# Stackelberg Game Model of Railway Freight Pricing Based on Option Theory

Authors: Jingwei Guo; Zhongqi Xie; Qinglin Li
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In recent years, although rail transport has contributed significantly to the productivity of the Chinese economy, it has also been faced with the fierce competition and challenge from other modes of transportation, and therefore, freight-pricing issue has received more attention by researchers. In this paper, the rail freight option (RFO) based on option theory is proposed to study the optimal pricing decision of the railway transportation enterprise and contract customers’ optimal purchase decisions. To obtain an effective RFO contract, the railway freight contract transaction process is first analyzed. Then, the theoretical framework for the RFO contract trading is put forward in the railway freight market. Next, a two-stage Stackelberg game theoretic approach is presented based on the principle of utility maximization to achieve the optimal decision of RFO contract. Subsequently, the reverse reasoning method in dynamic programming is used to solve the optimal combination decision of the contract customer. Finally, the optimal pricing decision of RFO is discussed using Kuhn–Tucker conditions and Lagrangian function. The result shows that the railway transportation enterprise should pay more attention to the option strike pricew1 in terms of maximizing system utility and achieving Pareto optimal.


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## Body

## 1. Introduction

Railway is one of the most efficient and environmental-friendly ways for transportation industry in China, and it has contributed significantly to the productivity of the Chinese economy [1]. However, in the process of the continuous growth of other modes of transportation and the increasing market competitiveness, the market share of the railway freight industry has declined year by year [2]. The marketization management mode of railway freight transport is the reform trend of China’s railway transport industry. Hence, the transformation to market-oriented operation must be accelerated for railway freight transport operations to adapt to changes in the transportation market, which will also benefit and help develop railway transport enterprises.Nowadays, China’s railway freight has the coexistence of contract market and spot market. In the contract market, railway transport enterprises sell part of their transport capacities by signing a contractual agreement (usually six months in advance) with the contract customer. The signing of these contracts has certain advantages in terms of stable supply and business relative to the spot market with poor transaction stability. More precisely, the contract provides stable bulk supply for railway transportation and requires transportation enterprises offering transportation guarantee for contract customers at the same time. However, during the contract period, both parties need to implement a fixed contract price and cannot profit from trading in market price fluctuations, which is unfavorable for maximizing the utility of both parties. Meanwhile, railway transport enterprises will not sell all their capacities in the contract market, and the remaining capacities can only be sold through the spot market. In the spot market, the relationship between supply and demand directly determines the freight rate, and some transactions failed due to large price fluctuations. All of these will restrict the railway transportation enterprises to formulate a market competitive freight pricing system. Therefore, the problems that how to price the freight rate of railway freight transportation enterprises and improve the competitiveness of railway transportation have to be settled urgently. Pricing freight rate is a complicated problem involving a range of issues; for example, firstly, there is a noncooperative game between railway transportation enterprise and contract customer. Secondly, as shown in Figure1 [3], competitive pricing should be used that helps railway transportation enterprises gain more market share and business. Thirdly, the essence of railway transportation service requires flexibility and risk avoidance in pricing mechanism.Figure 1
Total freight volume and rail freight volume in China.In real life, consider a company that uses two channels for trading: contract and spot, and several flexibility contracts were concerned by many scholars, such as return contract [4, 5], quantity flexibility (QF) contract [6–9], multiperiodic supply chain contracts [10–12], and risk-sharing contract [13, 14]. These flexibility contracts are widely applied in trade agreement for its flexibility and versatility. However, asymmetric information existing in trade activities is nothing new, and the above contracts cannot help the stakeholders make rational decisions. Then, investigating the role of options (contingent claims) in a buyer-supplier system has attracted great attention from researchers [15–18]. Dawn et al. illustrated how options provide flexibility to a buyer to respond to market changes in the second period [19]. Bester and Krähmer analyzed bilateral contracting in an environment with contractual incompleteness and asymmetric information using a simple deterministic exit option contract [20]. Luca et al. studied how exit options can affect bidding behavior and the buyer’s and the seller’s expected payoffs in multidimensional procurement auctions [21]. Furthermore, an option tool will improve the economic efficiency of the partners in the discrete environment [22], hedge market risk [23], and promote fair trade [24]. Through numerical examples, Barnes-Schuster et al. verified the role of option contracts in improving the utility of supply chain [25]. Additional applied researches have investigated the application of options in supply chain management. Cheng et al. established a single-cycle secondary supply chain option contract model to determine the optimal pricing and ordering strategies [26]. Xing et al. derived the Seller’s optimal bidding and Buyers’ optimal contracting strategies in a von Stackelberg game with the Seller as the leader [27]. Wang and Tsao assumed that the option contract executed in the second stage is not equal to the option purchase amount in the first stage, and the optimal strategy is obtained from the perspective of the option buyer [28]. Cai et al. investigated the relationship between the option contract and the subsidy contract and found supply chain coordination and Pareto improvement can be achieved by introducing the option contract [29]. Liu et al. investigate the coordination of both the supplier-led and the retailer-led supply chains under option contract [30]. All of those have largely promised the potential for option contracts’ applications to develop freight derivatives.Like options on stocks, options on freight provide stakeholders with protection against adverse freight rate movements. Increased globalization and increased demand for transportation has resulted in freight itself becoming a volatile commodity. In the freight transaction area, Rolf et al. established a capacity option pricing model and applied option contract into air cargo industry [31]. Koekebakker et al. set up the theoretical framework for the valuation of the Asian-style options traded in the shipping market [32]. Soltani et al. considered a commodity processor and developed models to determine the ocean freight firm’s optimal hedging policy [33]. Kyriakou et al. developed an accurate valuation setup for freight options, featuring an exponential mean-reverting model for the freight rate with distinct reversion scales for its jump and diffusion components [34]. Although freight options has been the primary tool used to manage fluctuations in the shipping due to volatile ship prices, the contracts were never traded [35]. For sea transport enterprises, the purpose of buying options is to hedge risks rather than exercise options [36]. By contrast, railway transportation enterprises adopt option trading to improve the competitiveness of railway transportation. As railway transportation market is relatively new, not much scientific research has been done in this area. Guo and Peng applied the option theory into railway freight pricing activities and established a multiphase trigeminal tree pricing model [37]. Meanwhile, a multiphase trigeminal tree pricing model with jump diffusion process was established to depict the fluctuations in RFO pricing strategy caused by the nonmarketing uncertainty which arrive at random discrete time points [38]. These have a certain reference value for studying the pricing of RFO. However, these studies are independent of the optimal decentralized decision of the supply chain system, and its pricing strategy is not conducive to the long-term stability of the supply chain.In this paper, a two-stage Stackelberg game model of railway transport enterprise and contract customer is developed based on the perspective of coordinating supply chain. The main contributions of this paper are summarized as follows:(1)
Considering the coexistence of contract market and spot market, a new tradable RFO is designed and the transaction process is elaborated(2)
A two-stage Stackelberg game theoretic approach based on the principle of supply chain utility maximization is proposed to achieve the optimal decision of RFO contract(3)
A new reverse reasoning algorithm is proposed to deal with the optimal decision of contract customer(4)
Combined Lagrangian function with Kuhn–Tucker conditions, a new method is put forward to obtain the optimal pricing decision of RFOThe remainder of this article is organized as follows: Section2 details the methodology; Section 3 presents the method for obtaining the contract customers’ optimal decision, and the optimal pricing decision of RFO is derived in Section 4; and Section 5 will conclude our work.


