Optimization of fuzzy demand distribution supply chain using modified sequence quadratic programming approach

Abstract

This paper applies the credibility distribution of fuzzy variables in uncertainty theory to formulate a fuzzy optimization model, so as to study the coordination mechanism of a distribution supply chain system with fuzzy demand. This practical system, focusing on the demand side of the supply chain, comprises multi-period, multi-manufacturer, multi-retailer, and multi-consumer. Next, using a novel modified sequence quadratic programming algorithm, the optimal order quantity, sales volume of the retailers, and the maximum expected return of the distribution supply chain are obtained. The effects of the wholesale price, retail price, and inventory cost on the expected return are also analyzed. Finally, the usfulness of the proposed modified sequence quadratic programming algorithm is compared against the conventional MATLAB optimal toolbox and genetic algorithm. Our computational results validate in favour of the proposed algorithm in terms of the computational time, number of iterations, and convergence rate.

Keywords

Distribution supply chain fuzzy demand uncertainty theory algorithm

1 Introduction

Supply chain design and planning invariably need to entertain numerous uncertainties. Conventional supply chain modeling tends to focus on the randomness aspect of uncertainty. For that matter, a number of stochastic modeling techniques have been developed to deal with uncertainty and have been successfully applied to supply chain design [1 –12].

One of the most difficult challenges faced by the decision makers in practical supply chain design and planning is in forecasting consumer demand.

Previous studies have been conducted under the assumption that consumer demand obeys a certain probability distribution. However, in today’s highly competitive market, increasingly short product life cycles render demand to be extremely volatile and historical demand statistics less useful. Put simply, past data on consumer demand may not always be reliable or even available.

Therefore, standard probability modeling may not be the best choice under such circumstances. Rather, the use of fuzzy demand better suits the new reality, and the parameters in the fuzzy demand can be estimated accordingly through the changes in actual consumer demand. This would then suggest that fuzzy theory is more appropriate for uncertainty modeling than the conventional probability theory, and fuzzy set theory provides an acceptable alternative approach to dealing with this uncertainty [13].

The extant literature on fuzzy supply chain modeling is replete, focusing mostly on two aspects: (1) how fuzzy uncertainty affects supply chain operations, and (2) how to apply various optimization tools to solve for fuzzy uncertainty in a distribution supply chain.

Specific to supply chain uncertainty, Giannoccaro et al. [14] have investigated a three-tier supply chain and highlighted the suitability of simulating market demand and inventory cost using fuzzy set theory. Xu et al. [15] proposed an optimization technique for single-period supply chain problems with fuzzy demand. The external demand was modeled using triangular fuzzy numbers; a decision model for independent retailers and an integrated supply chain was established, and a cooperative strategy to promote the optimization of the entire supply chain was proposed. Peidro et al. [16] considered the uncertainties of supply, demand and process, and proposed a fuzzy programming model for supply chain planning. Zhao et al. [17] studied the pricing decision for two complementary products in a two-level fuzzy supply chain involving two manufacturers and one co-retailer. Chou et al. [18] established an inventory model of a single period utility product under fuzzy demand. Wang et al. [19] developed a dual-target supplier managed inventory model with a single supplier and multiple retailers involving fuzzy demand as random variables. Table 1 lists some of the representative literature in the fuzzy supply chain domain and their research focus.

Table 1
Research on fuzzy demand supply chains

Reference Research focus

[9 , 30] Fuzzy demand

[20 , 28] Inventory

[25 , 33] Supplier selection

[4 , 31] Green supply chain

[17, 24] Pricing decision

[16] Supply uncertatinties

[15 , 24] Single period

[8, 21] Multi-period

[23] Forecasting

[20, 29] Supply chain performance

Reference	Research focus
[9 , 30]	Fuzzy demand
[20 , 28]	Inventory
[25 , 33]	Supplier selection
[4 , 31]	Green supply chain
[17, 24]	Pricing decision
[16]	Supply uncertatinties
[15 , 24]	Single period
[8, 21]	Multi-period
[23]	Forecasting
[20, 29]	Supply chain performance

To solve the problem of fuzzy uncertainty in the supply chain, scholars have proposed numerous optimization methods, such as genetic algorithm [20 , 27], ant colony optimization, particle swarm optimization [22, 28], fuzzy Analytic Hierarchy Process (AHP) [26, 29], hybrid memetic algorithm [30], fuzzy VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) [31], and fuzzy DEMATEL [32]. Among them, Wang et al. [20] proposed a fuzzy supply chain model based on possibility theory, and applied genetic algorithm to determine the optimal inventory to order for a supply chain. Aliev et al. [21] developed an integrated, multi-period, multi-product fuzzy production and distribution model for a supply chain. The model was formulated using fuzzy programming, and the solution was found through genetic algorithm. Sadeghi et al. [22] developed a vendor-managed inventory model in a multi-retailer single-vendor under a consignment stock environment, and an improved particle swarm optimizer was proposed to solve the model.

Clearly, consumer demand uncertainty leads to an uncertainty in the product quantity demanded at each stage of the distribution supply chain. This then logically increases the inventory level and other related costs. Most studies, as highlighted earlier, tend to focus on the production-distribution problem under the situation of a single period, manufacturer or retailer, or a specific demand distribution. In reality, however, many distribution supply chain management problems are not of a unitary type. Furthermore, to the best of our knowledge, there is a dearth of research on the multi-period, multi-manufacturer, multi-retailer, and multi-consumer distribution supply chain under fuzzy demand. Thus, to fill this gap, in this paper, we consider the coordination mechanism of a distribution supply chain system comprising multi-period, multi-manufacturer, multi-retailer, and multiple consumer. A fuzzy programming model is established to determine the optimal production and distribution decisions under fuzzy demand. The optimal strategy is obtained through a modified sequential quadratic programming (SQP) algorithm, and the effectiveness of the algorithm is verified numerically. The influence of fuzzy uncertainty on the profit and decision of each component of the distribution supply chain is analyzed. We illustrate the performance of the proposed model by comparing the effect of inventory cost and expected profitability on the decisions made, under the same conditions.

The rest of this paper is organized as follows. Section 2 briefly describes the mathematical preliminaries on fuzzy variables. The mathematical formulation of the problem is established in Section 3. Section 4 then presents an algorithm for solving the model. The computational results are presented and discussed in Section 5. Section 6 provides the concluding remarks and offers some future research directions.

2 Preliminaries

Converting fuzzy variables into an equivalent deterministic form has an important role in solvingthe fuzzy optimization problem, which is often encountered in real-life situations. Motivated by the need for an axiomatic approach to perform this step, Liu [34] has suggested the credibility measure and proposed the credibility theory. Through this theory, a fuzzy variable (fuzzy consumer demand) is transformed into its deterministic equivalent (expected consumer demand), denoted as the expected value. This Section provides the definition of the credibility distribution function and its expected value on the basis of the credibility measure. To do so, we need to define the credibility measure.

Definition 1. [34] Let Θ be a nonempty set, and P (Θ) be the power set of Θ. For any A ∈ P (Θ) , if Cr {A} satisfies the following conditions: (1) Cr {Θ} =1, Cr {∅} =0; If A ⊂ B, then Cr {A} ≤ Cr {B}. (2) Cr is self-dual, i.e., Cr {A} + Cr {A^c} =1 for any A ∈ P (Θ). (3) $Cr {⋃_{i} A_{i}} ⋀ 0.5 = sup_{i} Cr {A_{i}}$ for any A_i with Cr {A_i} ≤0.5, then Cr {A} is a credibility measure for event A. The tuple (Θ, P (Θ) , Cr) is defined as a credibility space. Cr {A} describes the credibility of the occurrence of event A. From the credibility measure, we can derive the credibility distribution of the fuzzy variables.

