Distribution network design of a decentralized supply chain with fuzzy committed distributors

Abstract

We develop a distribution network model for a decentralized supply chain to investigate the optimal decisions of participants in an uncertain demand environment. The supply chain consists of one manufacturer and set of independent candidate distributors that distribute products to geographically dispersed markets in a competitive manner. All chain’s participants are assumed risk-sensitive. The stability of a distribution network depends on the fuzzy commitment of its distributors. The novelty of the paper lies in the manufacturer concerns about the preferences of the distributors to ensure stability of his distribution network. Hence, the manufacturer considers criteria of expected profit, variance of profit, and stability level to evaluate distribution alternatives. A decision-making methodology is presented for evaluation and selection the distribution network design alternatives. The tactical decisions in each distribution alternative are obtained from a Stackelberg game formulated as a bi-level distributed programming problem. We show that the methodology is able to capture a principal trade-off between the preferences of manufacturer and contractor distributors. Managerial insights are obtained through an illustrative example and its sensitivity analyses.

Keywords

Game theory decentralized supply chain bi-level distributed programming distribution network design demand uncertainty

1 Introduction

Conventionally, a supply chain (SC) is composed of a broad variety of collaborative agreements and contracts among independent firms, which integrates them as collaborative networks [1]. Therefore, an SC is often supposed as a decentralized decision making system consists of several independent firms [2]. These firms pursue conflicting objectives respecting production, purchasing, inventory, transportation, and marketing contexts. From the viewpoint of system management, the decisions of SC’s participants can be categorized into three levels of strategic, tactical, and operational decisions [3]. The decisions of categories are distinctive according to their frequency and the time frame during which these decisions should be taken. Long-range decisions such as distribution network configuration are high level problems which need to be planned at the strategic level. Distribution network configuration deals with the selection of warehouses (distributors) locations and capacities specifying the production level of each product at each plant so as to minimize production, transportation, and inventory costs [4].

In the real distribution network configuration, the manufacturer is often faced with a number of candidate distributors to supply various geographically dispersed markets. He regularly has the authority to select the best subset of these distributors to make a long-term contract with them. That is, the manufacturer tries to build a stable coalition with these contractor distributors. Different contractor distributors lead to various configurations of distribution network design (DND). However, the manufacturer should meticulously consider that an increase in the number of distributors that distribute uniform products to common markets, raises the competition level among them in those markets. The best DND should be selected by analyzing the possible competition among the correspondingdistributors.

The conventional researches on distribution network configuration have often focused on the centralized decision making. However, the real world distribution problems regularly conform to the decentralized SC where participants pursue their own objectives. The independent essence of the distributors adds an additional dimension to the relationship between the manufacturer and distributors in each coalition. In general, according to Gale and Shapley theory [5], the “stability” of a coalition depends on transferable utility which incentivize individuals to make long-term commitments to the coalition. Thus, in a decentralized SC, the manufacturer should concern overall distributors’ objectives to insure their long-term commitment. On the other hand, the distributors should also regard the manufacturer objective, seeing that the greedy distributors may be substituted in the final distribution network. To model such interactions, the game theory approach can be an ideal technique.

We formulate each alternative of DND as a bi-level distributed programming problem (BLDPP). A BLDPP is a hieratical decision making problem where one decision center at the top level first makes the decision and then several divisions or decision centers at the bottom level take corresponding decisions with regard to top level’s decision [6]. The lower level divisions (or followers) are assumed independent and under control of the upper level.

In the proposed model, the manufacturer first determines a subset of contractor distributors to form a coalition. Afterwards, in the obtained DND alternative, the tactical decisions are taken according to BLDPP procedural. That is, the manufacturer as a Stackelberg leader makes the decision first, and then the set of contractor distributors as Stackelberg followers should take decisions regarding the manufacturer’s decision. Having analyzed possible sets of the followers (possible coalitions), the manufacturer will be able to contract with a set of distributors which maximizes his utility. Consequently, our model covers two additional aspects regarding the traditional BLDPP. First, the distributors (followers) compete for the markets; hence, their decision may not be independent. Second, the manufacturer (leader) has an authority to choose his followers such that his objectives become optimized. The objective of the present study is to model fuzzy commitment of distributers in order to guarantee the stability of DND. We suggest a decision-making methodology for evaluation and selection of optimal DND alternatives. The suggested methodology is systematic and comprehensive to be utilized in the real DND problems with independent decision makers.

The paper is organized as follows. The relevant literature is reviewed in Section 2. Section 3 consists of a discussion of the problem and its notations. Section 4 formulates our general Stackelberg model as a BLDPP. The methodology of DND selection is given in Section 5. Section 6 presents some computational results and their sensitive analyses in order to draw some meaningful managerial insights. Eventually, Section 7 presents the conclusions and several directions for future research.

2 Literature review

This research is directly related to price–service–advertising competition, competitive facility location and design problem (CFLDP), and risk management. In the economics literature, price and service have been regarded as two important factors influencing the buying decisions of customers [7 –12]. Iyer [8] modeled an SC consisting of two competing retailers that received products from a manufacturer. Similarly, Tsay and Agrawal [7] showed that the intensity of competition regarding price or service factors has a key role, as does the degree of cooperation between two rival retailers. Besides price and service, the advertising factor has also been regarded as a decisive factor affecting the buying decisions of customers [13, 14]. Gasmi et al. [13] studied customers’ demands of Coca-Cola and Pepsi drinks which were influenced by price and advertising. Similar to Hafezalkotob et al. [14], we assume that markets’ demands are contingent upon the prices and services and also the advertising effort.