## 2. Model Description

### 2.1. Definition of RFO

Freight has to be contracted, just like commodities. The only difference is that most commodities are real products, while freight is a service instead of a physical product [39]. So when freight is “bought,” the service of products being transported is contracted. Due to nonstorability of freight, it should be traded in time. To protect railway transport enterprises and contract customers against market risks, a new option contract related to freight is provided according to the concept of options in the financial market, called rail freight option (RFO). Unlike other options, RFO helps railway transport enterprises selling nonstorable freight in advance and also insures stakeholders against freight rates moving beyond a specified price level.Definition 1.
A RFO is a call option contract which states that the contract customer (holder) has the right to pay/receive the average of the values of the freight rates during some period on or before the expiration date and receive/pay strike price. The railway transport enterprise (writer) then has the obligation to receive/pay this average and pay/receive the strike price when the holder decides to exercise.
In fact, a RFO is an option contract for an asset that is subject to the railway freight transport service. The railway transport enterprise is the option that writer has the right, but not the obligation, to sell the option at the option price. And the contract customer is the holder who is a purchaser of option. If the execution price of the expiration date is higher than the spot market price, the contract customer will pay the execution fee and execute RFO; otherwise, the contract customer will abandon the execution right of RFO and choose to purchase the capacity in the spot market. With an option, the contract customer has no risk of losing any money more than strike price due to freight price volatility because he always has the possibility not to exercise RFO. For the railway transportation enterprise, if contract customer gives up the execution of RFO, they will sell the capacities in the spot market without refunding the option fees. This is a way the railway transportation enterprise can spread risk and schedule the freight train plans rationally. Under such circumstances, although the risk avoidance is realized effectively, the feasibility of decision-making process is another issue that should be considered.


### 2.2. Transaction Process Description

In practice, both railway transportation enterprise and contract customer are mostly partial to risk aversion just to have different degree. Therefore, the degree of risk acceptability determines the amount of options purchased. The vast majority of existing studies deal with how to use option for hedging. The optimal decision is a decentralized decision that is independent of the utility of the supply chain. Excessive pursuit of maximization of its own utility based on decentralized decision is not conducive to the long-term stability of the entire supply chain. The expectation of maximizing returns is not the optimal decision point in terms of transportation system. Compared with the existing studies focus on the expected returns of stakeholders, this paper studies the decision-making in the RFO trading to help maximize system utility, maintain the long-term development of the entire supply chain, and achieve Pareto optimal.Consider a railway transportation enterprise who is looking to protect the company against a possible decrease in the freight rates. To this extent, the railway transportation enterprise writes and sells the RFO by speculating contract customer buying behavior so as to formulate reasonable option prices and circulation. Then, contract customer determines the purchase amount of RFO according to the pricing announced by the railway transportation enterprise. And in the second stage, the contract customer will decide whether or not to exercise his RFO. If only the total amount that the contract customer has to add up to the option strike price is less than the spot freight rate at that point, the contract customer will exercise his RFO, and of course, the contract customer will abandon his RFO in reverse. The transaction process actually conforms to the two-stage dynamic game model and is shown in Figure2.Figure 2
The transaction process of RFO.


