Discrete choice model-based supply chain network equilibrium models

This subsection presents discrete choice-based supply chain network equilibrium models based on Assumption 1 in Section 1.3. Furthermore, to be useful, the equilibrium conditions are formulated as a discrete choice-based variational inequality problem, which allows firms or supply chain managers’ decision-making in terms of individual customers’ behavior choice.

A link flow is the sum of all chain flows on the link in which it participates. This work defines

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the link-chain incidence matrix ∆= (δ_𝑎𝑠)_{𝑎∈𝐴,𝑠∈𝑆} to be a |𝐴| × |𝑆| zero-one matrix, where δ_𝑎𝑠 = {1 𝑖𝑓 𝑙𝑖𝑛𝑘 𝑎 𝑝𝑎𝑟𝑡𝑖𝑐𝑖𝑝𝑎𝑡𝑒𝑠 𝑖𝑛 𝑐ℎ𝑎𝑖𝑛 𝑠,

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (3.8)

Therefore, one flow has the following conservation equation for link flow and chain flow variables:

𝑥_𝑎= ∑

𝑠∈𝑆𝑋_𝑠∙ δ_a𝑠∙ 𝜑_a𝑠, ∀𝑎 ∈ 𝐴. (3.9)

where 𝑋_𝑠 denote the flow of chain 𝑠 and 𝑥_𝑎 denote the flow on link 𝑎, and 𝜑_𝑎𝑠 is the absorption coefficient (Erengüç, Simpson, & Vakharia, 1999).

Next, the definition of a chain-imputable link cost is introduced which is the cost of link 𝑎 participating in chain 𝑠, denoted by 𝐶_𝑎𝑠. This chain-imputable link cost is the cost accountable for chain 𝑠, attributed to the work processed for chain 𝑠 on link 𝑎. It depends on the flow on link 𝑎 or a chain 𝑠 induced flow on link 𝑎, 𝜑_𝑎𝑠𝑋_𝑠, which is the amount of work contributed by chain 𝑠 to link 𝑎. Namely,

𝐶_𝑎𝑠 = 𝐶_a𝑠(𝑥_𝑎, 𝜑_𝑎𝑠𝑋_𝑠). (3.10)

The chain cost is the aggregated generalized cost incurred in all the links of this chain in delivering the final product to the markets. Therefore, the chain cost function is:

𝐶_𝑠= 𝐶_𝑠(𝑋_𝑠) = ∑

𝑎∈𝐴𝐶_𝑎𝑠∙ δ_𝑎𝑠. (3.11)

In a supply chain network equilibrium, a feasible chain 𝑋_𝑠^∗ constitutes a supply chain network equilibrium if and only if the following equality holds true:

𝑁_𝑘∙ 𝑃_𝑖 = 𝑋_𝑠^∗, 𝑖𝑓 𝑤_𝑖𝑗> 0, ∀𝑖, 𝑗, 𝑘, 𝑠 (3.12)

for all end consumer markets 𝑘, 𝑘 = 1, ⋯ , 𝐾; and products 𝑖, 𝑖 = 1, ⋯ , 𝐼.

This study defines 𝑈_𝑠 as the profit of chain 𝑠 and assumes each chain in a supply chain network is a profit-maximizer. The profit of chain 𝑠 is equal to the price multiplied by the demand minus the chain cost:

𝑈_𝑠= ∑^𝐼

𝑖=1𝑝_𝑖× 𝑁_𝑘∙ 𝑃_𝑖− ∑

𝑠∈𝑆𝐶_𝑠, ∀𝑖, 𝑘, 𝑠. (3.13)

According to Theorem 3.1, functions (3.12) and (3.13) for all chains can be expressed as the following variational inequality formulation (Lions & Stampacchia, 1967; Nagurney & Dong, 2002;

Nagurney, 2013). Variational inequality (3.14) characterizes the equilibrium condition assumption

30 several corollaries are provided to understand the effects of customer preference, decision variables, and strategies of competitors on the probability of choosing product 𝑖. The mathematical expression of the above equilibrium condition with chain flow may be different for different product characteristic attributes, for example, retail price, processing time, and quality. The following section is the proof of variational inequalities with chain flows.