Definition 2. [34] The credibility distribution Φ : R → [0, 1] of a fuzzy variable ξ is defined as $Φ (x) = Cr {θ \in Θ | ξ (θ) \leq x},$ In other words, Φ (x) is the credibility of the fuzzy variable ξ, which is less than or equal to x. Next, the definition of credibility density function is given.

Definition 3. [34] Suppose ξ is a fuzzy variable, and Φ (·) is the credibility distribution of ξ. If the function φ : R → [0, + ∞) satisfies the following equation: $Φ (x) = \int_{- \infty}^{x} φ (y) dy, \int_{- \infty}^{+ \infty} φ (y) dy = 1, \forall x \in R,$ then φ is the credibility density function of the fuzzy variable ξ.

With the above concept, the expected value of a fuzzy variable can be defined as follows.

Definition 4. [34] Let ξ be a fuzzy variable defined on the credibility space (Θ, P (Θ) , Cr). The expected value of a fuzzy variable ξ, E [ξ], is defined as $E [ξ] = \int_{0}^{+ \infty} Cr {ξ \geq r} dr - \int_{- \infty}^{0} Cr {ξ \leq r} dr,$ provided that at least one of the two integrals isfinite.

The expected value of a fuzzy variable can also be obtained from the credibility density function.

Lemma 1. [34] Let ξ be a fuzzy variable with credibility density function φ (·). If the expected value E [ξ] exists, then $E [ξ] = \int_{- \infty}^{+ \infty} x φ (x) dx .$

For this paper, in order to represent the expected sales volume of the retailers, we compute the expected value of the min function. In this regard, Wang et al. [35] have obtained the next lemma.

Lemma 2. [35] Let ξ be a fuzzy variable with credibility distribution function Φ (·) and credibility density function φ (·). Suppose the support of ξ is denoted by Θ = [u, v]. We have $E [min (z, ξ)] = z - \int_{u}^{z} (z - x) φ (x) dx, 0 \leq u \leq z \leq v .$

Through the above definitions and lemmas, we can successfully transform a fuzzy optimization problem into a deterministic optimization problem through the expected value function, making it a simple and convenient to solve supply chain uncertainty problems. This shows that the expected value function is an important link between the fuzzy optimization problem and the deterministic optimization problem. In Section 3, we describe the problem studied in this paper, and provide the process of transforming the fuzzy optimization problem into an exact optimization problem. Furthermore, we obtain the profit function of the distribution supplychain.

3 Problem definition and modeling

In this section, we formulate the model for a multi-period, multi-manufacturer, multi-retailer, and multi-consumer distribution supply chain network, as our emphasis is on the demand side of the supply chain. The distribution supply chain network structure is depicted in Figure 1. In particular, we consider I manufacturers, J retailers, and K consumers. We denote a typical manufacturer by i, a typical retailer by j, and a typical consumer by k. The links in the distribution supply chain network denote the transaction links. The manufacturers make a single product which is sold to the retailers, who have to contend with fuzzy consumer demand.

Fig.1

Network structure of distribution supply chain.

For simplicity, we use the following symbols for this model. All vectors are assumed to be column vectors.

3.1 Notation

Indices

i Manufacturer index; i = 1, 2, . . . , I

j Retailer index; j = 1, 2, . . . , J

k Consumer index; k = 1, 2, . . . , K

t An index for a time period; t = 1, 2, . . . , T

Parameters

${\tilde{D}}_{tjk}$ Fuzzy demand of consumer k for retailer j in period t

${\tilde{D}}_{tk}$ Consumer k’s total fuzzy demand in period t, ( ${\tilde{D}}_{tk} = \sum_{j = 1}^{j = J} {\tilde{D}}_{tjk}$ )

q_tij Quantity shipped from manufacturer i to retailer j in period t

f_ti (q) Production cost function of manufacturer i in period t

e_tij (q_tij) Transportation cost function from manufacturer i to retailer j in period t

p_ti Wholesale price of unit product of manufacturer i in period t

a_tjk Quantity of product sold by retailer j to consumer k in period t

q_tj Sales volume of retailer j in period t

c_tj Inventory cost of retailer j in period t

b_tj Retail price of unit product of retailer j in period t

h Shortage cost per unit

π_m Manufacturer’s profit function

π_r Retailer’s profit function

π Profit function of distribution supply chain

E (·) Expected revenue

3.2 Model development

Assumptions (1) The production cost function f_ti (q) and transportation cost e_tij (q_tij) are continuous and convex. (2) Retailer j sets the retail price concurrently with the retailer’s stocking decision. The retail price is fixed throughout the period. (3) The credibility distribution function of ${\tilde{D}}_{tjk}$ is Φ_tjk (x), and its credibility density function is φ_tjk (x). The credibility distribution has support on [u_tjk, v_tjk] with 0 ≤ u_tjk < v_tjk.

In period t-th, when the sales volume of retailer j is $q_{tj} = \sum_{k = 1}^{K} a_{tjk}$ , the expected sales volume S (q_tj), the desired stock quantity I (q_tj), and the desired shortage quantity R (q_tj) of retailer j are: $S (q_{tj}) = \sum_{k = 1}^{K} E [min (a_{tjk}, {\tilde{D}}_{tjk})]$ (1) $= q_{tj} - \sum_{k = 1}^{K} \int_{u_{tjk}}^{a_{tjk}} (a_{tjk} - x) φ_{tjk} (x) dx,$ $I (q_{tj}) = \sum_{k = 1}^{K} E [max (a_{tjk} - {\tilde{D}}_{tjk}, 0)]$ (2) $\begin{matrix} = & q_{tj} - S_{t} (q_{tj}) \\ = & \sum_{k = 1}^{K} \int_{u_{tjk}}^{a_{tjk}} (a_{tjk} - x) φ_{tjk} (x) dx, \end{matrix}$ $R (q_{tj}) = \sum_{k = 1}^{K} E [max ({\tilde{D}}_{tjk} - a_{tjk}, 0)]$ (3) $\begin{matrix} = & \sum_{k = 1}^{K} E [{\tilde{D}}_{tjk}] - S (q_{tj}) \\ = & \sum_{k = 1}^{K} \int_{a_{tjk}}^{v_{tjk}} (x - a_{tjk}) φ_{tjk} (x) dx . \end{matrix}$ So, the retailers’ expected profit function can be written as $π_{r} = \sum_{t = 1}^{T} \sum_{j = 1}^{J} (b_{tj} S (q_{tj}) - c_{tj} \sum_{t = 1}^{t} I (q_{tj})$ (4) $- hR (q_{tj})) - \sum_{t = 1}^{T} \sum_{j = 1}^{J} \sum_{i = 1}^{I} q_{tij} p_{ti} .$ Similarly, the manufacturers’ expected profit function can be written as $π_{m} = \sum_{t = 1}^{T} \sum_{j = 1}^{J} \sum_{i = 1}^{I} q_{tij} p_{ti} - \sum_{t = 1}^{T} \sum_{i = 1}^{I} f_{ti} (q)$ (5) $- \sum_{t = 1}^{T} \sum_{j = 1}^{J} \sum_{i = 1}^{I} e_{tij} (q_{tij}) .$ Then, the distribution supply chain’s expected profit can be expressed as $π = π_{m} + π_{r} = \sum_{t = 1}^{T} \sum_{j = 1}^{J} (b_{tj} S (q_{tj}) - c_{tj} \sum_{t = 1}^{t} I (q_{tj})$ (6) $- hR (q_{tj})) - \sum_{t = 1}^{T} \sum_{i = 1}^{I} f_{ti} (q) - \sum_{t = 1}^{T} \sum_{j = 1}^{J} \sum_{i = 1}^{I} e_{tij} (q_{tij}) .$ In distribution supply chain management, the supply chain decision makers will choose the optimal order quantity and sales volume to maximize the distribution supply chain’s total expected profit π (q_tij, a_tjk), i.e., $max π (q_{tij}, a_{tjk})$ (7) $\begin{matrix} s . t . \sum_{i = 1}^{I} q_{tij} \geq \sum_{k = 1}^{K} a_{tjk} \forall j, \\ q_{tij} \geq 0, a_{tjk} \geq 0 \forall t, i, j, k . \end{matrix}$ The first constraint ensures that the quantity ordered by retailer j from the manufacturers, are greater than or equal to that supplied to the consumers. The second constraint ensures that all the decision variables are non-negative.

