Supply Chain Coordination with Wholesale contract in an Uncertain Environment
Subject Areas : Stochastically Optimization
Alireza Ghaffari-Hadigheh
1
*
,
Nassim Mongeri
2
1 - Azarbaijan Shahid Madani University
2 - Dept. Applied Math, Azarbaijan Shahid Madani University, Tabriz, Iran
Keywords: Supply chain, coordinating contract, wholesale contract, uncertainty theory,
Abstract :
Supply chain coordination is one of the most recently studied fields. One of the coordination contracts is the wholesale contract that aims to balance the order quantity by offering a lower wholesale price to the retailer. The primary indeterminate player in this contract is the demand. The goal is to find the order amount so that both parties in the channel are satisfied with a reasonable profit concerning the proposed price. When a new product is presented to the market, the main challenge would be a sensible demand prediction such that either stock over or shortage stays under a moderate level. In such a new product, either there is insufficient sample space to be dealt with in probability theory, or the existing data for similar products does not apply to this new case. Referring to an expert would be an appropriate approach in this situation. Uncertainty theory is one of the mathematical paradigms which deals with this problem well. Using this paradigm, we assume that the demand is an uncertain variable with known uncertainty distribution. We mainly consider the linear uncertain demand and investigate optimal policy for both parties with and without coordination. An illustrative example verifies the proposed approach.
_____________________________________________________
Original Research .
Supply Chain Coordination with Wholesale Contract in an Uncertain Environment
Alireza Ghaffari-Hadigheh1*. Nassim Mongeri2
Received: 08 Oct 2022 / Accepted: 22 Oct 2024 / Published online: 05 Nov 2024
*Corresponding Author, hadigheha@azaruniv.ac.ir
1- Azarbaijan Shahid Madani University
2- Dept. Applied Math, Azarbaijan Shahid Madani University, Tabriz, Iran
Abstract
Supply chain coordination is one of the most recently studied fields. One of the coordination contracts is the wholesale contract, which aims to balance the order quantity by offering a reduced price for the large quantity order by the retailer. The goal is to determine the order amount so that both parties in the channel are satisfied with a reasonable profit concerning the proposed price. When a new product is presented to the market, the main challenge would be a sensible demand prediction such that either stock over or shortage stays under a moderate level. There is insufficient sample space in such a new product, or the existing data for similar products does not apply to this new case. These drawbacks prevent one from applying probability theory. Referring to an expert would be an appropriate approach in this situation. Uncertainty theory is one of the mathematical paradigms which deals with this problem well. Using this paradigm, we assume that the demand is an uncertain variable where its uncertainty distribution is presented. We mainly consider the linear uncertain demand and investigate optimal policy for both parties with and without coordination. An illustrative example verifies the appropriateness of the suggested approach. We also compare the results with the case when different scenarios are assumed, and probability theory is the base for the results.
Keywords- Supply chain; Coordinating contract; Wholesale contract; Uncertainty theory
INTRODUCTION
The economic world is facing additional complications nowadays. Due to rapid technological innovation and customer behaviour alterations, many products have short life cycles, and most of the time, their price drops at midlife. This phenomenon forces managers to act more effectively and preemptively than before. The main challenge is that the market integrates with inevitable indeterminacy, and appropriate modelling of problems in such an environment is essential. When one does not recognize the precise estimates of the involved parameters, and the model cannot handle this indeterminacy correctly, irrational decisions would be made, resulting in irrecoverable consequences. This research aims to provide practical solutions to these challenges, offering a roadmap for effective decision-making in uncertain supply chain environments. Supply Chain Management (SCM) has been established to handle many indeterminacies. Optimal interactions between a supplier and a retailer are one of the study fields in SCM. It is expected that proper coordination overcomes potential business complexities. Without coordination between the distribution channel players, each would pursue his benefits without considering the other. This competitive behaviour would disrupt the whole channel but not result in one’s gain.