Various industries such as publishing, music, electronics, and pharmaceutical businesses perform in a deeply uncertain environment. Hence, it is often supposed that the retailers are confronted with a stochastic or fuzzy demand in the operational and marketing fields [15 –19]. Uncertainty in demand causes a rugged trend in the companies’ profit. As a result, firms often take some degree of risk aversion attitude toward the profit uncertainty. Tsay [15] studied how risk sensitivity affects the relationship between a manufacturer and his retailer under different scenarios of strategic power. Gan et al. [20] shed light on the possibility of coordination in SCs with risk-averse agents. Xiao and Yang [9] showed that the risk sensitivity of one retailer towards demand uncertainty greatly influences his decisions such as pricing, purchasing, and service investment. Yang et al. [10] studied the effect of risk sensitivity of one retailer on the lot size decisions of rival retailer. Subulan et al. [21] and Ghezavati and Mrakabatchian [22] focused on reverse logistics and DND. Ghezavati and Mrakabatchian [22] showed that waste collection system is improved by considering fuzzy customer satisfaction level during the optimization process.

The negative consequences of demand uncertainty often extend across echelons of an SC and they may be intensified due to bullwhip effect. For example, Dell (device manufacturer) and Intel (processors supplier) constitute a SC. It takes several weeks for Intel to manufacture processors. However, Dell is not able to wait that long after receiving a customer order. Hence, both Dell and Intel should devise their corresponding production and ordering plans under demand uncertainty [3]. Similarly, in a distribution network, demand uncertainty poses risk not only to retailers but also to the manufacturer. We consider the risk sensitivity of the retailers and manufacturer.

Robinson and Swink [23] emphasized that several real DND problems faced by managers are often dealt with scenario evaluation method. They employed scenario evaluation technique for a real world problem. Besides, Robinson and Swink [24] experimentally studied the real decision makers’ abilities in evaluation of distribution scenarios. We focus on designing a decentralized SC under demand uncertainty where tactical decisions are taken in a competitive manner. On the other hand, the manufacturer as a Stackelberg leader determines strategic decisions of SCND using scenario evaluation method.

A stream of facility location and design problem literature emerges that deals with competitive facilities [25]. In the competitive facility location models, the decisions of any facility affect not just its own profit, but also their competitors. Meng et al. [26] investigated competitive facility location of a decentralized SC via variational inequality. Table 1 provides a detailed comparison among researches on competitive location of decentralized SCs. This table shows the key features of our proposed model. Because each retailer distributes products to the markets, we use the term “distributer” instead of “retailer”.

According to Table 1, there are four contributions in this research. First, the manufacturer enjoys the capability to contract with a set of independent distributors for serving geographically dispersed markets. To the best of the authors’ knowledge, no research was found in the context of competitive distributors (or retailers) with decentralized decision-making structure where the manufacturer’s authority to make decisions on DND would be investigated. Second, for structuring a distribution network, the manufacturer regards his distributors’ utility to insure that they are satisfied enough under the contract. Thus, the corresponding DND would be stable through a long period of time. We particularly use fuzzy number to represent the stability level of each distribution design alternative. Third, based on Arrow–Pratt measure of absolute risk aversion [20], both manufacturer and his distributors are presumed risk averse trying to maximize their utility other than expected profit. Ultimately, we develop a scenario evaluation based methodology for the selection of DND alternatives. In the methodology, the expected profit, risk, and stability degree of the alternatives are supposed in the role of selection criteria.

3 Problem statement and notation

We consider a decentralized SC composed of one manufacturer and a set of candidate distributors. The distributors supply a single type of products to n markets as shown in Fig. 1. The participants of the SC make their decisions before the market demands’ uncertainties are resolved, thus, their profit are transformed into uncertain. The manufacturer has the authority to structure his distribution network. That is he is able to contract with a subset of candidate distributors. Let | • | represents the cardinality of a set.

3.1 Sets and indices

The set of candidate distributors; I = { 1, … i…, |I| },

The set of demands markets; J = { 1, … j … , |J|},

The set of possible scenarios for DND (design alternatives); K ={ 1, … k … , |K| }.