### 2.3. Symbol and Assumption

To better understand the model, the list of all the notations used in our work is presented in Table1. Some notations will be more precisely defined as they appear in later sections of this paper.Table 1
Summary of notations.
NotationDescriptionsw0Option price (unit capacity)w1Option strike price (unit capacity)piSpot market freight rate (unit capacity) in timehpiProbability density function ofpiHpiDistribution function ofpiNPurchase amount of RFOCUnit capacity fixed costq1Execution amount of RFOq2Capacity purchases amount through spot marketb1Long-term preparation cost of transportation (unit capacity)b2Short-term preparation cost of transportation (unit capacity)KTotal capacity provided by railway freight transportD1Contract market capacity demandD2Spot market capacity demandfDProbability density function of freight demandμMean deviationσStandard deviationIn order to build the mathematical model, this study makes the following assumptions:Assumption 1.
The RFO covered in this article is European call option; that is to say, RFO can only be executed on the expiration date.Assumption 2.
Both railway transport enterprises and contract customers are completely rational and risk averse. In order to maximize system utility and achieve Pareto optimal, the optimal decision is based on the principle of maximizing supply chain utility as much as possible.Assumption 3.
The freight rate of the spot market in this model is an exogenous variable, which is completely dominated by the external market economic conditions, and is not affected by the railway transportation enterprises and contract customers.Assumption 4.
Both contract market and spot market have a normal distribution of freight demand;μ and σ represent the mean and standard deviation, respectively.


### 2.4. Stackelberg Model Construction

In the case of contract market coexisting with spot market, two-stage Stackelberg game model is established, in which the railway transportation enterprise is the leader and the contract customer is the follower. The game sequence is shown as the following:Step 1: in the contract market of timeT0, the railway transport enterprise writes the RFO including the option price w0 and the option strike price w1.Step 2: according to the published price strategy of RFO (N), the contract customer decides the purchase amount of RFO to maximize their expected returns.Step 3: in the spot market of timeT1, the contract customer will decide whether or not to exercise his RFO based on the spot market freight rate pi and the option strike price w1. Furthermore, determine the execution amount of RFO (q1) and the capacity purchases amount through spot market (q2). Thus, the utility function of contract customer π1q1,q2,pi,N can be expressed as follows:(1)π1q1,q2,pi,N=Uq1+q2−w0N−w1q1−piq2,whereUq1+q2 stands for the market return expected by the contract customer and Uq1+q2=eα−eα−βq1+q2/β is assumed based on risk aversion theory, in which α and β are undetermined parameters. w0N is the payment for the contract customer to purchase RFO, w1q1 stands for the option strike price paid by the contract customer exercising RFO, and piq2 represents the cost of purchasing the capacity from spot market.Step 4: similarly, the utility function of railway transportation enterpriseπ2w0,w1,b0,b1,N can be easily gained as follows:(2)π2w0,w1,b0,b1,N=w0N+w1q1−b1q1+K−N∧D2pi−b2+−KC,whereb1q1 stands for the long-term preparation cost of transportation, K−N∧D2pi−b2+ stands for the proceeds from the sale of the remaining capacity in the spot market, and KC stands for the fixed production cost [40].Step 5: then, the utility function of systemπ is obtained as follows:(3)π=π1+π2=Uq1+q2−piq2−b1q1+K−N∧Dpi−b2+−KC.It is worth noting how the game reaches the Nash equilibrium. Both misjudgment of the market and the behavior of the opponent will lead to the deviation of the decision. Once the deviation occurs, one party will inevitably obtain excess profits. The other party will inevitably adjust the decision or withdraw from the market to choose alternatives. It is extremely unfavorable to the supply chain relationship. In order to maximize system utility, the long-term development of the supply chain is maintained and Pareto optima is achieved, and optimal decision should be investigated from the perspective of maximizing the utility of the supply chain.


## 2.1. Definition of RFO

Freight has to be contracted, just like commodities. The only difference is that most commodities are real products, while freight is a service instead of a physical product [39]. So when freight is “bought,” the service of products being transported is contracted. Due to nonstorability of freight, it should be traded in time. To protect railway transport enterprises and contract customers against market risks, a new option contract related to freight is provided according to the concept of options in the financial market, called rail freight option (RFO). Unlike other options, RFO helps railway transport enterprises selling nonstorable freight in advance and also insures stakeholders against freight rates moving beyond a specified price level.Definition 1.
A RFO is a call option contract which states that the contract customer (holder) has the right to pay/receive the average of the values of the freight rates during some period on or before the expiration date and receive/pay strike price. The railway transport enterprise (writer) then has the obligation to receive/pay this average and pay/receive the strike price when the holder decides to exercise.
In fact, a RFO is an option contract for an asset that is subject to the railway freight transport service. The railway transport enterprise is the option that writer has the right, but not the obligation, to sell the option at the option price. And the contract customer is the holder who is a purchaser of option. If the execution price of the expiration date is higher than the spot market price, the contract customer will pay the execution fee and execute RFO; otherwise, the contract customer will abandon the execution right of RFO and choose to purchase the capacity in the spot market. With an option, the contract customer has no risk of losing any money more than strike price due to freight price volatility because he always has the possibility not to exercise RFO. For the railway transportation enterprise, if contract customer gives up the execution of RFO, they will sell the capacities in the spot market without refunding the option fees. This is a way the railway transportation enterprise can spread risk and schedule the freight train plans rationally. Under such circumstances, although the risk avoidance is realized effectively, the feasibility of decision-making process is another issue that should be considered.