For the sake of simplicity, we represent the objective functions separately, i.e., $π (q_{tij}, a_{tjk}) = π_{1} (a_{tjk}) + π_{2} (q_{tij}),$ where $π_{1} (a_{tjk}) = \sum_{t = 1}^{T} \sum_{j = 1}^{J} (b_{tj} S (q_{tj}) - c_{tj} \sum_{t = 1}^{t} I (q_{tj}) - hR (q_{tj})),$ $π_{2} (q_{tij}) = - \sum_{t = 1}^{T} \sum_{i = 1}^{I} f_{ti} (q) - \sum_{t = 1}^{T} \sum_{j = 1}^{J} \sum_{i = 1}^{I} e_{tij} (q_{tij}) .$ In addition, $\frac{d^{2} π_{1}}{{da}_{tjk}^{2}} = (- b_{tj} - h - c_{tj}) φ_{tjk} (a_{tjk}) .$ (8) From Equation (8), we have $\frac{d^{2} π_{1}}{{da}_{tjk}^{2}} < 0$ , i.e., function π₁ is strictly concave. From Assumption (1), function π₂ is concave. So, the objective function π is strictly concave and has an optimal solution to maximize the total expected profit.

4 Solution algorithm

Solving Equation (7) is equivalent to solving the following problem: $min θ (z)$ (9) $s . t . g (z) \geq 0,$ where z = (q_tij, a_tjk) is a decision variable, θ (z) = - π (q_tij, a_tjk) is the objective function, g (z) = (g₁ (z) , g₂ (z)) ^⊤ is the constraint function, $g_{1} (z) = \sum_{i = 1}^{I} q_{tij} - \sum_{k = 1}^{K} a_{tjk}, and g_{2} (z) = z, \forall i, j, k, t .$

The class of algorithms available to solve constrained optimization problems can roughly be divided into two categories: intelligent optimization algorithm and traditional optimization algorithm. The intelligent optimization algorithm is suitable for solving the multi-extremum optimization problems, which can be qualitatively analyzed but not quantitatively proved. Most of the traditional optimization algorithms belong to the category of convex optimization and have neat, global optimal solutions, and their convergence can be theoretically analyzed such as [5 –7].

As Problem (9) is a constrained convex optimization problem, we can design a traditional optimization algorithm to solve it. For this, we call on the Sequence Quadratic Programming (SQP) algorithm, which is currently recognized as one of the most effective methods for solving constrained nonlinear optimization problems. Compared to the other optimization algorithms, its obvious advantages are in its fast convergence, high computational efficiency, and strong boundary search ability, and these have been studied by many scholars and the references therein [37 –40].

This algorithm is developed following the Lagrange-Newton algorithm for the Kuhn-Tucker (KT) point in constraint optimization. Using the Karush-Kuhn-Tucker (KKT) conditions, the optimal solution and Lagrange multipliers of the original programming problem can be derived. From the Lagrangian of Problem (9), we obtain $L (z, λ) = θ (z) - λ^{⊤} g (z) .$

The KKT conditions can be expressed as

${\begin{matrix} \nabla θ (z^{*}) = λ^{* ⊤} \nabla g (z^{*}) \\ λ^{*} \geq 0, g (z^{*}) \geq 0, λ^{* ⊤} g (z^{*}) = 0 \end{matrix} .$

If the KKT conditions are established, then z^* is termed as the KT point of the constrained optimization problem, and λ^* is the optimal value of the Lagrange multipliers of the constrained optimization problem at point z^*.

Based on this, the quadratic programming subproblem of the inequality constrained optimization problem (9) is established:

$min \frac{1}{2} d^{⊤} W_{k} d + \nabla θ (z_{k})^{⊤} d$ (10) $s . t . g (z_{k}) + A_{k} d \geq 0,$

where $W_{k} = \nabla_{z}^{2} L (z_{k}, λ_{k}),$ A_k = ∇ g (z_k) .

The solution d_k of the quadratic programming subproblem is the search direction of Problem (9). According to the properties of Newton’s algorithm, the above algorithm has the property of local convergence. However, for an effective algorithm for nonlinear constrained optimization problems, it is not enough to have local convergence that we hope it has global convergence. To ensure the global convergence of the SQP algorithm and the descent of the search direction, we use the following value function to determine the stepsize of the search. $Φ (z_{k}, σ_{k}) = θ (z_{k}) + σ_{k} ∥ (g (z_{k}))_{-} ∥_{1},$ where (g (z_k)) _- = max {0, - g (z_k)} , $σ_{k} = {\begin{matrix} σ_{k - 1}, σ_{k - 1} \geq ∥ λ_{k} ∥_{\infty} + δ \\ ∥ λ_{k} ∥_{\infty} + δ, σ_{k - 1} < ∥ λ_{k} ∥_{\infty} + δ \end{matrix},$ and δ > 0 .

Next, we use the approximate matrix B_k of $\nabla_{z}^{2} L (z_{k}, λ_{k})$ to replace matrix W_k, and extend the correction formula of B_k+1 in [36] to ensure the stability of our algorithm.

Let s_k = α_kd_k, y_k = ∇ _zL (z_k+1, λ_k+1) - ∇ _zL (z_k, λ_k+1) , $B_{k + 1} = B_{k} + \frac{r_{k} r_{k}^{⊤}}{s_{k}^{⊤} r_{k}} - \frac{B_{k} s_{k} s_{k}^{⊤} B_{k}^{⊤}}{s_{k}^{⊤} B_{k} s_{k}},$ where r_k = ϱ_ky_k + (1 - ϱ_k) B_ks_k, $ϱ_{k} = {\begin{matrix} 1, s_{k}^{⊤} \geq τ s_{k}^{⊤} B_{k} s_{k} \\ \frac{(1 - τ) s_{k}^{⊤} B_{k} s_{k}}{s_{k}^{⊤} B_{k} s_{k} - s_{k}^{⊤} y_{k}}, s_{k}^{⊤} < τ s_{k}^{⊤} B_{k} s_{k} \end{matrix},$ and τ ∈ (0, 1) .

This then leads us to a new quadratic programming subproblem, $min \frac{1}{2} d^{⊤} B_{k} d + \nabla θ (z_{k})^{⊤} d$ (11) $s . t . g (z_{k}) + A_{k} d \geq 0,$

Thus, we obtain the modified SQP algorithm. The corresponding modified algorithm framework is as follows.

Algorithm 1: Modified SQP Algorithm (MSQP)

Step 0. Let (z₀) ∈ R^T(IJ+JK), λ₀ ∈ R^T(IJ+JK)+TJ . Choose parameters η, σ₀, ρ ∈ (0, 1) , ɛ₁, ɛ₂ ∈ [0, 1) , k = 0 .

Step 1. Compute d_k by solving Subproblem (11).