Proof. Assume that, for chain 𝑠: 𝑠 = 1, ⋯ , 𝑆, and product 𝑖: 𝑖 = 1, ⋯ , 𝐼, the profit function 𝑈_𝑠(𝑋) of chain 𝑠 is concave with respect to the variables in 𝑋_𝑠, and is continuously differentiable.

Therefore, equation (3.14) in this model for all products 𝐼 is equivalent to variational inequality (3.1), based on the classic VIP (see Zhang (2006) and Nagurney, Dong, and Nagurney (2013)):

where 𝛻𝑈_𝑠(𝑋) is the gradient of 𝑈_𝑠(𝑋) with respect to 𝑋_𝑠. Therefore, the derivative of equation

Considering flow conservation (equation (3.12)) and the relationship between chain and link flows (equation (3.9) and equation (3.11)), variational inequality (3.14) can also be expressed in variational inequality (3.16). Variational inequality (3.16) also characterizes the equilibrium condition assumption defined in Section 1.3.

∑

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term 𝜆_𝑖 is the Lagrange multiplier associated with the constraint of flow conservation for demand markets 𝑘. The mathematical expression of the above equilibrium condition with link flow may be different for different supply chain network topology and product characteristic attributes. The following section is the proof of variational inequalities with link flows.

Proof. To solve equation (3.13), a vector of Lagrange multipliers, 𝜆_𝑖∈ 𝐾, is associated with the flow conservation constraints in (3.12). The variational inequality (3.14) can be further rewritten in terms of chain flows as variational inequality (3.17):

∑

For chain 𝑠 and its participant link 𝑎, the following relationship between 𝑐_𝑎 and 𝐶_𝑎𝑠 illustrates that a link cost is the sum of all cost of link 𝑎 participating in all chains. The chain-imputable link cost of link 𝑎 with respect to chain 𝑠 should aggregately cover the link cost, 𝑐_𝑎. That is,

𝑐_𝑎= ∑

𝑠∈𝑆

𝐶_a𝑠∙ δ_𝑎𝑠. (3.18)

The link cost, 𝑐_𝑎, is uniform, if this link charges its participating chains at a uniform rate. That is 𝜕𝐶_𝑎𝑠⁄𝜕𝑋_𝑠 = 𝜕𝑐_𝑎⁄𝜕𝑥_𝑎. In Chapter 5, the scenario that the link cost is not uniform is discussed in detail. In light of formula (3.18), (3.11), and the relationship between link cost and chain cost, the first term in formula (3.17) can be rewritten as

∑ equilibrium pattern satisfying variational inequality. This method uses Lagrangian functions to

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convert constrained problems into an unconstrained one. The resulting variational inequality subproblem is then transformed into a quadratic programming problem in the modified projection method and projection method finally (Korpelevich, 1976; Nagurney, 2013). The algorithm of the modified projection method is stated in the following steps.

Step 0: Initialization

Start with an 𝑥⁰∈ Ω. Set 𝑘 ≔ 1 then select 𝜌, such that 0 < 𝜌 ≤¹_𝐿, where 𝐿 is the Lipschitz constant for function 𝐹(𝑥) in the variational inequality problem.

Step 1: Construction and computation

Compute 𝑥̅^𝑘−1 by solving the variational inequality subproblem:

[𝑥̅^𝑘−1+ (𝜌𝐹(𝑥^𝑘−1) − 𝑥^𝑘−1)]^𝑇∙ [𝑥^′− 𝑥̅^𝑘−1] ≥ 0, ∀𝑥^′∈ Ω.

The above variational inequality subproblem is equivalent to the following quadratic programming problem:

𝑚𝑖𝑛𝑥∈Ω

1

2𝑥̅^𝑘−1𝐻𝑥̅^𝑘−1+ (𝜌𝐹(𝑥^𝑘−1) − 𝑥^𝑘−1)^𝑇∙ 𝑥̅^𝑘−1. Step 2: Adaptation

Compute 𝑥^𝑘 by solving the variational inequality subproblem:

[𝑥^𝑘+ (𝜌𝐹(𝑥̅^𝑘−1) − 𝑥^𝑘−1)]^𝑇∙ [𝑥^′− 𝑥^𝑘] ≥ 0, ∀𝑥^′∈ Ω.