3.2 Parameters

{\tilde{d}}_{ijk} :

The stochastic demand rate function of market j from distributor i in DND alternative k,

{\tilde{D}}_{ik} :

The stochastic demand rate seen by distributor i in DND alternative k,

{\tilde{α}}_{ijk} :

The stochastic part of jth market’s demand from distributor i in DND alternative k with mean ${\bar{α}}_{ijk} > 0$ and variance $σ_{ijk}^{2}$ ,

η_i :

The service investment efficiency coefficient of distributor i;η _i > 0. The larger the coefficient η _i, the lower the service investment efficiency of distributor i will be,

ρ_ij :

The demand sensitivity of market j to retail price of distributor i;ρ _ij > 0,

θ_i′j :

The demand sensitivity of market j to retail price of rival distributor i′; θ _i′j ≥ 0 for all i′ ≠ i,

β _ij :

The demand sensitivity of market j to service level of distributor i; β _ij > 0,

γ_i′j :

The demand sensitivity of market j to the service level of rival distributor i′; ν _i′j ≥ 0 for all i′ ≠ i,

ν_j :

The demand sensitivity of market j to the manufacturer’s advertising expenditure; ν _j > 0,

TC_i :

The transportation cost of a product unit between manufacturer and ith distributor; TC _i > 0,

TC_ij :

The transportation cost of a product unit between distributor i and market j; TC _ij > 0,

λ :

The constant absolute risk aversion (CARA) of the manufacturer which is defined in the Arrow–Pratt sense,

λ_i :

The CARA of distributer i that is defined in the Arrow–Pratt sense,

C :

The unit production cost of the manufactuer.

3.3 Decision variables

m_ik :

The product unit marginal profit of distributor i in DND alternative k,

w_k:

The product unit wholesale price of the manufacturer in DND alternative k,

p_ik :

The price of the product offered by distributor i in DND alternative k; p _ik = w _k + TC _i + m _ik,

p_ijk :

The price of distributor i’s products in market j in DND alternative k; p _ijk = w _k + TC _i + m _ik + TC _ij,

s_ik :

The service level of distributor i in DND alternative k,

I_k :

The partition of set I that indicates which candidate distributors are selected by the manufacturer (to become contractor distributors) in DND alternative k,

a_k :

The advertising expenditure of the manufacturer in DND alternative k.

3.4 Assumptions

DND alternatives and the scope of decision making. Each DND alternative consists of a subset of candidate distributors. The manufacturer has an initiative to select the most appropriate alternative and contract with the corresponding group of candidate distributors. The contractor distributors spread the products throughout the markets in a competitive manner. In trade with the distributors, the manufacturer takes decisions about wholesale price and advertising expenditure; on the other hand, the contractor distributors set retail prices and service levels.

Geographical properties. The markets and candidate distributors are geographically dispersed. Transportation costs in the distribution networks are exogenous parameters. They depend on the structural futures of the networks such as geographic locations, distances, infrastructures, and transportation equipment. The transportation costs are added to the product price and should be carried by the buyers.

Sequences of decision making. In regard to time sequence of strategic and tactical decisions of an SC, we consider of the following decision making stages:

Stage 1: The manufacturer’s strategic decision regarding DND. The manufacturer selects appropriate distribution scenario from the set of possible DND alternative. The final utility of alternatives for the manufacturer and for his distributors would be the selection criteria. The manufacturer makes contract with the corresponding distributors.

Stage 2: The tactical decisions of participants. In the DND alternative identified in Stage 1, each candidate distributor sets her best retail price and service level in the face of competition with other distributors. Knowing the actions of the distributors, the manufacturer then optimizes the wholesale price and advertising expenditure to maximize his utility function and reserve an accepted utility level for his distributors.

Markets’ demand functions. All markets’ demands consist of two sections: deterministic and stochastic. The deterministic section is contingent upon the service level and retail price of the distributors and advertising expenditure of the manufacturer. The stochastic part has pre-specified mean and variance. With regard to linear demand function sensitive to price and service level [8, 10] and well-known demand function sensitive to price and advertising expenditure [13] as well as demand function sensitive to price, service level, and advertising expenditure [14], we assume the following demand rate function:

${\tilde{d}}_{ijk} = {\begin{matrix} \begin{matrix} {\tilde{α}}_{ijk} - ρ_{ij} p_{ijk} + \sum_{\binom{i^{'} \in I_{k}}{i^{'} \neq i}} θ_{ij} p_{i^{'} jk} \\ + β_{ij} s_{ik} - \sum_{\binom{i^{'} \in I_{k}}{i^{'} \neq i}} γ_{i^{'} j} s_{i^{'} k} + ν_{j} a_{k}^{1 / 2} \end{matrix}; if i \in I_{k}, \\ 0; otherwise, \end{matrix} ∥$ (1) where p _ijk = w _k + TC _i + m _ik + TC _ij, ∀ i ∈ I _k, j ∈ J, k ∈ K.

Equation (1) states that the demand rate of market j from distributor i is positive if distributor i is selected in DND alternative k. Distributor i with large ${\bar{α}}_{ij}$ implies the appropriate position and reputation of the distributor in market j. The market demand rate from each distributor increases with the distributor’s service level, rivals’ prices, and manufacture’s advertisement expenditure. However, it declines with the distributor’s price and rivals’ service levels. The demand rate seen by distributor i is equal to the sum of the markets’ demands, i.e. ${\tilde{D}}_{ik} = \sum_{j \in J} {\tilde{d}}_{ijk}$ .

4 The model formulation

The manufacturer and contractor distributors are assumed in the role of the Stackelberg leader and followers, respectively. The objective of the leader is to determine his move in such a way that maximize his profit after considering all rational moves that the followers may devise. Let us first formulate the distributors and manufacturer problems.