## 2.2. Transaction Process Description

In practice, both railway transportation enterprise and contract customer are mostly partial to risk aversion just to have different degree. Therefore, the degree of risk acceptability determines the amount of options purchased. The vast majority of existing studies deal with how to use option for hedging. The optimal decision is a decentralized decision that is independent of the utility of the supply chain. Excessive pursuit of maximization of its own utility based on decentralized decision is not conducive to the long-term stability of the entire supply chain. The expectation of maximizing returns is not the optimal decision point in terms of transportation system. Compared with the existing studies focus on the expected returns of stakeholders, this paper studies the decision-making in the RFO trading to help maximize system utility, maintain the long-term development of the entire supply chain, and achieve Pareto optimal.Consider a railway transportation enterprise who is looking to protect the company against a possible decrease in the freight rates. To this extent, the railway transportation enterprise writes and sells the RFO by speculating contract customer buying behavior so as to formulate reasonable option prices and circulation. Then, contract customer determines the purchase amount of RFO according to the pricing announced by the railway transportation enterprise. And in the second stage, the contract customer will decide whether or not to exercise his RFO. If only the total amount that the contract customer has to add up to the option strike price is less than the spot freight rate at that point, the contract customer will exercise his RFO, and of course, the contract customer will abandon his RFO in reverse. The transaction process actually conforms to the two-stage dynamic game model and is shown in Figure2.Figure 2
The transaction process of RFO.


## 2.3. Symbol and Assumption

To better understand the model, the list of all the notations used in our work is presented in Table1. Some notations will be more precisely defined as they appear in later sections of this paper.Table 1
Summary of notations.
NotationDescriptionsw0Option price (unit capacity)w1Option strike price (unit capacity)piSpot market freight rate (unit capacity) in timehpiProbability density function ofpiHpiDistribution function ofpiNPurchase amount of RFOCUnit capacity fixed costq1Execution amount of RFOq2Capacity purchases amount through spot marketb1Long-term preparation cost of transportation (unit capacity)b2Short-term preparation cost of transportation (unit capacity)KTotal capacity provided by railway freight transportD1Contract market capacity demandD2Spot market capacity demandfDProbability density function of freight demandμMean deviationσStandard deviationIn order to build the mathematical model, this study makes the following assumptions:Assumption 1.
The RFO covered in this article is European call option; that is to say, RFO can only be executed on the expiration date.Assumption 2.
Both railway transport enterprises and contract customers are completely rational and risk averse. In order to maximize system utility and achieve Pareto optimal, the optimal decision is based on the principle of maximizing supply chain utility as much as possible.Assumption 3.
The freight rate of the spot market in this model is an exogenous variable, which is completely dominated by the external market economic conditions, and is not affected by the railway transportation enterprises and contract customers.Assumption 4.
Both contract market and spot market have a normal distribution of freight demand;μ and σ represent the mean and standard deviation, respectively.


## 2.4. Stackelberg Model Construction

In the case of contract market coexisting with spot market, two-stage Stackelberg game model is established, in which the railway transportation enterprise is the leader and the contract customer is the follower. The game sequence is shown as the following:Step 1: in the contract market of timeT0, the railway transport enterprise writes the RFO including the option price w0 and the option strike price w1.Step 2: according to the published price strategy of RFO (N), the contract customer decides the purchase amount of RFO to maximize their expected returns.Step 3: in the spot market of timeT1, the contract customer will decide whether or not to exercise his RFO based on the spot market freight rate pi and the option strike price w1. Furthermore, determine the execution amount of RFO (q1) and the capacity purchases amount through spot market (q2). Thus, the utility function of contract customer π1q1,q2,pi,N can be expressed as follows:(1)π1q1,q2,pi,N=Uq1+q2−w0N−w1q1−piq2,whereUq1+q2 stands for the market return expected by the contract customer and Uq1+q2=eα−eα−βq1+q2/β is assumed based on risk aversion theory, in which α and β are undetermined parameters. w0N is the payment for the contract customer to purchase RFO, w1q1 stands for the option strike price paid by the contract customer exercising RFO, and piq2 represents the cost of purchasing the capacity from spot market.Step 4: similarly, the utility function of railway transportation enterpriseπ2w0,w1,b0,b1,N can be easily gained as follows:(2)π2w0,w1,b0,b1,N=w0N+w1q1−b1q1+K−N∧D2pi−b2+−KC,whereb1q1 stands for the long-term preparation cost of transportation, K−N∧D2pi−b2+ stands for the proceeds from the sale of the remaining capacity in the spot market, and KC stands for the fixed production cost [40].Step 5: then, the utility function of systemπ is obtained as follows:(3)π=π1+π2=Uq1+q2−piq2−b1q1+K−N∧Dpi−b2+−KC.It is worth noting how the game reaches the Nash equilibrium. Both misjudgment of the market and the behavior of the opponent will lead to the deviation of the decision. Once the deviation occurs, one party will inevitably obtain excess profits. The other party will inevitably adjust the decision or withdraw from the market to choose alternatives. It is extremely unfavorable to the supply chain relationship. In order to maximize system utility, the long-term development of the supply chain is maintained and Pareto optima is achieved, and optimal decision should be investigated from the perspective of maximizing the utility of the supply chain.