The above variational inequality subproblem is equivalent to the following quadratic programming problem:

𝑚𝑖𝑛𝑥∈Ω

1

2𝑥^𝑘𝐻𝑥^𝑘+ (𝜌𝐹(𝑥̅^𝑘−1) − 𝑥^𝑘−1)^𝑇∙ 𝑥^𝑘. Step 3: Convergence verification

If |𝑥^𝑘− 𝑥^𝑘−1| ≤ 𝜀 for 𝜀 > 0, a pre-specified tolerance, then stop; otherwise, set 𝑘 ≔ 𝑘 + 1 and return to Step 1.

3.4.2 The modified projection method introduced augmented Lagrangian function

In the modified projection method (Khanh & Vuong, 2014; Nagurney, 2013), the necessary condition for iterative convergence is that L(x) is strictly monotone, where L(x) is the function that enters the variational inequality problem. If such a condition is not met in practice, which is very common in supply chain management, the algorithm wouldn’t be iterative convergence (He &

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Yuan, 2012). The proposed method may still be appropriate if the augmented Lagrangian functions are negative definite. To improve the applied range and computational efficiency, step 4 is added to the algorithm stated above. The modified projection method introduced augmented Lagrangian function has some obvious advantages, including computational speed and total number times of iteration. The results of the comparison between the traditional modified projection method and the modified projection method introduced augmented Lagrangian function are shown in Table 3.1. In the following steps, the augmented Lagrangian functions are used to convert constrained problems into an unconstrained one. One can use the following steps to solve the variational inequalities.

Step 0: Initialization

Start with an 𝑥⁰∈ Ω. Set 𝑘 ≔ 1 then select 𝜌, such that 0 < 𝜌 ≤¹_𝐿, where 𝐿 is the Lipschitz constant for function 𝐹(𝑥) in the variational inequality problem.

Step 1: Construction and computation

Compute 𝑥̅^𝑘−1 by solving the variational inequality subproblem:

[𝑥̅^𝑘−1+ (𝜌𝐹(𝑥^𝑘−1) − 𝑥^𝑘−1)]^𝑇∙ [𝑥^′− 𝑥̅^𝑘−1] ≥ 0, ∀𝑥^′∈ Ω.

The above variational inequality subproblem is equivalent to the following quadratic programming problem:

𝑚𝑖𝑛𝑥∈Ω

1

2𝑥̅^𝑘−1𝐻𝑥̅^𝑘−1+ (𝜌𝐹(𝑥^𝑘−1) − 𝑥^𝑘−1)^𝑇∙ 𝑥̅^𝑘−1. Step 2: Adaptation

Compute 𝑥^𝑘 by solving the variational inequality subproblem:

[𝑥^𝑘+ (𝜌𝐹(𝑥̅^𝑘−1) − 𝑥^𝑘−1)]^𝑇∙ [𝑥^′− 𝑥^𝑘] ≥ 0, ∀𝑥^′∈ Ω.

The above variational inequality subproblem is equivalent to the following quadratic programming problem:

𝑚𝑖𝑛𝑥∈Ω

1

2𝑥^𝑘𝐻𝑥^𝑘+ (𝜌𝐹(𝑥̅^𝑘−1) − 𝑥^𝑘−1)^𝑇∙ 𝑥^𝑘. Step 3: Convergence verification

If |𝑥^𝑘− 𝑥^𝑘−1| ≤ 𝜀 for 𝜀 > 0, a pre-specified tolerance, then stop.

Step 4: Update the new iteration by

34 𝜆^𝑘=𝜆̅^𝑘−𝜃𝑔_𝑚(𝑧̅^𝑘)

Set 𝑘 ≔ 𝑘 + 1 and return to Step 1.

Table 3.1. Results for the modified projection method and the proposed method on linear constrained variational inequality problems1.