4.1 The distributors’ model formulation

Let m _k ={ m _ik, ∀ i ∈ I _k } and s_K ={ s _ik, ∀ i ∈ I _k } denote profit margins and service levels of the contractor distributors in DND alternative k. For a given I _k, w_k, and, a _k of the manufacturer, the contractor distributors obtain the best decisions m_K and s_K to maximize their profits. The distributor’s profit = sale revenue-purchase cost-service expenditure. Similar to [9 , 14], we presume that the service cost function of distributor i is $0.5 (η_{i} s_{ik}^{2})$ . That is, enhancing service level has a diminishing influence service expenditure. In consequence, the uncertain profit of distributor i for given (I _k, w _k, a _k), is expressed as

$\begin{matrix} {\tilde{π}}_{ik} (m_{k}, s_{k} | I_{k}, w_{k}, a_{k}) \\ = {\begin{matrix} \begin{matrix} p_{ik} {\tilde{D}}_{ik} - (w_{k} + {TC}_{i}) {\tilde{D}}_{ik} \\ - \frac{1}{2} η_{i} s_{ik}^{2} \end{matrix}; if i \in I_{k}, \\ 0; otherwise . \end{matrix} \end{matrix}$ (2)

Each contractor distributor regards the profit fluctuations based on individual preference. As a result, contractor distributor i (i ∈ I _k) takes her decisions in the light of the following nonlinear multi-objective functions: $max_{m_{ik}, s_{ik}} E ({\tilde{π}}_{ik} (m_{k}, s_{k} | I_{k}, w_{k}, a_{k})) = m_{ik} E ({\tilde{D}}_{ik}) - \frac{1}{2} η_{i} s_{ik}^{2},$ (3) $min_{m_{ik}, s_{ik}} var ({\tilde{π}}_{ik} (m_{k}, s_{k} | I_{k}, w_{k}, a_{k})) = m_{ik}^{2} var ({\tilde{D}}_{ik}) .$ (4)

Bar-Shira and Finkelshtain [30] expressed that the utility function approach which simultaneously raises the expected value and reduces the variance; i.e., $min E (\tilde{π}) - λ var (\tilde{π})$ , is a more robust approach than those founded on expected value. Such approach enjoys two significant preponderances. First, it transforms a multi-objective problem into a singleton. Second, the parameter λ or CARA represents the attitude of the decision maker towards uncertainty. For further information, we refer the reader to [8 , 20]. Thus, using the mean-variance approach, distributor i consider the following utility function: $\begin{matrix} max_{m_{ik}, s_{ik}} u_{i} ((m_{k}, s_{k} | I_{k}, w_{k}, a_{k})) \\ = E ({\tilde{π}}_{ik} (m_{k}, s_{k} | I_{k}, w_{k}, a_{k})) \\ - λ_{i} var ({\tilde{π}}_{ik} (m_{k}, s_{k} | I_{k}, w_{k}, a_{k})) \\ = m_{ik} E ({\tilde{D}}_{ik}) - \frac{1}{2} η_{i} s_{ik}^{2} - λ_{i} m_{ik}^{2} \sum_{j \in J} var ({\tilde{D}}_{ik})) . \end{matrix}$ (5)

The Hessian matrix of u _i (( m_K , s_K |I _k, w _k, a _k )) is $H_{ik} = [\begin{matrix} - 2 \sum_{j \in J} ρ_{ij} - 2 λ_{i} \sum_{j \in J} σ_{ijk}^{2} & \sum_{j \in J} β_{ij} \\ \sum_{j \in J} β_{ij} & - η_{i} \end{matrix}] .$ (6)

The distributor’s utility is a concave function on (m _ik, s _ik) if the Hessian matrix is negative definite. A symmetric two-dimensional matrix is negative definite when diagonal elements are negative and determinant of the matrix is positive [31]. The determinant of the Hessian matrix (6) is as follows $B_{ik} = 2 η_{i} (\sum_{j \in J} ρ_{ij} + λ_{i} \sum_{j \in J} σ_{ijk}^{2}) - {(\sum_{j \in J} β_{ij})}^{2}$ (7)

The condition B _ik > 0 intimates that the service level should not be too inexpensive. Otherwise, distributor i may excessively invest in the service level such that her utility function becomes negative. Such assumption was also made by other researches [8–10 , 14]. Therefore, we henceforth assume B _ik > 0 (∀ i ∈ I _k and k ∈ K). This condition results in the concavity of all distributors’ utility function in all distribution design alternatives.

4.2 The manufacturer’s model formulation

Now, let us formulate the manufacturer’s profit function. The manufacturer’s profit = total sale revenue- advertisement expenditure. On this account, the manufacturer’s uncertain profit is given by ${\tilde{π}}_{k} (I_{k}, w_{k}, a_{k}, m_{k}, s_{k}) = (w_{k} - c) \sum_{i \in I_{k}} {\tilde{D}}_{ik} - a_{k} .$ (8)

Applying the mean-variance approach for random profit (8), the manufacturer’s objective would be $\begin{matrix} max_{I_{k}, w_{k}, a_{k}} u ({\tilde{π}}_{k} (I_{k}, w_{k}, a_{k}, m_{k}, s_{k})) \\ = E ({\tilde{π}}_{k} (I_{k}, w_{k}, a_{k}, m_{k}, s_{k})) \\ - λ var ({\tilde{π}}_{k} (I_{k}, w_{k}, a_{k}, m_{k}, s_{k})) . \end{matrix}$ (9)

In a DND alternative, the long-term commitments of contractor distributors are contingent upon their obtained profits from the contracts. In non-cooperative game, it is traditionally assumed that there is no interaction between the players. However, such assumption may not be reasonable when a manufacturer concerns over long-term commitment of his contractor distributors. The long-term commitment of contractor distributors ensures stability of manufacturer’s distribution network. In such circumstances, the manufacturer should not only regard the utility of the distribution network for himself but also it’s utility from the candidate distributors’ viewpoint.