## 3. Optimal Decision of Contract Customer

According to the order of decision, this paper uses the reverse reasoning method in dynamic programming to solve the optimal combination decision of the contract customer firstly. The problem of the contract customer optimal decision is decomposed into the following two stages:Step 1: the constraint optimization method is used to solve the optimal combinationq1∗,q2∗ of the RFO execution amount and the spot market purchase amountStep 2: the optimal RFO order quantityN∗ of contract customer is solved according to the optimal combination q1∗,q2∗ obtained in the first stage


### 3.1. Calculation of the Optimal Combination

The goal of the contract customer is to maximize its own expected utility, which can be expressed as the following optimization problem with constraints:(4)maxq1,q2Eπq1,q2,pi,Ns.tq2≥0,q1≤N,0≤q2≤D2.According to formula (4), the RFO execution amount and the spot market purchase amount of the contract customer are obtained as described in Theorem 1.Theorem 1.
The optimal choice of RFO execution can be expressed as follows:(5)q1∗=N,w1≤pi,0,w1>pi.The optimal choice of the capacity purchases amount through spot market can be expressed as follows:(6)q2∗=D2,w1>p,D2−N,∂U∂N>pi≥w1,0,pi≥∂U∂N>w1.Proof.
The Kuhn–Tucker conditions of formula (4) are constructed as follows:(7)τ=eα−βq1+q2−w0N−w1q1−piq2+γ1q2+γ2N−q1+γ3q1+r4D2−q2,γi≥0i=1,2,3,4,q2≥0,N−q1≥0,q2≥0,D2−q2≥0,γ1q2=0,γ2N−q1=0,γ3q2=0,r4D2−q2=0.
Next, the first-order partial derivatives ofq1 and q2 for function τ are obtained on the basis of Lagrange multiplier γ as follows:(8)∂τ∂q1=eα−βq1+q2−w1−γ2+γ3=0,(9)∂τ∂q2=eα−βq1+q2−pi+γ1−r4=0.
Then, the optimal combinationq1∗,q2∗ is discussed as follows:(a)
Ifγ1>0,γ2=0,γ3=0,γ4=0, w1≤pi is gained, and the optimal combination q1∗,q2∗ can be discussed as the follows:(1)
Supposing that the marginal payment ability∂U/∂N of the contract customer is greater than the spot market freight rate pi, it is profitable for option customer to purchase capacity from the spot market. Hence, the optimal combination q1∗,q2∗ can be calculated as follows:(10)q1∗=N,q2∗=D2−N.(11)q1∗=N,q2∗=0.(b)
Ifγ1=0,γ2>0,γ3=0,andγ4=0, w1≤pi is gained, and the optimal combination q1∗,q2∗ can be calculated as follows:(12)q1∗=N,q2∗=0orD2−N.(c)
Ifγ1=0,γ2=0,γ3>0,andγ4=0, w1≥pi is gained, and the optimal combination q1∗,q2∗ can be calculated as follows:(13)q1∗=0,q2∗=D2.(d)
Ifγ1=0,γ2=0,γ3=0,andγ4>0, w1≥pi is gained, and the optimal combination q1∗,q2∗ can be calculated as follows:(14)q1∗=N,q2∗=0.(2)
Conversely, if∂U/∂N−pi≤0, the contract customer blindly purchasing the capacity in the spot market will reduce its profit. The optimal choice is to stop the repurchase of capacity. Hence, the optimal combination q1∗,q2∗ can be calculated as follows:
Therefore, Theorem1 is proved.
In addition, this article finds an interesting conclusion that the contract customer’s RFO execution is greatly affected by the option strike pricew1. If only the option strike price w1 is smaller than spot market freight rate pi and the market demand is much larger than its RFO purchase amount N, the contract customer will choose to execute all RFO purchased. On the contrary, it will choose to abandon RFO and instead purchase the capacity in the spot market to obtain higher profits.