Modified Projection Method ^1,2 Modified Projection Method introduced Augmented Lagrangian function

Name³ n Iter. CPU⁴ Iter. CPU

NLP3 3 1579 10.7609 391 2.9844

Mathiesen 3 729 15.0781 249 3.7656

KojimaSh 4 396 8.1406 393 6.7969

Nash10 10 1230 9.9375 1229 9.3281

NLP30 30 2600 17.5469 801 7.1875

1 Parameters set (ρ = 0.001, ε = 0.001, σ = 100). And for all methods, this proposed method used 𝑥⁰=(1,⋯,1). (Iter. Denotes the total number times of iteration)

2 Modified projection method as described in (MV Solodov & Tseng, 1996).

3 The Mathiesen problem, used first by Mathiesen (Mathiesen, 1987), and others in (Mikhail V Solodov & Svaiter, 1999 and the references therein). Then, the first NLP3 problem is extended to NLP30.

4 CPU denotes the average time (in seconds) obtained in 10 times computation using MATLAB 2018a function CPUtime and run on a computer with CPU: Intel(R) Core(TM) i7-8550U 1.80GHz 1.99GHz. RAM: 8.0GB. System type:64-bit Windows 10 operating system at the University of Calgary.

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Chapter 4. A Supply Chain Network Economic Model with Time-cost Tradeoff for Heterogeneous Customers

This chapter presents an integrated model for time-cost competition between supply chains with heterogeneous customers. The firms in the model of Chapter 4 can offer various time options for their production/service to time-sensitive customers. This gives rise of a new concept of time-based supply chain, which is called T-chain in this chapter, to be the basic element in the competition and extends the inter supply chain competition to a new dimension of time. Assuming the customers are heterogeneous in time-cost bi-criteria decision making, this chapter integrates the discrete choice theory into supply chain network competition and formulate the equilibrium conditions as a multinomial logit-based variational inequality problem. The demand in the proposed model is specified and calculated at the level of individual customers. The use of individual customer data in industry will bring many different estimations and subsequent equilibrium solutions. Numerical examples are presented for model illustration and managerial insights such as profit maximization for a firm who participates in this supply chain network.

4.1 Introduction

Time-based competition in coordinating and controlling production and transportation supply chain network is an important strategic weapon that considers operations management academics and decision-makers in firms. For example, time-based competition has been studied to improve inventory management, on-time delivery, supply chain management, production and distribution planning, process integration, new product development, and even quality management to increase competitiveness (Li, 1992; Lederer & Li, 1997; Beliën & Forcé, 2012; Yu & Nagurney, 2013;

Nagurney, Besik, & Yu, 2018). In practice, time is used as a marketing tool. Retailers, manufacturers, and third-party logistics (3PL) firms also use timely service as a competitive advantage (Aiello, La Scalia, & Micale, 2012). Some retailers, such as the German company Globus, use freshness as a key competitive advantage to attract customers (Entrup, Günther, Van Beek, Grunow, & Seiler, 2005). Federal Express provides next day delivery service in many countries. One major supermarket chain in California, Lucky, guarantees quick service for their

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customers (So, 2000). With the increase of time-based products (or services) provided by firms in supply chains, it is essential to understand the relationship between operational cost and processing time in a supply chain network. The operations management of time-cost competition in a supply chain network affects not only people’s lives but also firms’ profitability. Time-based competition has become a strategy, as important as productivity, quality, and even innovation. However, after increasing investments in human and financial capital toward time-based competitiveness, many firms admitted they did not achieve competitive advantage or higher profits. Ignoring customers’

preferences and needs, the capabilities of competitors, and segmenting customers by their sensitivity to time, Stalk and Webber (1993) report many firms in Japan found themselves caught in both time and acceleration traps (Bruch & Menges, 2010). Additionally, there is a serious problem with waste. Approximately 40–50% of all US food produce is thrown away (Smithers, 2013), and 37% of fruits and vegetables in Asia go to waste in supply chains every year (Gunders, 2012). This is the result of poor temperature control, natural volatility, expiration, and other factors associated with poor time management and ignoring customer preference. Furthermore, the structure of a supply chain network is increasingly complex and related to the superposition of many types of networks (Ahumada & Villalobos, 2009), including transportation, finance, information, and time. In considering this, a question is raised. “What is the role of time in supply chain network competition from a customer perspective?” Although significant progress regarding time-cost competition issues has been made, the practicability and effectiveness of time-cost competition methods in a supply chain network remain to be established. The results ultimately answer the below questions. “How are the company’s equilibrium flows, profits, and prices of the variants affected by competitors and customer preference?”, “Under what conditions should firms increase processing time beyond market equilibrium?” and “which potential supply chain will win in a time-cost competition?”.