The decision makers naturally have fuzzy goals for their objective functions when they consider the fuzziness of human judgments [32]. We assume that each distributor has fuzzy goal such that utility function $u_{ik} ({\tilde{π}}_{ik})$ should be considerably more than specific value $u_{i}^{min}$ . Otherwise, the distributor possibly withdraws from the trade and the corresponding DND alternative will be unfeasible.

Xiao and Yang [9] and Yang et al. [10] assumed that a retailer would like to maintain a long-term relationship if the manufacturer provides a reservation utility for her in the contract. Contradictorily, we adopt fuzzy approach for the commitment of the distributors. It is assumed that satisfaction of distributor i has membership function $μ_{i} (u_{ik} ({\tilde{π}}_{ik}))$ , which monotonically increase with $u_{ik} ({\tilde{π}}_{ik})$ . It is established upon the variation ratio of satisfactory degree in the interval between $u_{i}^{min}$ and $u_{i}^{max}$ . The satisfactory degree of the distributor is 0 if her obtained utility value is less than $u_{i}^{min}$ and the satisfactory degree is 1 if it is higher than $u_{i}^{max}$ . For the sake of simplicity, the linear membership function is used to characterize the fuzzy goal which is formulate as follows:

$\begin{matrix} μ_{i} (u_{ik} ({\tilde{π}}_{ik})) \\ = {\begin{matrix} 0 & if u_{ik} ({\tilde{π}}_{ik}) < u_{i}^{min} \\ \frac{u_{ik} ({\tilde{π}}_{ik}) - u_{i}^{min}}{u_{i}^{max} - u_{i}^{min}} & if u_{i}^{max} \leq u_{ik} ({\tilde{π}}_{ik}) < u_{i}^{min} \\ 1 & if u_{ik} ({\tilde{π}}_{ik}) \geq u_{i}^{max}, \end{matrix} \end{matrix}$ (10) and it is also depicted in Fig. 2.

As illustrated in Fig. 2, the high level of the distributor’s satisfactory guarantees her long-term commitment. To obtain the overall satisfactory degree for all contractor distributors, we adopt the following maxmin-operator proposed by Bellman and Zadeh [33]: $\max min_{i \in I_{k}} {μ_{i} (u_{ik} ({\tilde{π}}_{ik}))} .$ (11)

Using an auxiliary variable Λ _k as the overall satisfactory level of DND alternative k (the least degree of all contractor distributors’ fuzzy memberships; i.e., $Λ_{k} = min_{i \in I_{k}} {μ_{i} (u_{ik} ({\tilde{π}}_{ik}))}$ ), the following equivalent model is achieved $\begin{matrix} \max Λ_{k}, \\ s . t . Λ_{k} \leq μ_{i} (u_{ik} ({\tilde{π}}_{ik})), \forall i \in I_{k} . \end{matrix}$

For cementing a cooperative relationship, the manufac

turer needs to regard the utility of distribution network design for contractor distributors as well. The high degree of satisfactory level Λ _k of DND alternative ensures the long-term commitment of distributors and the stability of the DND. Therefore, in the light of overall satisfactory level and objective function (9), the manufacturer problem can be formulated as follows $\begin{matrix} max_{I_{k}, w_{k}, a_{k}} u ({\tilde{π}}_{k} (I_{k}, w_{k}, a_{k}, m_{k}, s_{k})) \\ = (w_{k} - c) \sum_{i \in I_{k}} E ({\tilde{D}}_{ik} (p_{k}, s_{k}, a_{k})) \end{matrix}$ (12) $- a_{k} - λ {(w_{k} - c)}^{2} \sum_{i \in I_{k}} \sum_{j \in J} var ({\tilde{D}}_{ik} (p_{k}, s_{k}, a_{k})),$ (13) $max_{Λ_{k}} Λ_{k},$ subject to: $\begin{matrix} m_{ik} E ({\tilde{D}}_{ik} (p_{k}, s_{k}, a_{k})) - \frac{1}{2} η_{i} s_{ik}^{2} \\ - λ_{i} m_{ik}^{2} \sum_{j \in J} σ_{ij}^{2} \geq \\ Λ_{k} (u_{i}^{max} - u_{i}^{min}) + u_{i}^{min}, \forall i \in I_{k} . \end{matrix}$ (14)