Step 2. If ∥d_k ∥ ₁ ≤ ɛ₁ and ∥ (g (z_k)) _- ∥ ₁ ≤ ɛ₂, Stop. We obtain an approximate KT point $(z_{k}^{*}, λ_{k}^{*})$ for Problem (9).

Step 3. Compute α_k = max {ρ^m|m = 0, 1, 2, ⋯} such that $Φ (z_{k} + ρ^{m} d_{k}, σ_{k}) \leq Φ (z_{k}, σ_{k})$ $+ η ρ^{m} D (Φ (z_{k}, σ_{k}), d_{k}),$ set z_k+1 = z_k + α_kd_k .

Step 4. Compute $A_{k + 1} = \nabla g (z_{k + 1})^{⊤}, λ_{k + 1} = (A_{k + 1}^{⊤})^{- 1} \nabla θ (z_{k + 1}) .$

Step 5. Update B_k to get B_k+1.

Step 6. Set k = k + 1 . Goto Step 1.

Remark 1. We use the Levenberg-Marquardt type Algorithm ([41]) to solve the subproblems.

The following theorem proves the feasibility of Algorithm 1.

Theorem 1. Let d_k and λ_k+1 be generated by solving subproblem (11). Then, the directional derivative of Φ in the direction d_k satisfies $D (Φ (z_{k}, σ_{k}), d_{k}) = \nabla θ (z_{k})^{⊤} d_{k} - σ_{k} ∥ (g (z_{k}))_{-} ∥_{1} .$ Furthermore, $\begin{matrix} D (Φ (z_{k}, σ_{k}), d_{k}) \leq - d_{k}^{⊤} B_{k} d_{k} \\ - (σ_{k} - ∥ λ_{k + 1} ∥_{\infty}) ∥ (g (z_{k}))_{-} ∥_{1} < 0, \end{matrix}$ i.e. d_k is a descent direction of the function Φ atz_k .

Proof. Applying Taylor^′s theorem to θ (z_k) and g (z_k), we have $\begin{matrix} Φ (z_{k} + α d_{k}) & = & θ (z_{k}) + α \nabla θ (z_{k})^{⊤} d_{k} \\ + & \frac{1}{2} α^{2} d_{k}^{⊤} \nabla^{2} θ (\in_{k}) d_{k}, \end{matrix}$ $\begin{matrix} g (z_{k} + α d_{k}) = g (z_{k}) + α A (z_{k}) d_{k} . \end{matrix}$

As ∥ · ∥ ₁ is a convex function and g (z_k) + A (z_k) d_k ≥ 0, we have $\begin{matrix} Φ (z_{k} + α d_{k}) - Φ (z_{k}) = θ (z_{k}) + α \nabla θ (z_{k})^{⊤} d_{k} \\ + \frac{1}{2} α^{2} d_{k}^{⊤} \nabla^{2} θ (\in_{k}) d_{k} + σ_{k} ∥ (g (z_{k}) + α A (z_{k}) d_{k})_{-} ∥_{1} \\ - θ (z_{k}) - σ_{k} ∥ (g (z_{k}))_{-} ∥_{1} \\ \geq α \nabla θ (z_{k})^{⊤} d_{k} + \frac{1}{2} α^{2} d_{k}^{⊤} \nabla^{2} θ (\in_{k}) d_{k} \\ + (1 - α) σ_{k} ∥ (g (z_{k}))_{-} ∥_{1} - σ_{k} ∥ (g (z_{k}))_{-} ∥_{1} \\ = α (\nabla θ (z_{k})^{⊤} d_{k} - σ_{k} ∥ (g (z_{k}))_{-} ∥_{1}) \\ + \frac{1}{2} α^{2} d_{k}^{⊤} \nabla^{2} θ (\in_{k}) d_{k} . \end{matrix}$ We note that g (z_k + αd_k) = g (z_k) + αA (z_k) d_k ≥ g (z_k) - αg (z_k) = (1 - α) g (z_k) , and (·) _- has the following properties, $t_{1} \geq t_{2}, (t_{1})_{-} \leq (t_{2})_{-} .$

Therefore, $\begin{matrix} Φ (z_{k} + α d_{k}) - Φ (z_{k}) \\ = θ (z_{k}) + α \nabla θ (z_{k})^{⊤} d_{k} + \frac{1}{2} α^{2} d_{k}^{⊤} \nabla^{2} θ (\in_{k}) d_{k} \\ + σ_{k} ∥ (g (z_{k} + α d_{k}))_{-} ∥_{1} - θ (z_{k}) - σ_{k} ∥ (g (z_{k}))_{-} ∥_{1} \\ \leq α \nabla θ (z_{k})^{⊤} d_{k} + \frac{1}{2} α^{2} d_{k}^{⊤} \nabla^{2} θ (\in_{k}) d_{k} \\ + (1 - α) σ_{k} ∥ (g (z_{k}))_{-} ∥_{1} - σ_{k} ∥ (g (z_{k}))_{-} ∥_{1} \\ = α (\nabla θ (z_{k})^{⊤} d_{k} - σ_{k} ∥ (g (z_{k}))_{-} ∥_{1}) \\ + \frac{1}{2} α^{2} d_{k}^{⊤} \nabla^{2} θ (\in_{k}) d_{k} . \end{matrix}$ Taking limits, we conclude that the directional derivative of Φ in the direction d_k is given by $D (Φ (z_{k}, σ_{k}), d_{k})$ (12) $= \nabla θ (z_{k})^{⊤} d_{k} - σ_{k} ∥ (g (z_{k}))_{-} ∥_{1} .$ From the KKT conditions of Subproblem (11), leads us to

${\begin{matrix} \nabla θ (z_{k}) + B_{k} d_{k} = \nabla g (z_{k})^{⊤} λ_{k + 1}, \\ λ_{k + 1} \geq 0, g (z_{k}) + \nabla g (z_{k})^{⊤} d_{k} \geq 0, \\ λ_{k + 1}^{⊤} (g (z_{k}) + \nabla g (z_{k})^{⊤} d_{k}) = 0 . \end{matrix}$ (13)

By (13), we have

$\nabla θ (z_{k})^{⊤} d_{k} = - d_{k}^{⊤} B_{k} d_{k} - g (z_{k})^{⊤} λ_{k + 1} .$ (14)

By Schwartz inequality, we have $- g (z_{k})^{⊤} λ_{k + 1} \leq λ_{k + 1}^{⊤} (g (z_{k}))_{-}$ (15) $\leq ∥ λ_{k + 1} ∥_{\infty} ∥ (g (z_{k}))_{-} ∥_{1} .$

Using (12), (14), (15), we obtain $\begin{matrix} D (Φ (z_{k}, σ_{k}), d_{k}) = \nabla θ (z_{k})^{⊤} d_{k} - σ_{k} ∥ (g_{k})_{-} ∥_{1} \\ = - d_{k}^{⊤} B_{k} d_{k} - g (z_{k})^{⊤} λ_{k + 1} - σ_{k} ∥ (g (z_{k}))_{-} ∥_{1} \\ \leq - d_{k}^{⊤} B_{k} d_{k} - (σ_{k} - ∥ λ_{k + 1} ∥_{\infty}) ∥ (g (z_{k}))_{-} ∥_{1} . \end{matrix}$

Using Step 5 of Algorithm 1, it is easy to show that $s_{k}^{⊤} z_{k} \geq τ s_{k}^{⊤} B_{k} s_{k}$ , From [36], we know that, when B_k is positive, B_k+1 is positive. Through Step 3 of Algorithm 1, we have σ_k ≥ ∥ λ_k+1 ∥ _∞ . So, D (Φ (z_k, σ_k) , d_k) <0 .