There are different coordinating contracts in the literature of SCM. Wholesale price, buy-back, and revenue-sharing contracts are some to name [1]. A coordinating contract is designed to boost the whole channel’s performance and satisfy customers’ unknown demands as much as possible. The main objective in coordinating contracts is to propose decisions that benefit the entire supply chain channel. The study of these contracts with different points of view includes a vast study range. The wholesale contract is one of the most studied contracts between the supplier and the retailer. The manufacturer places the discounted wholesale price in a returns-discount contract to secure the most profits for both sides and increase the supply chain’s performance. Even though it is not a coordinating one from the probability theory perspective but is popular nowadays, this fact may be concluded to the impression that the probability theory perspective in the study of this contract does not make sense in many diverse environments.
Several approaches have been employed to diminish the impacts of such indeterminacy in analyzing coordinating contracts. Historically, probability theory came first to interpret frequencies when enough reliable data exists. Some others have been suggested for other situations. Most of them claim to model the unknowns based on expert opinion. An approach is the fuzzy theory. While it successfully models many problems, some paradoxical conclusions still exist in applying the fuzzy theory. Let us clarify why fuzzy theory is unsuitable for modelling and analyzing some problems. Suppose that the variable of demand is considered as a fuzzy variable
with the membership function 
. Here, the triangular membership function is used for the fuzzy demand amount, which would be something like follows.
It means that the fuzzy number referring to uncertain demand mount is (150, 170, 210). Considering the membership function 
and the meaning of the possibility measure, we have Pos
This means that for the demand amount of 170, we have Pos{Demand=170}=1, and Pos{Demand≠170}=1. Therefore, we conclude that (a) Demand is “precisely 170” with possibility measure 1, and (b) Demand is “not 170” with possibility measure 1. This means that “Exactly 170” is as possible as “not 170” for the demand level. Therefore, one has to consider them equally likely, which is not a logical inference. For more details, consult [2], see also [3] and [4].
There are different methods for predicting the demand level. One of them is the trend projection that applies previous sales data to plan future sales. It is the most clear-cut demand forecasting method. Another method is Market research demand forecasting based on customer survey data. Observe that this method is an after-production method and is not applicable for the new-introduced-to-market products. The econometric method can also be exploited but necessitates some number processing. This quantitative forecasting mode merges sales data with information on outside forces that affect demand level. Then, a mathematical modus operandi is devised to forecast future customer demand. These methods need some reliable data and are generally based on probability theory. It is important to mention that such data rarely exists for new innovative products. Consequently, using probability theory to predict the level of demand would not be efficient. The Delphi method would also be used as a qualitative method of demand forecasting that forces expert opinions on the market forecast. This method requires involving outside experts and an experienced facilitator. This is also impractical for innovative products since producers prefer to keep some existing information as their assets and not reveal it. As mentioned above, Fuzzy theory also has problems in some situations that may lead to misleading results. We emphasize that one of the main challenges is that the market always faces inevitable uncertainty, and proper modelling is necessary for these issues in such environments. When the person does not know the exact values of the parameters in the problem and the model cannot correctly manage this uncertainty, irrational decisions lead to irreparable consequences. Here, we apply uncertainty theory for modelling and analyzing such a situation.
The main drawback of existing studies is that probability theory-based findings are applicable when enough data exist, the probability distribution of demand is provided even approximately, and axioms of probability theory govern the unknown environment. The fuzzy theory also has intuitive contradictions in theory and practice, as mentioned above. None of these paradigms would produce sensible results when an expert provides data. Uncertainty theory was established in 2007 as a solid mathematical structure that formalizes human reasoning in an axiomatic framework. It has been used and denoted its capability in many practical problems.