As noted by Nagurney, Masoumi and Yu (2015), time plays a critical role in time-sensitive product supply chains and, therefore, should be a fundamental element in time-sensitive product supply chain equilibrium models. Stalk (1990) coins the term time-based competition, and Stonich (1990) points out that the important points to outline a time-based strategy is segmenting the market that value timeliness, estimating the value of meeting customer needs, and assessing the impact of the strategy. The objective of time-cost competition is gaining an improved distribution time (process time-based competition) and/or a higher frequency of introducing new products to the markets or improving the existing ones (product time-based competition) (Toni & Meneghetti, 2000). Meanwhile, a growing body of literature has focused on time-cost competition issues in

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supply chains and a supply chain network. Operational cost and processing time are often conflicting objectives when coordinating and controlling production and transportation in a supply chain network. Some literature has focused on the multicriteria decision-making issue in a supply chain network to optimize benefits regarding time, price, and cost (Dong, Zhang, Yan, & Nagurney, 2005; Nagurney & Matsypura, 2005) and minimize loss through implementing a loss function in the objective functions to handle the time cost (Widodo, Nagasawa, Morizawa, & Ota, 2005;

Ahumada & Villalobos, 2011). Relatedly, a product’s marginal value of time, the rate at which a product loses value over time, is used to minimize lost value in perishable product supply chains (Blackburn & Scudder, 2009). In addition, the rate of the total quality decrease from the original quality is improved with time control in frozen-food chains (Zhang, Habenicht, & Spieß, 2003).

Moreover, a network model with arc multipliers is introduced into a supply chain network to handle the perishability of pharmaceutical and blood products (Masoumi, Yu, & Nagurney, 2012;

Nagurney, Masoumi, & Yu, 2012). These studies have found that time control contributes to the quality of perishable products and benefits their supply chains or a supply chain network. Although some studies recognize the importance of time competition in supply chains or a supply chain network, the processing time is treated as an influence factor that can be converted into some kind of cost at the firm level. In this study, time is treated as not only some kind of cost, but also a competitive tool, a new dimension. It appears that there is an increase in customers who are willing to pay more for timely products (or services). In the 2011 Customer Experience Impact (CEI) Report, 86% of consumers indicated they would pay more for better customer experience. This study is specifically interested in customer purchase decisions. In other words, to seamlessly satisfy customer demand to avoid time traps and reduce waste in a supply chain network, this chapter focuses on customer choices, mainly through the lens of the customer discrete choice and supply chain network economic models. Relatedly, in a disaster relief network, Gossler, Wakolbinger, Nagurney, and Daniele (2019) develop a game-theoretic model to allocate limited supplies to different groups of beneficiaries. Therefore, it is necessary to understand how customer demand is affected by the conflicting objectives of optimizing selling price and processing time. Because the competition is no longer between stand-alone companies but rather supply chain against supply chain (Christopher, 1992), this chapter focuses on inter supply chain competition. The objective in this chapter is to develop a network economic model in consideration of time-cost competition in a supply chain network to satisfy customer demand with customer characteristic attributes in time-sensitive competing markets, while improving a supply chain’s ability to connect seamlessly with customers and avoiding time traps and unnecessary waste.

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4.2 Model formulation

This section presents a supply chain network economic model with time-cost competition in competing markets. Additionally, a multinomial logit-based variational inequality is given to express the equilibrium conditions in a supply chain network with time-cost competition.

4.2.1 Definitions

This section introduces two definitions essential to understanding time-cost competition in a supply chain network. These definitions are illustrated mathematically in detail in subsequent sections.

Consider a general network with time attribute 𝐻 = [𝐺, 𝐿, 𝑇], where 𝐺 denote the set of nodes in the network, 𝐿 denote the set of directed links, and 𝑇 denote the set of processing time on directed links. Without loss of practicability and generality, a processing time set is added on each directed link. Each processing time set represents the processing time options needed for each activity on its link. That is, each link has a number of alternative processing times.