The global convergence result of the above algorithm is as follows.

Theorem 2. Suppose θ (z) and g (z) are continuously differentiable,and that there exist two positive constants m, M such that $m ∥ d ∥^{2} \leq d^{⊤} B_{k} d \leq M ∥ d ∥^{2}$ holds for all k and d ∈ R^T(IJ+JK). If σ_k ≥ ∥ λ_k ∥ _∞ for all k, then any accumulation point of z_k generated by Algorithm 1 is a KKT point of (9).

Proof. The proof is similar to the global convergence result of the method in [42].

5 Numerical results

In this section, we apply the MSQP algorithm to solve a numerical example and to provide a discussion of the results.

All the program codes are written on MATLAB R2014a running an ACER computer with 2GB DDR3 memory. Throughout the computational experiments, the parameters in Algorithm 1 are set as: η = 0.1, σ₀ = 0.8, ρ = 0.5, τ = 0.3, δ = 0.05, ɛ₁ = 10^-6, ɛ₂ = 10^-5 .

The stopping criterion for the algorithm are ∥d_k ∥ ₁ ≤ 10^-6 and ∥ (g_k) _- ∥ ₁ ≤ 10^-5, or k_max = 100 .

In the respective tables of the numerical results, $π_{m}^{*}$ denotes the manufacturers’ profit in two periods, $π_{r}^{*}$ denotes the retailers’ profit in two periods, V^* denotes the total distribution supply chain’s profit, where $π^{*} = π_{m}^{*} + π_{r}^{*}$ , z^* denotes the optimal order quantity and sales volume, with $z^{*} = (q_{111}^{*}, q_{112}^{*}, q_{121}^{*}, q_{122}^{*},$ $a_{111}^{*}, a_{112}^{*}, a_{121}^{*}, a_{122}^{*},$ $q_{211}^{*}, q_{212}^{*}, q_{221}^{*}, q_{222}^{*},$ $a_{211}^{*},$ $a_{212}^{*},$ $a_{221}^{*},$ $a_{222}^{*}$ ). Itr denotes the number of iterations, TI denotes the computing time in seconds. In what follows, we provide a description of the test problem.

The example considers a two-period, two-manufacturer, two-retailer, and two-consumer synthetic problem, as shown in Figure 2. The data for this example are constructed for the purpose of easy interpretation. The production cost functions for the manufacturers are given by: $f_{11} (q) = 10 (q_{111} + q_{112}), f_{12} (q) = 10.5 (q_{121} + q_{122}),$ $f_{21} (q) = 11 (q_{211} + q_{212}), f_{22} (q) = 12 (q_{221} + q_{222}) .$

Fig.2

Distribution supply chain network structure of numerical example.

The transportation cost functions from the manufacturers to the retailers are given by:

$e_{111} (q_{111}) = 0.005 q_{111}^{2} + 1.5 q_{111},$

$e_{112} (q_{112}) = 0.004 q_{112}^{2} + 1.5 q_{112},$

$e_{121} (q_{121}) = 0.005 q_{121}^{2} + 1.5 q_{121},$

$e_{122} (q_{122}) = 0.004 q_{122}^{2} + 1.5 q_{122},$

$e_{211} (q_{211}) = 0.005 q_{211}^{2} + 1.5 q_{211},$

$e_{212} (q_{212}) = 0.004 q_{212}^{2} + 1.5 q_{212},$

$e_{221} (q_{221}) = 0.005 q_{221}^{2} + 1.5 q_{221},$

$e_{222} (q_{222}) = 0.004 q_{222}^{2} + 1.5 q_{222} .$

The inventory costs of the retailers are given by: $c_{11} = 0.5, c_{12} = 0.6, c_{21} = 0.5, c_{22} = 0.6 .$

The wholesale prices of the unit product of these manufacturers are given by: $p_{11} = 18, p_{12} = 18, p_{21} = 18, p_{22} = 18 .$

The retail prices of the unit product of the retailers are given by: $b_{11} = 24, b_{12} = 24, b_{21} = 24, b_{22} = 24 .$

The fuzzy demands of consumers are shown in Table 2.

Table 2

Consumers’ fuzzy demand

	T=1		T=2
	j=1	j=2	j=1	j=2
k=1	$(\frac{2500}{p_{211}} - 1, \frac{2500}{p_{211}}, \frac{2500}{p_{211}} + 1)$	$(\frac{2400}{p_{212}} - 1, \frac{2400}{p_{212}}, \frac{2400}{p_{212}} + 1)$	$(\frac{2600}{p_{221}} - 1, \frac{2600}{p_{221}}, \frac{2600}{p_{221}} + 1)$	$(\frac{2500}{p_{222}} - 1, \frac{2500}{p_{222}}, \frac{2500}{p_{222}} + 1)$
k=2	$(\frac{2600}{p_{211}} - 1, \frac{2600}{p_{211}}, \frac{2600}{p_{211}} + 1)$	$(\frac{2700}{p_{212}} - 1, \frac{2700}{p_{212}}, \frac{2700}{p_{212}} + 1)$	$(\frac{2700}{p_{221}} - 1, \frac{2700}{p_{221}}, \frac{2700}{p_{221}} + 1)$	$(\frac{2800}{p_{222}} - 1, \frac{2800}{p_{222}}, \frac{2800}{p_{222}} + 1)$

The credibility density function of a triangular fuzzy variable ξ = (a, b, c) is as follows: $φ (x) = {\begin{matrix} \frac{1}{2 (b - a)}, a \leq x \leq b \\ \frac{1}{2 (c - b)}, b \leq x \leq c \\ 0, otherwise . \end{matrix}$

We test this problem with z₀ = (1, 1, ⋯ , 1) ^⊤ and λ₀ = (0, 0, ⋯ , 0) ^⊤ as the initial point. The numerical results are given in Tables 3, 4, 5, 6, 7, 8, and Figure 3, respectively.

Fig.3

Numerical results for distribution supply chain problem using Algorithm 1 and fmincon.

Table 3

Profit of distribution supply chain under fuzzy demand and expected demand

Demand	z*	π_m*	π_r*	π*
	(131.4128, 137.6929, 81.4128, 75.1929,
Fuzzy	104.3294, 108.4961, 100.1929, 112.6929,	4.5183e+03	5.1280e+03	9.6464e+03
	160.5447, 173.0794, 60.5447, 48.0794,
	108.4614, 112.6280, 104.3294, 116.8294)
	(131.2506, 137.4991, 81.2494, 75.0009,
Expected	104.1667, 108.3333, 100.0000, 112.5000,	4.5122e+03	5.0500e+03	9.5622e+03
	160.4164, 172.9170, 60.4169, 47.9163,
	108.3333, 112.5000, 104.1667, 116.6667)

Table 4

Comparison of Algorithm 1, fmincon, and Genetic Algorithm

Algorithm	z ^∗	λ ^∗	$π_{m}^{*}$	$π_{r}^{*}$	π ^∗	Itr	TI
	(131.4128, 137.6929, 81.4128, 75.1929,	(12.8141, 12.6015,
Algorithm 1	104.3294, 108.4961, 100.1929, 112.6929,	14.1054, 13.8846,	4.5183e+03	5.1280e+03	9.6464e+03	33	5.0270
	160.5447, 173.0794, 60.5447, 48.0794,	0, 0, 0, 0, 0, 0,
	108.4614, 112.6280, 104.3294, 116.8294)	0, 0, 0, 0, 0, 0)
	(131.4127, 137.6928, 81.4127, 75.1928,	(12.8141, 12.6015,
fmincon	104.3293, 108.4960, 100.1928, 112.6928,	14.1054, 13.8846,	4.5183e+03	5.1280e+03	9.6464e+03	38	5.4080
	160.5447, 173.0795, 60.5447, 48.0793,	0, 0, 0, 0,0, 0,
	108.4614, 112.6280, 104.3294, 116.8294)	0, 0, 0, 0, 0, 0)
	(106.5017, 94.8145, 106.9064, 119.0556,
Genetic	104.8540, 108.5512, 100.5811, 113.2837,	//	4.4540e+03	5.1091e+03	9.5631e+03	564	82.9734
Algorithm	111.6629, 112.4563, 109.9609, 108.7687,
	108.5802, 113.0442, 104.3753, 116.8481)