In this paper, we consider the wholesale contract in an uncertain environment. An expert provides her opinion about the unknown data on the channel’s coordination, and the analysis is based on uncertainty theory. To keep the modelling simple, we assume that the channel compromises one supplier and one retailer and only the demand level in an uncertain variable. We restrict the argument to when uncertain demand has a linear distribution. Analyzing the model shows that this contract would coordinate the channel’s players’ decisions, which is in accord with the exercising reality in the nowadays deals. This paper is structured as follows. We first review some published results in this field with different views. Sunsequent section is devoted to reviewing some necessary concepts from uncertainty theory. Then, we presents the problem model and solution approach. We also solve the problem with a special selection of the demand’s uncertain level as linear. A simple toy example is presented in the next section to clarify the methodology. We also compare our approach and scenario-based probability theory point of view to denote the superiority of our findings. In the final section, some concluding remarks are mentioened and additional work directions are sketched.
Here in this review, we focus on recent works with different points of view on supply chain coordination, especially on the wholesale contract. Besides other contracts, the wholesale price contract has been studied using probability theory. In [5], a newsvendor problem involving one retailer and one manufacturer was investigated. It was shown how the manufacturer puts the discounted wholesale price in a return-discount contract to get the most profits and improve the supply chain’s performance. A stochastic demand, dependent on the retailer’s promotional and the manufacturer’s innovation efforts, is assumed. The paper proposed a new compensation-based wholesale price contract to encourage actors to engage in the joint decision-making scheme actively [6]. In another study, a wholesale contract was applied in a supply chain with one supplier and one buyer; both are risk-neutral. They face fixed demand for a single-selling period, and the study focuses on the consequence of supply indeterminacy. They found that contract performance monotonically improves with the supplier's negotiating power for random capacity, regardless of the purchase process mode [7]. A recent study considers two supply chain members: the supplier and the manufacturer [8]. The authors also compared the bare wholesale price contracts in centralized and decentralized frameworks.
For one manufacturer that distributes multiple products to multiple retailers, the authors in [9] proposed a multi-product, multi-period wholesale price coordination procedure in a decentralized supply chain. A bi-level wholesale price contract is presented in another recent research [10]. The authors considered three sections (one manufacturer, one distributor, and one retailer) in a supply chain with a scenario-based demand, and corporate social responsibility was treated as the coordinated decision in the channel. Coordinating contracts are also studied using fuzzy theory; see, e.g. [10] for the buy-back contract, [11] for computational complexities in practice, [12] for the buy-back contract with different risk attitudes. The wholesale contract has also been investigated by considering some elements as fuzzy numbers. For instance, a bilateral wholesale price in the supply chain has been proposed in a recent study [13]. Their coordination model improved pricing, quality, after-sales service, and service level performance. As a result, this contract could effectively coordinate the cell phone supply chain and enhance their profits compared to individual decisions for their specific goals.
The supply chain in robust optimization has been the subject of innovative research. A bi-objective model was recently explored for perishable products, aiming to minimize network costs and reduce greenhouse gas emissions. The results revealed a meaningful disparity between the mean of two objective functions and the run time. The general findings indicated that the weighted sum method is highly effective in obtaining the expected value of the first objective function. The Torabi-Hassini method, on the other hand, yields superior results based on the second objective [14]. The game theory introduced a practical approach to the wholesale contract. For instance, a game theory model of an altruistic retailer and a self-interested supplier in a wholesale price contract has been developed. The authors asserted that their model offers a robust optimal solution even when the demand information is lost. Their findings suggest that complete altruism, rather than partial altruism, can effectively coordinate the supply chain in the wholesale price contract [15]. Another model, in combination with robust optimization [16], has been devised. It considers a buyer’s purchasing policy from a supplier with stochastic capacity. The study demonstrates how buyers can settle the optimal order quantity and the wholesale price, with the main result being that the parameter uncertainty degree rises as the optimal robust wholesale price goes up.