Now, let us evaluate the concavity property of the objective function (12). Hessian matrix of $u ({\tilde{π}}_{k})$ for DND alternative k with respect to variables w_k and a _k is $H_{k} = [\begin{matrix} - 2 \sum_{i \in I_{k}} \sum_{j \in J} (ρ_{ij} - \sum_{\binom{i^{'} \in I_{k}}{i^{'} \neq i}} θ_{i^{'} j}) - 2 λ \sum_{i \in I_{k}} \sum_{j \in J} σ_{ijk}^{2} & \sum_{i \in I_{k}} \sum_{j \in J} \frac{ν_{j}}{2 \sqrt{a_{k}}} \\ \sum_{i \in I_{k}} \sum_{j \in J} \frac{ν_{j}}{2 \sqrt{a_{k}}} & - \frac{(w_{k} - c)}{\sqrt{a_{k}^{3}}} \sum_{i \in I_{k}} \sum_{j \in J} \frac{ν_{j}}{4} \end{matrix}] .$ (15)

We bring up the following condition $ρ_{ij} > \sum_{i^{'} \in I_{k}, i^{'} \neq i} θ_{i^{'} j}, \forall j \in J, \forall k \in K .$ (16)

Inequality (16) is called “dominant diagonal” condition for the price competition which is highly intuitive and satisfied in most industries [34]. This condition expresses that demand of market j from each distributor diminishes if all distributors simultaneously raise their prices by the same amount. Moreover, it implies that a price increase by any one of the distributors leads to a decrease of total demands of markets. See [35], Proposition 1, for the detailed descriptions. Accordingly, we assume condition (16) for all markets throughout this paper. Determinant of H _k matrix is as follows $\begin{matrix} A_{k} = \sum_{i \in I_{k}} \sum_{j \in J} (ρ_{ij} - \sum_{\binom{i^{'} \in I_{k}}{i^{'} \neq i}} θ_{i^{'} j}) + 2 λ \sum_{i \in I_{k}} \sum_{j \in J} σ_{ijk}^{2} \\ - {(\sum_{i \in I_{k}} \sum_{j \in J} \frac{ν_{j}}{2})}^{2} . \end{matrix}$

Condition (16) along with A _k > 0 ensures that H _k is negative definite and $u_{ik} ({\tilde{π}}_{ik})$ is jointly concave on (w _k, a _k). Condition A _k > 0 means that the markets’ sensitivity to advertising expenditure should not be too high. Otherwise, the manufacturer may excessively invest in the advertising such that it incurs a negative utility to him.

4.3 The Stackelberg model formulation as a BLDPP

The DND problem in a decentralized SC can be considered as a BLDPP where a set of contractor distributors, the followers, take their decisions with regard to the decisions of the manufacturer as the leader. The approach of Stackelberg solution has been often utilized as a solution approach for the bi-level programming problems [36]. In the Stackelberg solution, it is supposed that there is no communication between the leader and follower(s), or they do not have any binding agreement even if there exists such communication. However, such assumptions may not be always reasonable when decentralized firms constitute a leader and followers hierarchical structure where top management establishes a cooperative relationship with his low-level divisions [32]. Top management often pursues overall management policy and objective. On the other hand, it should concern the followers’ goals and their long-term commitment. We formulate the following cooperative BLDPP for distribution network design problem: $max_{I_{k}, w_{k}, a_{k}} u ({\tilde{π}}_{k} (I_{k}, w_{k}, a_{k}, m_{k}, s_{k})) =$ (17) $\begin{matrix} E ({\tilde{π}}_{k} (I_{k}, w_{k}, a_{k}, m_{k}, s_{k})) - λ var ({\tilde{π}}_{k} (I_{k}, w_{k}, a_{k}, m_{k}, s_{k})), \\ max_{Λ_{k}} Λ_{k} \end{matrix}$ (18) subject to: $\begin{matrix} u_{i} ({\tilde{π}}_{ik} (m_{k}, s_{k} | I_{k}, w_{k}, a_{k})) \geq Λ_{k} (u_{i}^{max} - u_{i}^{min}) \\ + u_{i}^{min}, \forall i \in I_{k}, \end{matrix}$ (19) $0 \leq Λ_{k} \leq 1, \forall k \in K,$ (20) $w_{k}, a_{k} \geq 0, \forall k \in K,$ (21) $max_{m_{ik}, s_{ik}} u_{i} ({\tilde{π}}_{ik} (m_{k}, s_{k} | I_{k}, w_{k}, a_{k})) =$ (22) $E ({\tilde{π}}_{ik} (m_{k}, s_{k})) - λ_{i} var ({\tilde{π}}_{ik} (m_{k}, s_{k})), \forall i \in I_{k},$ subject to: $m_{ik}, s_{ik} \geq 0, \forall i \in I_{k}, \forall k \in K,$ (23) where p _ijk = w _k + TC _i + m _ik + TC _ij, for all i ∈ I _k, j ∈ J, and k ∈ K.

At the first level of the BLDPP and in objective function (17), the manufacturer specifies wholesale price and advertising expenditure in order to optimize his utility function. Moreover, in another first level objective function; i.e., Equation (18) and Constraint (19), the manufacturer maximizes the smallest degree of contractor distributors’ satisfaction. The second level objective functions (22) are utility functions of the contractor distributors. They determine margin profit as well as service level expenditure such that maximize their utility function. The decisions of each distributor depend on those of other distributors because they compete with each other in different markets.