Table 5

Effect of retail price on distribution supply chain profit

(b₁₁, b₁₂, b₂₁, b₂₂)	z ^∗	$π_{m}^{*}$	$π_{r}^{*}$	π ^∗
(21, 24, 24, 24)	(146.4976, 137.6929, 96.4976, 75.1929, 119.1166, 123.8785, 100.1929, 112.6929,	4.6725e+03	4.5837e+03	9.2563e+03
	160.5447, 173.0794, 60.5447, 48.0794, 108.4614, 112.6280, 104.3294, 116.8294)
(22, 24, 24, 24)	(141.0117, 137.6929, 91.0117, 75.1929, 113.7389, 118.2844, 100.1929, 112.6929,	4.6170e+03	4.7817e+03	9.3987e+03
	160.5447, 173.0794, 60.5447, 48.0794, 108.4614, 112.6280, 104.3294, 116.8294)
(23, 24, 24, 24)	(136.0033, 137.6929, 86.0033, 75.1929, 108.8294, 113.1772, 100.1929, 112.6929,	4.5657e+03	4.9624e+03	9.5282e+03
	160.5447, 173.0794, 60.5447, 48.0794, 108.4614, 112.6280, 104.3294, 116.8294)
(24, 24, 24, 24)	(131.4128, 137.6929, 81.4128, 75.1929, 104.3294, 108.4961, 100.1929, 112.6929,	4.5183e+03	5.1280e+03	9.6464e+03
	160.5447, 173.0794, 60.5447, 48.0794, 108.4614, 112.6280, 104.3294, 116.8294)
(24, 23, 24, 24)	(131.4128, 142.2860, 81.4128, 79.7859, 104.3294, 108.4961, 104.5142, 117.5577,	4.5678e+03	4.9624e+03	9.5301e+03
	160.5447, 173.0794, 60.5447, 48.0794, 108.4614, 112.6280, 104.3294, 116.8294)
(24, 22, 24, 24)	(131.4128, 147.2971, 81.4128, 84.7971, 104.3294, 108.4961, 109.2289, 122.8653,	4.6213e+03	4.7816e+03	9.4029e+03
	160.5447, 173.0794, 60.5447, 48.0794, 108.4614, 112.6280, 104.3294, 116.8294)
(24, 21, 24, 24)	(131.4127, 152.7862, 81.4129, 90.2860, 104.3295, 108.4961, 114.3933, 128.6790,	4.6795e+03	4.5836e+03	9.2631e+03
	160.5446, 173.0796, 60.5448, 48.0793, 108.4614, 112.6280, 104.3294, 116.8294)
(24, 24, 21, 24)	(131.4128, 137.6929, 81.4128, 75.1929, 104.3294, 108.4961, 100.1929, 112.6929,	4.6380e+03	4.5622e+03	9.2002e+03
	176.2224, 173.0795, 76.2225, 48.0794, 123.8415, 128.6034, 104.3294, 116.8294)
(24, 24, 22, 24)	(131.4128, 137.6929, 81.4128, 75.1929, 104.3294, 108.4961, 100.1929, 112.6929,	4.5951e+03	4.7680e+03	9.3630e+03
	170.5209, 173.0794, 70.5209, 48.0794, 118.2482, 122.7936, 104.3294, 116.8294)
(24, 24, 23, 24)	(131.4128, 137.6929, 81.4128, 75.1929, 104.3294, 108.4961, 100.1929, 112.6929,	4.5553e+03	4.9558e+03	9.5111e+03
	165.3157, 173.0794, 65.3157, 48.0794, 113.1417, 117.4896, 104.3294, 116.8294)

Table 6

Effect of retail price on distribution supply chain profit

(b₁₁, b₁₂, b₂₁, b₂₂)	z ^∗	$π_{m}^{*}$	$π_{r}^{*}$	π ^∗
(24, 24, 24, 24)	(131.4128, 137.6929, 81.4128, 75.1929, 104.3294, 108.4961, 100.1929, 112.6929,	4.5183e+03	5.1280e+03	9.6464e+03
	160.5447, 173.0794, 60.5447, 48.0794, 108.4614, 112.6280, 104.3294, 116.8294)
(24, 24, 24, 23)	(131.4128, 137.6929, 81.4128, 75.1929, 104.3294, 108.4961, 100.1929, 112.6929,	4.5575e+03	4.9558e+03	9.5133e+03
	160.5447, 177.8529, 60.5447, 52.8529, 108.4614, 112.6280, 108.8312, 121.8747)
(24, 24, 24, 22)	(131.4128, 137.6929, 81.4128, 75.1929, 104.3294, 108.4961, 100.1929, 112.6929,	4.5997e+03	4.7679e+03	9.3676e+03
	160.5447, 183.0610, 60.5447, 58.0610, 108.4614, 112.6280, 113.7428, 127.3792)
(24, 24, 24, 21)	(131.4128, 137.6929, 81.4128, 75.1929, 104.3294, 108.4961, 100.1929, 112.6929,	4.6455e+03	4.5621e+03	9.2076e+03
	160.5447, 188.7658, 60.5447, 63.7658, 108.4614, 112.6280, 119.1229, 133.4086)
(22, 22, 24, 24)	(141.0117, 147.2971, 91.0116, 84.7970, 113.7389, 118.2844, 109.2289, 122.8653,	4.7199e+03	4.4352e+03	9.1552e+03
	160.5448, 173.0795, 60.5446, 48.0793, 108.4614, 112.6280, 104.3294, 116.8294)
(23, 23, 24, 24)	(136.0033, 142.2859, 86.0033, 79.7859, 108.8294, 113.1772, 104.5142, 117.5577,	4.6152e+03	4.7967e+03	9.4119e+03
	160.5447, 173.0794, 60.5447, 48.0794, 108.4614, 112.6280, 104.3294, 116.8294)
(24, 24, 23, 23)	(131.4128, 137.6929, 81.4128, 75.1929, 104.3294, 108.4961, 100.1929, 112.6929,	4.5944e+03	4.7836e+03	9.3780e+03
	165.3157, 177.8530, 65.3156, 52.8529, 113.1417, 117.4896, 108.8312, 121.8747)
(24, 24, 22, 22)	(131.4128, 137.6929, 81.4128, 75.1929, 104.3294, 108.4961, 100.1929, 112.6929,	4.6764e+03	4.4078e+03	9.0842e+03
	170.5209, 183.0610, 70.5209, 58.0610, 118.2482, 122.7936, 113.7428, 127.3792)
(22, 22, 22, 22)	(141.0117, 147.2971, 91.0117, 84.7971, 113.7389, 118.2844, 109.2289, 122.8653,	4.8780e+03	3.7150e+03	8.5930e+03
	170.5209, 183.0610, 70.5209, 58.0610, 118.2482, 122.7936, 113.7428, 127.3792)
(23, 23, 23, 23)	(136.0033, 142.2859, 86.0033, 79.7859, 108.8294, 113.1772, 104.5142, 117.5577,	4.6912e+03	4.4523e+03	9.1435e+03
	165.3157, 177.8530, 65.3157, 52.8529, 113.1417, 117.4896, 108.8312, 121.8747)