In a recent research [17], a collaborative game strategy has been devised. The authors considered a capacity allocation problem with an airline selling cargo paths to several shipment forwarders. It is supposed that the specific capacity from one route cannot secure the total orders of forwarders (referred to as 'hot selling routes'), while from the substitute route is much less than its capacity (referred to as 'underutilized routes'). To solve this disproportion problem, a sequential cooperation game is played between the airline and the freight forwarders, stressing the collaborative nature of the solution. In this game, the player's payoff is the anticipated gain from exercising a mixed-wholesale-option contract between airlines and forwarders. The model result indicates that the demand on the underutilized routes follows self-replicating distributions. The mixed model yields the highest quotas on the underutilized routes, guiding to an improved demand balance amongst the alternative courses.
Another study considered the quantity of grain farmers sell, where setup costs of factories and warehouses are uncertain variables and proposes an uncertain grain supply chain [18]. An uncertain sustainable supply chain has been investigated in another research where cost factors, environmental impacts, and social benefits are uncertain. The authors developed a multi-objective chance-constrained model with an uncertain scenario to examine the effects of uncertainties on decision variables. The results express that the decision-maker should consider consuming more resources to cope with the system’s uncertainties at a higher confidence level [19]. An uncertainty theory to model demand distribution is applied [20], proposes the belief rate of order size is fewer than the supply chain optimal order amount and prepares the lower bound of the belief level. As a result, the relationship between the expected remaining inventory, the optimal order quantity, and the belief degree of this uncertain event is obtained. The pricing decision problem in which two producers compete to supply dissimilar but substitutable products through a typical retailer under other power configurations has been investigated using uncertainty theory [21]. Uncertain variables are manufacturing costs, sales costs, and demands. The authors derived how to decide about the optimal pricing on wholesale prices and retailer profits in three potential situations. In the study on strategic cooperation in supply chains [22], it was found that producers consistently desire long-term wholesale agreements. The study also revealed that a combined structure, when a supply chain exercises a long-term contract and the other a short-term contract, can result in a more stable system. Furthermore, the study emphasized that customers gain more from behavior-based pricing, while short-term wholesale contracts increase surplus.
Whether the supplier advantages from setting up nonlinear capacity reservation contracts as an alternative to wholesale price contracts has also been investigated [23]. From the supplier's standpoint, the capacity reservation contract achieves meaningfully better results than the wholesale price contract. However, the supplier's profit from applying capacity reservation is much superior at low margins than at high margins. Concerning supply chain performance, the supplier's positive effect outweighs the buyer's negative impact in low-margin settings. In contrast, the each consequence reduce the effect the other in high-margin settings. The authors affirm that even though nonlinear contract complexity leads to inferior performance than the theory anticipates, their study denotes that suppliers would even profit from installing them. Therefore, required management concepts are provided in deciding on the type of contract. Wholesale price contracts are also commonly exercized in closed-loop supply chains (CLSCs). A recent study by [24] found that when the inequality aversion of the is strong and the wholesale price contract is fixed, coordination is possible in a decentralized channel and a manufacturer-led CLSC. The study concludes that the producer's allocated portion is superior than the retailer's, and the retailer's share is above a smallest possible limit. The methodology exploited in this study involved solving a multistage successive move game under two set models and using the Karush-Cohen-Tucker condition for constrained optimization to find the bounds for the existence of perfect Nash equilibrium.
Wholesale contracts are also applied in the environmental discipline. It assesses pricing options, the extent of sustainability activities and the carbon cap considering wholesale price and cost-sharing contracts [25]. The author developed a Stackelberg game approach for both type of supply chain contracts and suggested the equilibrium solutions provided by the game actors. The conclusions assert that the existence of a carbon cap meaningfully influences the performance of the supply chain. Furthermore, the higher the carbon ceiling determined by the officals, the more sustainability improvement attempts the supply chain will make, and the more the supply chain can enhance its cost-effectiveness and sustainability under a cost-sharing contract. Wholesale price contracts with risk restrictions in a supply chain, incorporating a supplier and a capital-constrained retailer, were also studied [26]. A mean-variance model is derived to evaluate wholesale price contract design decisions under trade credit and bank credit finance support. It also stipulated the conditions in which the supplier is prepared to provide trade credit and the conditions in which the retailer selects bank or trade credit. It is also deduced that the risk aversion position of the supply chain members plays an important role in adjusting the financing balance. Finally, the results support the conclusion that trade credit financing leads to a win-win outcome only when the supplier's risk aversion tolerance is fair.
