}{T}_{ip}$ is shortage cost which producer must be tolerated.
In Eq. (2), the ordering cost depends on the number of orders for products and items and number of optimum stockpile of ith product. Also in the third term, the holding cost of products depends on the items which made up products, consumption rate of the aforementioned items in products and number of optimum stockpile of ith product. The wholesaler at the third level buys the produced products from the producer and sells them to multiple retailers. Hence, the annual total inventory cost of wholesaler, including the ordering, holding and shortage costs is obtained by Eq. (3):
In Eq. (3), the first term is the ordering cost of the wholesaler, the second term focuses on costs associated with the holding and the third term is related to shortage costs for wholesaler. Also, the shortage at the wholesaler level is the backlogged type, because this level same as previous levels is exclusive. Retailers at the fourth level buy the products from the wholesaler and sell them to customers. The annual total inventory cost of the retailers, including the ordering and the holding cost are determined using Eq. (4).
It is noted, in the retailer level, there are several retailers. If shortage occurs at this level, customers will supply their required products from another retailer. Because of the multiplicity of retailers, we considered shortage cost as independent of the time in this level. As stated before, four levels of supply chain interact and agree with each other on having the same period length and same number of stockpile for each product in order to make an integrated supply chain for minimizing the chain total cost. Based on this interaction, integration and agreement between levels, we will have:
In other words, λ_{is}, λ_{ip}, λ_{iw} values in the Equations 1 to 4 are replaced with λ_{i} in Equations 6, 22 to 36. Similarly, T_{is}, T_{ip}, T_{iw}, T_{irk} values in the Equations 1 to 4 are replaced with T_{i} in Equations 6, 22 to 36.
With regard to the integration in equation (5) and with respect to Equations (1) to (4), the objective function of a fourlevel supply chain in integrated status is identified as follows:
The stochastic constraints of this model have been stated in Section 2, where the problem was defined and the assumptions were stated. Stochastic constraints are as Equations (7) to (21).(8)(9)(12)(13)(16)(17)(20)
It is noted, the values of λ_{is}, Q_{is}, λ_{ip}, Q_{ip}, λ_{iw}, Q_{iw}, Q_{irk} are positive. Constraints (7) to (10) guarantee the limitation of procurement costs respectively for supplier, producer, wholesaler and retailers. Constraints (11) to (14) guarantee the number of orders limitation respectively for supplier, producer, wholesaler and retailers. Constraints (15) to (18) guarantee space cost limitation respectively for supplier, producer, wholesaler and retailers. Constraints (19) to (21) guarantee the limitation in number of stockpile for respectively supplier, producer and wholesaler. Based on Equation (5), by replacing λ_{i} and T_{i} parameters in Equations (7) to (21), the stochastic constrains of the problem can be rewritten as follow:(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34)(35)
It is noted, the values of λ_{i} , T_{i} are positive. The formulation given in Equation (6) as an integrated objective function and Equations (22) to (36) as problem stochastic constraints is a nonlinearprogramming type. However, SQP is a powerful and effective solution method for a wide range of nonlinear optimization problems. SQP method developed as into more effective algorithm with less iteration will be used in the next section to optimize the research model. Pasandideh et al. (2015) suggested that SQP has satisfactory performance in solving nonlinear problems while yielding optimum solutions, number of iterations to achieve the optimum solution, infeasibility, optimality error and complementarity. Also, Gharaei et al. (2015) confirmed satisfactory performance of this exact method for solving nonlinear problems. SQP with superlinear convergence rate helps us to find optimum solution in largescale nonlinear problems (Gharaei and Pasandideh, 2016; Gharaei et al., 2017a; Gharaei et al., 2017b; Gharaei et al., 2017c). SQP method will be used in the next section for optimization the model of this research.