Suppose that the supply chains in a supply chain network compete and cooperate on a time and cost basis in a selling season. In each selling season, firms in supply chains serve customers with various required delivery dates in demand markets. Thus, the model as presented in this chapter views the differentiated product with various processing times and selling prices attributes as variants. Each product has a certain number of variants in a supply chain network. Each customer hopes to choose a variant of a differentiated product to yield the greatest utility (Anderson, Palma,

& Thisse, 1992).

Definition 4.1 (T-operation link and T-chain).

Denote 𝐴 the set of all the operations and denote 𝑎 ∈ 𝐴 a particular operation. For any operation 𝑎 ∈ 𝐴, suppose in general there can be more than one process time option for an operation. Next, a T-operation link can be defined. A T-operation link, a time-based operation link, represents a substantial business function with a given processing time performed by a firm in a supply chain network. For example, FedEx provides several price options with different delivery dates for customers to choose from the same origin to destination. In the network model, one has a set of T-operation links associated with T-operation 𝑎, 𝑎^𝑇 = {𝑎^𝑡: 𝑡 ∈ 𝑇_𝑎} corresponding to the operation with processing time options, 𝑇_𝑎= {𝑡_𝑎¹, ⋯ , 𝑡_𝑎^|𝑇𝑎|}, where | | is the cardinality of 𝑇𝑎. The set of the T-operation links can be expressed by 𝐴^𝑇 = ∪

𝑎∈𝐴𝑎^𝑇. A T-operation link 𝑎^𝑡, 𝑡 ∈ 𝑇_𝑎 represents the operation of 𝑎 performed in process time 𝑡. The T-operation link flow 𝑥_𝑎^𝑡 on T-operation link

39

𝑎^𝑡, 𝑡 ∈ 𝑇_𝑎 is the work load processed by operation 𝑎 in time option 𝑡.

An interface link denotes a coordination function connecting two successive T-operation links.

Denote 𝐵 the set of all the interface links and a particular interface link 𝑏 ∈ 𝐵.

A time-based supply chain (T-chain) inducing a variant of a product (or service) is a chain of coordinated the processing time of business activities on each T-operation link and interface link.

These activities involve such things as production, storage, transportation, procurement associated with the delivery of a variant of a product (or service) within a given time requirement.

Zhang (2006) was the first to introduce the concepts of operation link and interface link into the modeling of a supply chain network. These concepts are adopted in this chapter in the context of time-cost competition between supply chains. In the proposed model, each interface link only belongs to one T-chain. The cost and processing time of an interface link reflect the effectiveness of coordination and integration of two T-operation links that it bridges (Zhang. 2006). For example, in potential T-chains constituted by firms initiating a working relationship, some connection costs are relatively higher, and time frames are longer. Therefore, the competition between T-chains accounts not only for the cost and processing time of the operations but also for the cost and processing time of their interface connection.

Each T-chain is ultimately driven by customers. To meet customer requirements, it is necessary to determine and activate a customer specific T-operation link. This can be oriented towards more than one T-chain. The proposed model allows a T-operation link to participate in multiple T-chains, even mutually competing ones. For example, a manufacturing link can connect with several T-transportation links (e.g., Kraft produces cheese for both Walmart and Safeway). In other words, the time-based competition of two competitors in local demand markets is embodied in the competition of the two T-chains they participate in. In addition, they often cooperate with each other in other aspects in the supply chain network, such as component procurement or transportation. From the above definitions and formulas, a supply chain network can be treated as a combination of its chains, each of which is pulled by the customers’ given requirement. The T-chains are competing against each other for customers.

In Fig. 4.1, a particular product is associated with 𝑖 ( 𝑖 = 1, ⋯ , 𝐼) processing times that is customer sensitive and provided by different T-chains. If customers are sensitive to the sum of

In Fig. 4.1, a particular product is associated with 𝑖 ( 𝑖 = 1, ⋯ , 𝐼) processing times that is customer sensitive and provided by different T-chains. If customers are sensitive to the sum of

In document Discrete Choice-based Equilibrium Modeling of Supply Chain Network with Conflicting Objectives and Demand Uncertainty (Page 42-151)