A supply chain is also investigated under a manufacturer's discount policy, assuming revenue-sharing and wholesale price contracts, where the Stackelberg equilibrium is evaluated for the supply chain respecting each contract [27]. It was deduced that the retailer's optimal retail price is identical in both the manufacturer's corresponding optimal decisions for revenue sharing and discounted wholesale price contracts. Moreover, sufficient prerequisites are obtained under which a revenue-sharing contract and a discount wholesale price contract can result in Pareto advance, i.e., both manufacturer and retailer earn a higher expected profit than the situation not considering discount. Appropriate requirements are also derived, where a revenue-sharing contract with a discount strategy can lead to Karldor-Hicks' improvement. Furthermore, it has been demonstrated that with a discount, a revenue-sharing contract can accomplish Karldor-Hicks' improvements over the wholesale price contract, too. The wholesale contract has also been considered in the framework of two-echelon e-commerce. In [28], the author considers a supply chain with a retailer and a supplier. Here, the retailer is a follower, and the focus theory of choice is considered for him. The effect of retailer pricing options on the supplier is examined for diverse emphasis and coordination priorities for the whole supply chain. The lower the parameter 
(degree of positivity) and the higher the parameter 
(level of confidence), the closer the gain of the whole supply chain to the coordination result and as for defining , the lower , the better the supply chain coordination. The conclusions of the proposed model can simultaneously present a theoretical basis for increasing cooperation between supply chain players and managerial insight for decision-makers to select cooperating bodies.
The study of the supply chain in uncertainty theory is still in its infancy. There are almost no published results on the analysis of contracts using uncertainty theory. Here, we mention some recent findings published in SCM using the uncertainty theory framework. A closed-loop supply chain network for producing and recovering button batteries under uncertainty has been modelled in which the demand, cost, and capacity are uncertain variables. The model evaluates the supply chain network’s environmental effects using a life cycle assessment approach. The results confirm that the expected value model tends to be decentralized while the chance-constrained model tends to be centralized [29].
Preliminaries
This section reviews some basic concepts in uncertainty theory. We refer the interested reader to [2] for more details. Let 
be a nonempty set, and 
be a 
-algebra over . Any element 
in is called an event. A set function 
from to 
is an uncertain measure if the following four axioms hold for this function.
Axiom 1 (Normality Axiom) 
.
Axiom 2 (Duality Axiom) 
for any event .
Axiom 3 (Subadditivity Axiom) For every countable sequence of events 
,
|
The triplet 
is called an uncertainty space. The product axiom, distinguishing the probability theory from the uncertainty theory, is defined as follows [30].
Axiom 4 (Product Axiom) Let 
be uncertainty spaces for 
. Then, the product uncertain measure on product -algebra 
satisfies
|
An uncertain variable 
is a measurable function from an uncertainty space to the set of real numbers in which 
is an event for any Borel set 
. Uncertainty distribution of an uncertain variable is defined as [2],
|
|
|
|
| (2) |
| ( 3) |
| (4) |
| (5) |
Notation | Description |
| Uncertain demand, |
| Wholesale’s production cost per unit |
| Retailer’s marginal cost per unit |
| Retailer price |
| Wholesale price |
| Salvage value |
| Wholesale shortage cost |
| Retailer’s shortage cost |
| Expected profit of the supplier |
| Expected profit of the retailer |
| Transfer payment |
| Demand uncertainty distribution function |
| ( 6) |