5.SOLUTION METHOD
Sequential (or successive) Quadratic Programming or SQP represents one of stateoftheart and most popular methods for nonlinear constraint optimization. It is also one of the robust methods. For a general nonlinear optimization problem, we have:(37)
The fundamental idea of SQP is to approximate the computationally extensive full Hessian matrix using a quasi Newton updating method. Subsequently, this method generates a subproblem of quadratic programming (Called QP subproblem) at iterations and the solution to it can be used to determine the search direction and next trial solution. Using Taylor expansion, the problem above can be approximated at iterations as the following problem:(38)(39)(40)
Subject to:
Where the Lagrange function, also referred to as merit function, is defined by:(41)
where $\lambda ={\left({\lambda}_{1},\text{\hspace{0.17em}}\cdots ,\text{\hspace{0.17em}}{\lambda}_{\text{p}}\right)}^{\text{T}}$ is the vector of Lagrange multipliers, and $\mu ={\left({\mu}_{1},\text{\hspace{0.17em}}\cdots ,\text{\hspace{0.17em}}{\mu}_{\text{q}}\right)}^{\text{T}}$ is the vector of KarushKuhn Tucker (KKT) multipliers.
Here, we have used the notation $h={\left({h}_{1}\left(x\right),\text{\hspace{0.17em}}\cdots ,\text{\hspace{0.17em}}{h}_{p}\left(x\right)\right)}^{T}$ and $g={\left({g}_{1}\left(x\right),\text{\hspace{0.17em}}\cdots ,\text{\hspace{0.17em}}{g}_{q}\left(x\right)\right)}^{T}$. To approximate the Hessian ${\nabla}^{2}L\left({x}_{k}\right)$ by a positive definite symmetricmatrix H_{k} , the standard BroydonFletcherGoldfarboShanno (BFGS) approximation of the Hessian can be used, and we have:(42)
where:(43)
The QP subproblem is solved to obtain the search direction.(44)
Using a line search method by minimizing a penalty function, also commonly called merit function,(45)
where ρ is the penalty parameter; It is worthy of note that any SQP method requires a good choice of H_{k} as the approximate Hessian of the Lagrangian L. Obviously, if H_{k} is exactly calculated as ∇^{2}L , SQP essentially becomes Newton’s method for solving the optimality condition. A popular way to approximate the Lagrangian Hessian is to use a quasiNewton scheme as we used the BFGS formula. The procedure below is the process of Sequential Quadratic Programming:
Choose a starting point x_{0} and approximation H_{0} to the Hessian.
Repeatk =1, 2, …
Solve a QP subproblem: QP_{k} to get the search direction s_{k}
Given s_{k}, find∝so as to determine x_{k+1}
Update the approximate Hessian H_{k+1} using the BFGS scheme.
k = k + 1, until stop criterion (Zhu, 2005)
A numerical example is solved in the next section to demonstrate the applicability of the proposed algorithm for solving the recent convex nonlinear model.
6.NUMERICAL EXAMPLE
In the numerical example presented here, there are 1 supplier, 1 producer, 1 wholesaler and 2 retailers. There are 2 products in this chain which have singlestage production process on the producer’s level. Also, each product consists of 2 items, and the consumption rate of 1st and 2nd items in the first product is 2 and 1 respectively. The consumption rate of 1st and 2nd items of the second product is 2. As stated earlier, the resources in the numerical example follow normal distributions with known means and variances, where Z_{∝s} , Z_{∝p} , Z_{∝w} and Z_{∝rk} are assumed as 2.75 for stochastic constraints of supplier, producer, wholesaler and retailers are displayed in the numerical example. The general data and parameters of “supplier, producer, wholesaler and retailers” in the numerical example is summarized respectively in Table 1, Table 2. The objectives are to find both the number of agreed optimum stockpile and agreed optimum period length for each product on each level while the model stochastic constraints are satisfied. The starting point of solution method in numerical example is:
The example has been solved using the SAS 9.2 computer software. Table 3 shows the optimum solution, number of iterations to achieve the optimum solution, infeasibility, optimality error, and complementarity obtained for the numerical example.