Table 7

Effect of wholesale price on manufacturers’ and retailers’ profit

(p₁₁, p₁₂, p₂₁, p₂₂)	$π_{m}^{*}$	$π_{r}^{*}$	π ^∗
(17, 18, 18, 18)	4.2492e+03	5.3972e+03	9.6464e+03
(18, 18, 18, 18)	4.5183e+03	5.1280e+03	9.6464e+03
(19, 18, 18, 18)	4.7875e+03	4.8589e+03	9.6464e+03
(18, 17, 18, 18)	4.3617e+03	5.2847e+03	9.6464e+03
(18, 19, 18, 18)	4.6750e+03	4.9714e+03	9.6464e+03
(18, 18, 17, 18)	4.1847e+03	5.4617e+03	9.6464e+03
(18, 18, 19, 18)	4.8520e+03	4.7944e+03	9.6464e+03
(18, 18, 18, 17)	4.4097e+03	5.2367e+03	9.6464e+03
(18, 18, 18, 19)	4.6270e+03	5.0194e+03	9.6464e+03
(17, 17, 18, 18)	4.0926e+03	5.5538e+03	9.6464e+03
(18, 18, 18, 18)	4.5183e+03	5.1280e+03	9.6464e+03
(19, 19, 18, 18)	4.9441e+03	4.7023e+03	9.6464e+03
(18, 18, 17, 17)	4.0761e+03	5.5703e+03	9.6464e+03
(18, 18, 19, 19)	4.9606e+03	4.6858e+03	9.6464e+03
(17, 18, 17, 18)	3.9156e+03	5.7308e+03	9.6464e+03
(19, 18, 19, 18)	5.1211e+03	4.5253e+03	9.6464e+03
(18, 17, 18, 17)	4.2531e+03	5.3933e+03	9.6464e+03
(18, 19, 18, 19)	4.7836e+03	4.8628e+03	9.6464e+03
(17, 17, 17, 17)	3.6504e+03	5.9960e+03	9.6464e+03
(19, 19, 19, 19)	5.3863e+03	4.2601e+03	9.6464e+03

Table 8

Effect of inventory cost on distribution supply chain profit

(c₁₁, c₁₂, c₂₁, c₂₂)	z ^∗	$π_{m}^{*}$	$π_{r}^{*}$	π ^∗
(0.5, 0.6, 0.5, 0.6)	(131.4128, 137.6929, 81.4128, 75.1929, 104.3294, 108.4961, 100.1929, 112.6929,	4.5183e+03	5.1280e+03	9.6464e+03
	160.5447, 173.0794, 60.5447, 48.0794, 108.4614, 112.6280, 104.3294, 116.8294)
(0.6, 0.6, 0.6, 0.6)	(131.4058, 137.6929, 81.4058, 75.1929, 104.3224, 108.4891, 100.1929, 112.6929,	4.5182e+03	5.1279e+03	9.6462e+03
	160.5414, 173.0794, 60.5413, 48.0794, 108.4580, 112.6247, 104.3294, 116.8294)
(0.7, 0.6, 0.7, 0.6)	(131.3989, 137.6929, 81.3989, 75.1929, 104.3155, 108.4822, 100.1929, 112.6929,	4.5182e+03	5.1279e+03	9.6461e+03
	160.5380, 173.0794, 60.5380, 48.0794, 108.4547, 112.6213, 104.3294, 116.8294)
(0.5, 0.7, 0.5, 0.7)	(131.4128, 137.6860, 81.4128, 75.1860, 104.3294, 108.4961, 100.1860, 112.6860,	4.5182e+03	5.1279e+03	9.6462e+03
	160.5447, 173.0761, 60.5447, 48.0761, 108.4614, 112.6280, 104.3261, 116.8261)
(0.5, 0.8, 0.5, 0.8)	(131.4128, 137.6791, 81.4128, 75.1791, 104.3294, 108.4961, 100.1791, 112.6791,	4.5181e+03	5.1278e+03	9.6460e+03
	160.5447, 173.0727, 60.5447, 48.0727, 108.4614, 112.6280, 104.3227, 116.8227)
(0.6, 0.7, 0.6, 0.7)	(131.4058, 137.6860, 81.4058, 75.1860, 104.3224, 108.4891, 100.1860, 112.6860,	4.5181e+03	5.1278e+03	9.6460e+03
	160.5413, 173.0761, 60.5413, 48.0761, 108.4580, 112.6247, 104.3261, 116.8261)
(0.7, 0.8, 0.7, 0.8)	(131.3989, 137.6791, 81.3989, 75.1791, 104.3155, 108.4822, 100.1791, 112.6791,	4.5179e+03	5.1276e+03	9.6456e+03
	160.5380, 173.0727, 60.5380, 48.0727, 108.4547, 112.6213, 104.3227, 116.8227)

In Table 3, we present the profit of each member of the distribution supply chain under expected and fuzzy demand, where the expected demand is equal to the value of the center point of the fuzzy demand. With fuzzy demand, the manufacturers’ profit, retailers’ profit, and the distribution supply chain profit are all greater than the profit under the expected demand.

Next, Table 4 provides a comparison of the numerical results for Algorithm 1, MATLAB’s Genetic Algorithm (GA) toolbox and fmincon (fmincon is a MATLAB toolbox for constrained optimization). We use the SQP algorithm in the fmincon toolbox to solve this example. Using Algorithm 1, the optimal order quantity $q_{111}^{*} + q_{121}^{*} + q_{211}^{*} + q_{221}^{*}$ and optimal sales volume $a_{111}^{*} + a_{112}^{*} + a_{211}^{*} + a_{212}^{*}$ for retailer 1 are 433.9150 and 433.9149, respectively. The optimal order quantity $q_{112}^{*} + q_{122}^{*} + q_{212}^{*} + q_{222}^{*}$ and optimal sales volume $a_{121}^{*} + a_{122}^{*} + a_{221}^{*} + a_{222}^{*}$ for retailer 2 are 434.0446 and 434.0446, respectively. Furthermore, the manufacturers’ profit $π_{m}^{*}$ is 4.5183e + 03, the retailers’ profit $π_{r}^{*}$ is 5.1280e + 03, and the distribution supply chain’s profit π^* is 9.6464e + 03. The shadow price at optimality is λ^* = (12.8141, 12.6015, 14.1054, 13.8846, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). The profit result using fmincon is the same as that of Algorithm 1, albeit the number of iterations needed to arrive at optimality exceeds that of Algorithm 1. With GA, the optimal order quantity and optimal sales volume of retailer 1 are 435.0319 and 435.0296, respectively. Likewise, the optimal order quantity and optimal sales volume of retailer 2 are 435.0951 and 435.0886, respectively. Similarly, the manufacturers’ profit $π_{m}^{*}$ is 4.4540e + 03, the retailers’ profit $π_{r}^{*}$ is 5.1091e + 03, and the distribution supply chain’s profit π^* is 9.5631e + 03. This profit figure is less than that of Algorithm 1, while the number of iterations and computing time are much more than that of Algorithm 1. In sum, from the analysis and Figure 3, Algorithm 1 requires fewer iterations to converge and it converges faster, suggesting that the Algorithm 1 is better suited for optimizing the distribution supplychain.

Next, we consider the effect of wholesale price, retail price, and inventory cost on the optimal strategy respectively.