Proposed model (17)–(23) is cooperative in two senses. Firstly, through the solving the problem, the manufacturer regulates his decisions trying to satisfy his contractor distributors and guarantee their long-term commitment. Secondly, the contractor distributors should concern about the overall manufacturer’s profit, because the low manufacturer’s profit derived from high-reservation utility of the distributor may debate the DND alternative from the manufacturer’s viewpoint. Consequently, the greedy distributors with high $u_{i}^{min}$ and $u_{i}^{max}$ may be omitted in the final manufacturer’s DND.

The optimal decisions of each DND alternative are evaluated at first; afterwards, the most appropriate alternative is selected by the manufacturer. Under the concavity condition of follower’s objective (22), first-order conditions $\partial u_{i} ({\tilde{π}}_{ik}) / \partial m_{ik} = 0$ and $\partial u_{i} ({\tilde{π}}_{ik}) / \partial s_{ik} = 0$ give the unique optimal decisions of each distributer (i.e., m _ik and s _ik) in response to the manufacturer’s decisions (i.e., I _k, w _k, a _k). Since the lower level problem (22) and (23) are concave and regular, they can be substituted by their Karush-Kuhn-Tucker (KKT) conditions (see [36]). Consequently, the problem (17)–(23) can be transformed into the following equlivant single level problem: $\begin{matrix} max_{I_{k}, w_{k}, a_{k}} L = (w_{k} - c) \sum_{i \in I_{k}} E [{\tilde{D}}_{ik} (p_{k}, s_{k}, a_{k})] \\ - a_{k} - λ {(w_{k} - c)}^{2} \sum_{i \in I_{k}} \sum_{j \in J} σ_{ij}^{2} + φ_{k} Λ_{k}, \end{matrix}$ (24) subject to: $\begin{matrix} m_{ik} E ({\tilde{D}}_{ik} (p_{k}, s_{k}, a_{k})) - \frac{1}{2} η_{i} s_{ik}^{2} - λ_{i} m_{ik}^{2} \sum_{j \in J} σ_{ijk}^{2} \\ \geq Λ_{k} (u_{i}^{max} - u_{i}^{min}) + u_{i}^{min}, \forall i \in I_{k}, \end{matrix}$ (25) $\begin{matrix} E ({\tilde{D}}_{ik} (p_{k}, s_{k}, a_{k})) + m_{ik} [\sum_{j \in J} (- ρ_{ij}) - 2 λ_{i} \sum_{j \in J} σ_{ijk}^{2}] \\ + ξ_{ik} = 0, \forall i \in I_{k}, \end{matrix}$ (26) $m_{ik} \sum_{j \in J} β_{ij} - η_{i} s_{ik} + ς_{ik} = 0, \forall i \in I_{k},$ (27) $ξ_{ik} m_{ik} = 0, \forall i \in I_{k},$ (28) $ς_{ik} s_{ik} = 0, \forall i \in I_{k},$ (29) Constraints (20) and (21).

Parameter φ _k is the marginal rate of substitution (MRS) between two upper-level objective functions. It implies how these functions can be compensated by each other from the manufacturer’s viewpoint. The low value of the parameter indicates the manufacturer concerns his utility function more than his distributers’ one. For instance, φ _k = 0 means that the manufacturer only considers utility of DND from his own point of view. φ _k =∞ implies that the manufacturer only take stability of DND into account.

The proposed model is a nonlinear programming problem (NLP) with a single level objective function and a set of equality and inequality constraints. The number of the constraints is directly contingent upon the number of the contractor distributors in corresponding DND alternative. Objective function and constraints are continuous and differentiable on w_k, a _k, m_K , and s _k; hence, a broad range of nonlinear solution algorithms such as penalty or barrier functions methods, or feasible directions methods can be employed for the problem [31]. In our case, once each DND alternative is formulated as a single objective NLP, we numerically solve the problem via interior-point algorithm developed in Matlab. In the subsequent section, we develop a methodology for choosing the most desirable DND alternative from a set of feasible alternatives.

5 The methodology for selection of distribution network design alternative

We describe a methodology for the selection of DND alternative in a decentralized SC. The main phases of the proposed methodology are depicted below.

Phase 1: Alternatives definition. The manufacturer should identify a set of potential markets for his products and obtain demographic and socioeconomic data about them. Afterwards, he should discern candidate distributors to supply the markets. Now, the manufacturer is able to generate various possible DND alternatives using different subsets of candidate distributors.

Phase 2: Alternatives evaluation. Each DND alternative should be evaluated separately. The manufacturer should first negotiate with the distributors for recognizing their fuzzy goals (i.e., $u_{i}^{min}$ and $u_{i}^{max}$ ). Once the distributors’ fuzzy goals are identified, a compromise Stackelberg equilibrium solution can be achieved by solving the problems (24)–(29). If all distributors are satisfied enough from the solution, the utility of the DND alternative for the manufacturer and the degree of satisfactory for the distributors have to be computed.

Phase 3: Alternatives selection. The manufacturer is now able to choose the most appropriate DND alternative by a tradeoff between two criteria, i.e. the alternative utility for him and the satisfactory degree of his distributors. The systematic changes of φ _k generate a Pareto frontier for the alternative; therefore, the most appropriate alternative may be obtained by analyzing the non-dominated solutions on the Pareto frontier.