### 3.2. Optimal RFO Order Quantity of Contract Customer

According to the optimal combinationq1∗,q2∗ obtained, the original utility function independent variable is changed based on the principle of utility maximization, and formula (4) can be updated as follows:(15)maxNEπw1,Ns.tN≥0.Then, solving mathematical program with constraints in formula (14), the optimal RFO order quantity N∗ of the contract customer is obtained as described in Theorem 2.Theorem 2.
Based on the option price written by railway transport enterprise, the contract customer decides the optimal option purchase amountN∗ satisfies the following formula:(16)U′N∗1−HU′N∗+∫0∂U/∂NHpidpi=b1+w1+pi−b2Hpi+pi.Proof.
Substituting formula (3) into formula (15), the objective function of formula (15) can be updated as follows:(17)π=Uq1+q2−piq2−b1q1+K−N∧D2pi−b2+−KC=EUq1+q2−piD2−q1+−b1∫w1∞Nhpidpi+pi−b2+∫0K−ND2fD2dD2+∫K−N∞K−NfD2dD22−KC=EUq1+q2−b1∫w1∞w1hpidpi∫0ND1fD1dD1−∫w1∞w1hpidpi∫N∞NfD1dD1−b1∫w1∞Nhpidpi−∫w1∞Nhpidpi∫N∞D1−NfD1dD1+pi−b2+∫0K−ND2fD2dD2+∫K−N∞K−NfD2dD22−KC.
AsED1=∫0NfD1dD+∫N∞D1fD1dD1, the expected utility function of the supply chain can be expressed equivalently as(18)π=EUq1+q2−∫w1∞w1hpidpi∫0ND1fD1dD1−∫w1∞w1hpidpi∫N∞NfD1dD1+∫w1∞pihpidpi∫N∞NfD1dD1−piED1+∫w1∞pihpidpi∫N∞D1fD1dD1−b1∫w1∞Nhpidpi+pi−b2+∫0K−ND2fD2dD2+∫K−N∞K−NfD2dD2−KC=EUq1+q2−piED1+∫w1∞pi−w1hpidpi∫0ND1fD1dD1+∫N∞NfD1dD1−b1∫w1∞Nhpidpi+pi−b2+∫0K−ND2fD2dD2+∫K−N∞K−NfD2dD2−KC=EUq1+q2−piED1+Epi−w1+D1∧N−b1q1+Epi−b2+K−N∧D2−KC.
Moreover, if formulas (5) and (6) are substituted into formula (18), the expected utility function of the entire supply chain can be calculated as follows:(19)π=∫Uq1+q2−piDhpidpi+∫pi−w1+Nhpidpi+∫pi−b2+K−Nhpidpi−b1∫w1∞Nhpidpi−KC=∫0∂U/∂NUDpi−piDpidhpi+∫∂U/∂N∞UN−piNdhpi+∫w1∞pi−w1Ndhpi+∫w1∞pi−b2K−Ndhpi−∫w1∞b1Ndhpi−KC.
Next, taking the first-order partial derivative with respect toN for formula (19), the optimality conditions of first order can be calculated as follows:(20)∂π∂N=∫∂U/∂N∞U′N−pidHpi+∫w1∞pi−w1dHpi−∫w1∞pi−b2dHpi−∫w1∞b1dHpi=∫∂U/∂N∞U′N−pidHpi+∫w1∞b2−b1−w1dHpi=EU′N−pi−∫∂U/∂N∞U′N−pidHpi+Eb2−b1−w1−b2−b1−w11−Hpi=U′N1−HU′N−∫0∂U/∂NpidHpi−pi+b2−b1−w1Hpi=U′N1−HU′N−piHpi−pi+∫0∂U/∂NHpidpi−pi+b2−b1−w1Hpi.
In addition, take the second derivative with respect toN for formula (20) to get ∂2π/∂N2=U″N.
In accordance with Assumption2, the article has ∂2π/∂N2=U″N≤0 as the result of the contract customer is a risk averse. That is to say, πw0,w1,N is a concave function with respect to N. Let ∂π/∂N=0, then the optimal option purchase amount N∗ satisfies the following formula:(21)U′N∗1−HU′N∗+∫0∂U/∂NHpidpi=b1+w1+pi−b2Hpi+pi.
Therefore, Theorem2 is proved.Theorem 3.
The optimal option purchase amountN∗ of the contract customer is monotonously decreasing with respect to the option strike price w1.Proof.
For formula (16), taking total differentiation with respect to the option strike price w1 using Newton–Leibniz formula, formula (22) is obtained:(22)∂N∗∂w1=HpiU″N∗<0.
Obviously, if the railway transport enterprise increases the option strike pricew1, the optimal option purchase amount N∗ of the contract customer will be reduced accordingly.