Tables 5 and 6 show the effect of retail price on the optimal strategy. The retailer’s optimal order quantity $q_{111}^{*} + q_{121}^{*} + q_{211}^{*} + q_{221}^{*}$ $(q_{112}^{*} + q_{122}^{*} + q_{212}^{*} + q_{222}^{*})$ and optimal sales volume $a_{111}^{*} + a_{112}^{*} + a_{211}^{*} + a_{212}^{*}$ $(a_{121}^{*} + a_{122}^{*} + a_{221}^{*} + a_{222}^{*})$ decrease with the increase in retail price b₁₁ (or b₁₂ or b₂₁ or b₂₂). For instance, in Table 5, when b₁₂, b₂₁, b₂₂ are unchanged and b₁₁ increases from 21 to 24, the retailer’s optimal order quantity $q_{111}^{*} + q_{121}^{*} + q_{211}^{*} + q_{221}^{*}$ decreases from 464.0846 to 433.9150, and optimal sales volume $a_{111}^{*} + a_{112}^{*} + a_{211}^{*} + a_{212}^{*}$ also decreases from 464.0845 to 433.9149. Further, the manufacturer’s profit $π_{m}^{*}$ decreases from 4.6725e + 03 to 4.5183e + 03, but the retailer’s profit $π_{r}^{*}$ increases from 4.5837e + 03 to 5.1280e + 03, and the distribution supply chain’s profit π^* also increases from 9.2563e + 03 to 9.6464e + 03. In Table 6, likewise, when (b₁₁, b₁₂, b₂₁, b₂₂) increases from (22, 22, 22, 22) to (24, 24, 24, 24), the retailer’s optimal order quantity $q_{111}^{*} + q_{121}^{*} + q_{211}^{*} + q_{221}^{*}$ decreases from 473.0652 to 433.9150, and optimal sales volume $a_{111}^{*} + a_{112}^{*} + a_{211}^{*} + a_{212}^{*}$ also decreases from 473.0651 to 433.9149. Besides, the manufacturer’s profit $π_{m}^{*}$ decreases from 4.8780e + 03 to 4.5183e + 03, but the retailer’s profit $π_{r}^{*}$ increases from 3.7150e + 03 to 5.1280e + 03, and the distribution supply chain’s profit π^* increases from 8.5930e + 03 to 9.6464e + 03.

In this synthetic example, the retailer’s optimal sales volume is near the fuzzy center. For example, when b₁₁ = 21, b₁₂ = 24, b₂₁ = 24, b₂₂ = 24, the retailer’s optimal sales volume $a_{111}^{*}$ , $a_{112}^{*}$ , $a_{121}^{*}$ , $a_{122}^{*}$ , $a_{211}^{*}$ , $a_{212}^{*}$ , $a_{221}^{*}$ , $a_{222}^{*}$ is approximately equal to the values of the fuzzy demand center points $\frac{2500}{b_{11}}$ , $\frac{2600}{b_{11}}$ , $\frac{2400}{b_{12}}$ , $\frac{2700}{b_{12}}$ , $\frac{2600}{b_{21}}$ , $\frac{2700}{b_{21}}$ , $\frac{2500}{b_{22}}$ , $\frac{2800}{b_{22}}$ , respectively. This shows that the retailer’s order quantity and sales volume are closely related to the retail price, and similarly, and correspond better to the estimated value of the fuzzy demand parameter. The retailers can influence consumer demand by adjusting their prices in the face of consumer demand uncertainty. As retail prices increase, the corresponding consumer demand will decrease, but the distribution supply chain profit will increase. In addition, Table 7 shows the effect of wholesale price on the manufacturers’ profit and retail’s profit. We note that the manufacturers can increase their profits by increasing the wholesale price. At the same time, the corresponding retailer’s profit will decrease. However, the distribution supply chain profit will not change. In this example, the manufacturers and retailers achieve their desired profits by setting the wholesale prices. Although the wholesale price can affect the profitability of the manufacturers and retailers, it can nevertheless assure income stability for the manufacturers and retailers. This could possibly point to the stronger stability for the distribution supply chain model established in this paper.

Table 8 shows the effect of inventory cost on the optimal strategy. The retailer’s optimal order quantity $q_{111}^{*} + q_{121}^{*} + q_{211}^{*} + q_{221}^{*}$ $(q_{112}^{*} + q_{122}^{*} + q_{212}^{*} + q_{222}^{*})$ and optimal sales volume $a_{111}^{*} + a_{112}^{*} + a_{211}^{*} + a_{212}^{*}$ $(a_{121}^{*} + a_{122}^{*} + a_{221}^{*} + a_{222}^{*})$ decrease with the increase in inventory cost c₁₁, c₂₁ (or c₁₂, c₂₂). For example, when c₁₂, c₂₂ are unchanged and c₁₁, c₂₁ increases from 0.5 to 0.7, the optimal order quantity of retailer 1 declines from 433.9150 to 433.8738, and the optimal sales volume also declines from 433.9149 to 433.8737. Likewise, the manufacturer’s profit $π_{m}^{*}$ decreases from 4.5183e + 03 to 4.5182e + 03, the retailer’s profit $π_{r}^{*}$ decreases from 5.1280e + 03 to 5.1279e + 03, and the distribution supply chain’s profit π^* also decreases from 9.6464e + 03 to 9.6461e + 03. When (c₁₁, c₁₂, c₂₁, c₂₂) increases from (0.5, 0.6, 0.5, 0.6) to (0.7, 0.8, 0.7, 0.8), the optimal order quantity of retailer 1 decreases from 433.9150 to 433.8738, and the optimal sales volume of the retailer also decreases from 433.9149 to 433.8737. The manufacturer’s profit $π_{m}^{*}$ decreases from 4.5183e + 03 to 4.5179e + 03, the retailer’s profit $π_{r}^{*}$ decreases from 5.1280e + 03 to 5.1276e + 03, and the distribution supply chain’s profit π^* also decreases from 9.6464e + 03 to 9.6456e + 03. In short, the increase in inventory cost has reduced the order quantity and sales volume; the profits of the manufacturers, retailers, and the distribution supply chain have also decreased. This shows that, in the distribution supply chain optimization problem, it is necessary to reduce the inventory cost. The retailers can reduce the inventory cost by reducing the inventory cycle and inventory quantity.

6 Conclusion

This paper has proposed an MSQP algorithm based model to analyze the distribution supply chain coordination mechanism through the credibility distribution of fuzzy variables using uncertainty theory, in the context of a fuzzy demand environment. Through numerical experimentation, the optimal order quantity and optimal sales quantity of the retailers are obtained. Furthermore, the maximum expected return of the manufacturers, retailers, and distribution supply chain can also be obtained, and the effect of the wholesale price, retail price, and inventory cost on the distribution supply chain entities are analyzed. Our numerical analysis suggests that retailers can influence consumer demand by adjusting the retail prices. As retail prices increase, the corresponding consumer demand will decrease, but the distribution supply chain profits will increase. The manufacturers and retailers can set the wholesale price to meet their desired profit. At the same time, the retailers can increase the distribution supply chain profit by lowering their inventory costs. The computational result shows that MSQP algorithm is better suited to a distribution supply chain optimization problem under fuzzy consumer demand.

Future research can consider the uncertainty of supply, demand, and lead-time in the supply chain simultaneously, seek to investigate the interaction among the various forms of uncertainty, and their effects on the supply chain system performance. Indeed, focusing on the supply side of the supply chain would serve as a natural extension to this paper. Finally, we can also examine the robustness of the distribution supply chain based on the structural reliability and flexibility of the supply chain under fuzzy uncertainty.

Footnotes

Acknowledgment

The work is supported by a research grant from the Natural Scientific Foundation of China (No. 71571055). The authors thank the editor and the reviewers for their highly constructive comments on the manuscript.

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