In Table 3, λ_{1} and λ_{2} are the number of agreed optimum stockpile for products 1 and 2 for supplier, producer and wholesaler. Also, T_{1} and T_{2} are the optimum agreed period length for products 1 and 2 in the objective function. The number of steps taken by the developed SQP algorithm to achieve the optimal solution is given in the column entitled “Number of Iterations.” As the computation is carried out in a finiteprecision environment, rounding errors prevents the algorithm from obtaining a solution exactly satisfying the preceding condition (shown in the Infeasibility column). Instead, we terminate the algorithms at some small threshold values which can be measured in an absolute or relative sense. The fifth column in the aforesaid Table refers to the “optimality error.” The last column (“complementarity”) is the relative Infeasibility defined as the maximum amount of constraint violation relative to the one in the “Infeasibility” column.
As was stated earlier, the calculations are done in SAS 9.2 computer software. For instance, in Table 3, written program in the SAS software based on developed SQP method is as Appendix (A). With the implementation of program in Appendix (A) in the SAS software, the final output to be displayed as Table 3 for this example.
7.SENSİTİVİTY ANALYSİS
A sensitivity analysis on the change rate of the objective function based on the change rate of the “λ_{i}” parameter is performed in this section for the mentioned numerical example with the initial data shown in Tables (1), (2). It involves increasing or decreasing parameter “λ_{i}” at ±20, ±30, ±35 and ±50 percent, Table 4 shows the results. Moreover, Figure 1 shows the change rate of the objective function based on the change rate of the agreed optimum stockpiles. Figure 1 is as inverse symmetric around the nochange point. In other words, the positive change rate of the parameter λ_{i} leads to the increase in objective function and vice versa. So, increasing the rate of parameter “λ_{i}” (greater than 1), leads to increase the objective function and decreasing the rate of parameter “λ_{i}” (less than 1), leads to decrease the objective function. It is clear that the same changes rate of “λ_{i}” parameter leads to the point that change rate of objective function be as inverse symmetric curve like Figure 1.
8.CONCLUSİON AND FUTURE RESEARCH/RESEARCH PERSPECTİVE
We modeled and optimized inventory costs in a fourechelon integrated supply chain with stochastic constraints under shortage condition. Every constraint has a probability greater than a separated certain value. There are stochastic limitations in procurement cost, number of ordered products organized on levels, space cost and number of orders for products on each level. In the foresaid SC, all levels interact and agree with each other on having the same period length and stockpiles for each product to minimize the total cost of the chain. The ultimategoal is to optimize the number of agreed optimum stockpile and period length for products while stochastic constraints are satisfied. The research model is nonlinear and large, and in it, the sequential quadratic programming (SQP) as an exact developed solution was utilized in order to find the number of optimum stockpile and optimum period length for each product on the chain levels. The results of the SQP implementation on the numerical example exhibited the optimum performance of the SQP exact method for solving nonlinear and large problemssuch as the one mentioned in the model. A sensitivity analysis is later performed on the change rate of the objective function versus change rate of the agreed optimum stockpiles for products based on thesuggested algorithm. The sensitivity analysis revealed excellent performance of thealgorithm for solving the present research model. The results of the sensitivity analysis further showed that there is a direct relationship with strong correlation between the change rate of parameter “λ_{i}” and that of the integrated objective function. Also, we found that the same change rate of the “λ_{i}” parameter does lead to that of the objective function, with their nature being aninverse symmetric curve. Based on the aforesaid results, it is more important for managers to try to convince other levels of the supply chain that the integration may improve the entire chain’s competitiveness and costeffectiveness. Furthermore, the design and optimization of an integrated supply chain in cooperation with the supply chain managers can prove beneficial and applicable for them to minimize the total inventory cost of the chain byfinding both the number of agreed optimum stockpiles and agreed optimum period length. Therefore, a higher level of supply chain integration formed by thecooperation bewteenchain managers leads to the improvement ofcosteffectiveness on levels. Finally, it can be said that managers who intend to remain in competitive markets have to integrate their inventory systems with the other levels of the chain. For the purpose of future researches, we will develop and enrich the research model into an integrated inventory model with high reliability in biobjective multilevel supply chain. Also, we will consider multiplesuppliers, producers and wholesalerswhile the damand is randomin the future suggested development.