## 3.1. Calculation of the Optimal Combination

The goal of the contract customer is to maximize its own expected utility, which can be expressed as the following optimization problem with constraints:(4)maxq1,q2Eπq1,q2,pi,Ns.tq2≥0,q1≤N,0≤q2≤D2.According to formula (4), the RFO execution amount and the spot market purchase amount of the contract customer are obtained as described in Theorem 1.Theorem 1.
The optimal choice of RFO execution can be expressed as follows:(5)q1∗=N,w1≤pi,0,w1>pi.The optimal choice of the capacity purchases amount through spot market can be expressed as follows:(6)q2∗=D2,w1>p,D2−N,∂U∂N>pi≥w1,0,pi≥∂U∂N>w1.Proof.
The Kuhn–Tucker conditions of formula (4) are constructed as follows:(7)τ=eα−βq1+q2−w0N−w1q1−piq2+γ1q2+γ2N−q1+γ3q1+r4D2−q2,γi≥0i=1,2,3,4,q2≥0,N−q1≥0,q2≥0,D2−q2≥0,γ1q2=0,γ2N−q1=0,γ3q2=0,r4D2−q2=0.
Next, the first-order partial derivatives ofq1 and q2 for function τ are obtained on the basis of Lagrange multiplier γ as follows:(8)∂τ∂q1=eα−βq1+q2−w1−γ2+γ3=0,(9)∂τ∂q2=eα−βq1+q2−pi+γ1−r4=0.
Then, the optimal combinationq1∗,q2∗ is discussed as follows:(a)
Ifγ1>0,γ2=0,γ3=0,γ4=0, w1≤pi is gained, and the optimal combination q1∗,q2∗ can be discussed as the follows:(1)
Supposing that the marginal payment ability∂U/∂N of the contract customer is greater than the spot market freight rate pi, it is profitable for option customer to purchase capacity from the spot market. Hence, the optimal combination q1∗,q2∗ can be calculated as follows:(10)q1∗=N,q2∗=D2−N.(11)q1∗=N,q2∗=0.(b)
Ifγ1=0,γ2>0,γ3=0,andγ4=0, w1≤pi is gained, and the optimal combination q1∗,q2∗ can be calculated as follows:(12)q1∗=N,q2∗=0orD2−N.(c)
Ifγ1=0,γ2=0,γ3>0,andγ4=0, w1≥pi is gained, and the optimal combination q1∗,q2∗ can be calculated as follows:(13)q1∗=0,q2∗=D2.(d)
Ifγ1=0,γ2=0,γ3=0,andγ4>0, w1≥pi is gained, and the optimal combination q1∗,q2∗ can be calculated as follows:(14)q1∗=N,q2∗=0.(2)
Conversely, if∂U/∂N−pi≤0, the contract customer blindly purchasing the capacity in the spot market will reduce its profit. The optimal choice is to stop the repurchase of capacity. Hence, the optimal combination q1∗,q2∗ can be calculated as follows:
Therefore, Theorem1 is proved.
In addition, this article finds an interesting conclusion that the contract customer’s RFO execution is greatly affected by the option strike pricew1. If only the option strike price w1 is smaller than spot market freight rate pi and the market demand is much larger than its RFO purchase amount N, the contract customer will choose to execute all RFO purchased. On the contrary, it will choose to abandon RFO and instead purchase the capacity in the spot market to obtain higher profits.


## 3.2. Optimal RFO Order Quantity of Contract Customer

According to the optimal combinationq1∗,q2∗ obtained, the original utility function independent variable is changed based on the principle of utility maximization, and formula (4) can be updated as follows:(15)maxNEπw1,Ns.tN≥0.Then, solving mathematical program with constraints in formula (14), the optimal RFO order quantity N∗ of the contract customer is obtained as described in Theorem 2.Theorem 2.
Based on the option price written by railway transport enterprise, the contract customer decides the optimal option purchase amountN∗ satisfies the following formula:(16)U′N∗1−HU′N∗+∫0∂U/∂NHpidpi=b1+w1+pi−b2Hpi+pi.Proof.
Substituting formula (3) into formula (15), the objective function of formula (15) can be updated as follows:(17)π=Uq1+q2−piq2−b1q1+K−N∧D2pi−b2+−KC=EUq1+q2−piD2−q1+−b1∫w1∞Nhpidpi+pi−b2+∫0K−ND2fD2dD2+∫K−N∞K−NfD2dD22−KC=EUq1+q2−b1∫w1∞w1hpidpi∫0ND1fD1dD1−∫w1∞w1hpidpi∫N∞NfD1dD1−b1∫w1∞Nhpidpi−∫w1∞Nhpidpi∫N∞D1−NfD1dD1+pi−b2+∫0K−ND2fD2dD2+∫K−N∞K−NfD2dD22−KC.
AsED1=∫0NfD1dD+∫N∞D1fD1dD1, the expected utility function of the supply chain can be expressed equivalently as(18)π=EUq1+q2−∫w1∞w1hpidpi∫0ND1fD1dD1−∫w1∞w1hpidpi∫N∞NfD1dD1+∫w1∞pihpidpi∫N∞NfD1dD1−piED1+∫w1∞pihpidpi∫N∞D1fD1dD1−b1∫w1∞Nhpidpi+pi−b2+∫0K−ND2fD2dD2+∫K−N∞K−NfD2dD2−KC=EUq1+q2−piED1+∫w1∞pi−w1hpidpi∫0ND1fD1dD1+∫N∞NfD1dD1−b1∫w1∞Nhpidpi+pi−b2+∫0K−ND2fD2dD2+∫K−N∞K−NfD2dD2−KC=EUq1+q2−piED1+Epi−w1+D1∧N−b1q1+Epi−b2+K−N∧D2−KC.
Moreover, if formulas (5) and (6) are substituted into formula (18), the expected utility function of the entire supply chain can be calculated as follows:(19)π=∫Uq1+q2−piDhpidpi+∫pi−w1+Nhpidpi+∫pi−b2+K−Nhpidpi−b1∫w1∞Nhpidpi−KC=∫0∂U/∂NUDpi−piDpidhpi+∫∂U/∂N∞UN−piNdhpi+∫w1∞pi−w1Ndhpi+∫w1∞pi−b2K−Ndhpi−∫w1∞b1Ndhpi−KC.
Next, taking the first-order partial derivative with respect toN for formula (19), the optimality conditions of first order can be calculated as follows:(20)∂π∂N=∫∂U/∂N∞U′N−pidHpi+∫w1∞pi−w1dHpi−∫w1∞pi−b2dHpi−∫w1∞b1dHpi=∫∂U/∂N∞U′N−pidHpi+∫w1∞b2−b1−w1dHpi=EU′N−pi−∫∂U/∂N∞U′N−pidHpi+Eb2−b1−w1−b2−b1−w11−Hpi=U′N1−HU′N−∫0∂U/∂NpidHpi−pi+b2−b1−w1Hpi=U′N1−HU′N−piHpi−pi+∫0∂U/∂NHpidpi−pi+b2−b1−w1Hpi.
In addition, take the second derivative with respect toN for formula (20) to get ∂2π/∂N2=U″N.
In accordance with Assumption2, the article has ∂2π/∂N2=U″N≤0 as the result of the contract customer is a risk averse. That is to say, πw0,w1,N is a concave function with respect to N. Let ∂π/∂N=0, then the optimal option purchase amount N∗ satisfies the following formula:(21)U′N∗1−HU′N∗+∫0∂U/∂NHpidpi=b1+w1+pi−b2Hpi+pi.
Therefore, Theorem2 is proved.Theorem 3.
The optimal option purchase amountN∗ of the contract customer is monotonously decreasing with respect to the option strike price w1.Proof.
For formula (16), taking total differentiation with respect to the option strike price w1 using Newton–Leibniz formula, formula (22) is obtained:(22)∂N∗∂w1=HpiU″N∗<0.
Obviously, if the railway transport enterprise increases the option strike pricew1, the optimal option purchase amount N∗ of the contract customer will be reduced accordingly.