The schematic diagram or algorithm for the selection of appropriate DND alternative for a given application is indicated in Fig. 3.

6 Numerical example and case study

In view of the fact that there is no benchmark model for the presented decentralized DND problem, our model’s results are evaluated for a real small-size problem. The case analysed refers to a leading Iranian company, which manufactures safe box and safety doors. Because the company has observed an unprecedented boom in demand in major cities in the recent past, re-examining the distribution network structure seemed ineluctable. The manufacturer’s products are voluminous, heavy, and costly to transport. Hence, the geographical structure of distribution network is a contributory factor influencing its efficiency.

In this problem, the manufacturer considers five candidate distributors to supply nine geographically dispersed markets. On the account of the fact that the candidate distributors are major independent stores which are capable to dispense the manufacturer’s products throughout the country, the competition among distributors would appear inevitable. The manufacturer encounters three DND alternatives represented by K ={ 1, 2, 3 }. The contractor distributors in these alternatives are I ₁ ={ 1 }, I ₂ ={ 1, 2, 3 }, and I ₃ ={ 4, 5 }. We compute the optimal decisions in these alternatives by solving model (24)–(29).

We showed that objective function (12) (and also objective function (24)) is a concave function. Moreover constraints (25)–(27) are linear. Therefore, the problem (24)–(29) is a quadratically constrained quadratic problem (QCQP). This problem can be efficiently solved by NLP algorithms. We use interior point algorithm implemented in Matlab software to solve the problem for each φ _k. Because both objectives (17) and (18) are integrated in objective function (24), the solution of QCQP would be a non-dominate solution. Consequently, we are able to generate a set of non-dominate solutions by systematically changing parameter φ _k and solving QCQP (24)–(29). These objective functions of these non-dominate solutions can give us an estimation of Pareto frontier for each DND alternative.

Tables 2, 3, and 4 display the detailed information concerning the optimal decisions for the DND alternatives. From the tables, we find that the higher the MRS φ _k, the lower the wholesale price and advertising expenditure of the manufacturer will be. Moreover, the optimal utility of contractor distributors ascends and their commitment becomes stronger as the MRS φ _k increases. The Pareto frontier for each DND alternative is delineated in Fig. 4. The figure illustrates how manufacturer’s utility depends on the stability of DND in Pareto extreme points.

The first insight we get from Fig. 4 refers to changing the preferences with regard to increasing the stability level. That is, as the stability level grows, the first DND alternative takes precedence over the second alternative. However, the third DND alternative is dominated by the second alternative throughout all range of stability level. As a result, the manufacturer should not choose alternative 3 when he is able to contract with distributors 1, 2, and 3 according to DND alternative 2. If distributors 4 and 5 wish to be considered as a potential DND alternative, they should revise their commitment level to the manufacturer by moderating their u ^min and u ^max.

We derive the following managerial insights from the numerical example.

The essential characteristic of the proposed model is that the manufacturer’s utility function and the stability level of his distribution network are conflicting. That is, as the manufacturer endeavoring to configure more stable distribution network, he may deteriorate the utility function of himself. When the manufacturer only concerns his utility function, the commitment of his contractor distributors reaches the minimum; however, the stability of the DND alternative may not be zero.

The decreasing rate of the manufacturer’s utility regarding the increase in stability in various DND alternatives may not be similar. Thanks to the fact that these differences can be well illustrated by Pareto frontier of alternatives, the most favorable alternative can be selected by evaluating these frontiers. If the manufacturer takes a specific stability level into account, the alternative with the most utility should be selected. On the other hand, if he considers a specific utility level, the most stable DND alternative should be preferred.

A compromise solution can be achieved by systematically changing the MRS parameter φ _k. This parameter should be carefully tuned according to the methodology to satisfy both the manufacturer and his distributors as much as possible.

7 Conclusion

We consider a game theoretic model to study the distributed DND of a risk-averse manufacturer who organizes a set of risk-averse distributors. These distributors supply geographically dispersed markets in a competitive fashion. Each DND alternative formulated as a BLDPP is transformed into a single level nonlinear problem. When the manufacturer considers the stability of distribution network in addition to his utility, the Pareto efficient solutions for each alternative are obtained by tuning the MRS value of these objectives. We show that considering the stability criteria for the problem may result in different desirable alternatives. We also propose a methodology for DND problem founded upon the compromise solutions of alternatives. We evaluate the results of the model and present some managerial insights by applying an industrial case. These insights help the practitioners to understand and employ the results.

There are several directions and suggestions for future research. First, while our linear demand relationships are analytically tractable, more general demand relationships could be considered. Second, future researchers could focus on developing a richer model for the manufacturer in view of production capacity, materials availability issues, and nonlinear production economics for reflecting economies of scale. Third, this paper assumes asymmetry of information in BLDPP; however, it can be extended to the case where this assumption is relaxed and private information exists. It is appealing to investigate how a manufacturer should design his distribution network when candidate distributors do not reflect real behavior such as their commitment or risk sensitivity. Lastly, our model contemplates a single period DND in a single selling season. Inasmuch as the DND is a strategic and long-term decision, multi period DND in which the Stackelberg game is played through the periods would be challenging but very interesting.

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