## 4. Optimal Pricing Decision of RFO

### 4.1. Optimal Option Strike Price of RFO

Under the circumstance of contract market coexisting with spot market, the article should not focus on the decentralized decision-making between contract customers and railway transport enterprises, but the equilibrium of supply chain system. Therefore, according to the principle of supply chain utility maximization and formula (13), a programming optimization model is established as follows:(23)Γ=maxw1EUq1+q2−piED1+Epi−w1+D1∧N∗−b1q1+Epi−b2+K−N∗∧D2−KC,s.tw1≥0.According to the proof of Theorem2, formula (23) can be rewritten as follows:(24)Γ=∫0∂U/∂NUDpi−piDpidhpi+∫∂U/∂N∞UN−piNdhpi+∫w1∞pi−w1Ndhpi+∫w1∞pi−b2K−Ndhpi−∫w1∞b1Ndhpi−KC=∫0∂U/∂NUDpi−piDpidhpi+∫∂U/∂N∞UN−piNdhpi+∫w1∞pi−w2K+b2−b1−w1dhpi−KC.Using Kuhn–Tucker conditions to solve formula (23), the optimal option strike price w1∗ of RFO can be obtained, as described in Theorem 4:Theorem 4.
The optimal option strike pricew1∗ of RFO satisfies the following formula:(25)b2K+b2N∗−piK−b1N∗−w1∗N∗hw1∗+b2−b1−w1∗1−ψzw1−pi1−ψzU‘HpiU″N∗−N∗1−ψzw1=0,l1w1∗=0,l1≥0.Proof.
The Kuhn–Tucker conditions of formula (23) are constructed as follows:(26)∂ϑ∂w1=∂Γ∂w1+l1=0,ϑ=Γ+l1w1,l1≥0,l1w1=0.
Next, take the first-order partial derivative with respect tow1 for the Lagrangian function ϑ as follows:(27)∂Γ∂w1=∂∂w1pi−b2K∫w1∞hpidpi+b2−b1−w1N∗∫w1∞hpidpi−∂N∗∂w1pi∫∂U/∂N1∞hpidpi+l1=b2−piKhw1−N∗∫w1∞hpidpi+b2−b1−w1N∗hw1+∂N∗∂w1b2−b1−w1∫w1∞hpidpi−∂N∗∂w1pi∫∂U/∂N1∞hpidpi+l1=b2K+b2N∗−piK−b1N∗−w1N∗hw1+b2−b1−w11−ψzw1−pi1−ψzU′∂N∗∂w1−N∗1−ψzw1,in which zw1=w1−μD2/σD2 and ψzw1=∫−∞w1ftdt.
Then, substituting formulas (22) and (27) into formula (26), formula (25) is obtained.
Therefore, Theorem4 is proved.


### 4.2. Optimal Option Price of RFO

Considering the optimal option purchase amountN∗ and optimal option strike price w1∗ calculated above, the original utility function independent variable is changed based on the principle of supply chain utility maximization. Therefore, the constraint problem of satisfying the optimal option price w0∗ can be updated as follows:(28)Π=w0N+w1∗q1−b1q1+K−N∗∧D2pi−b2+−KC,s.tw0≥0.According to the proof of Theorem2, formula (28) can be rewritten as follows:(29)Π=w0N+w1−b1∫w1∞Nhpidpi+pi−b2+∫0K−ND2fD2dD2+∫K−N∞K−NfD2dD2−KC.Using Kuhn–Tucker conditions to solve formula (28), the optimal option price w0∗ of RFO can be obtained, as described in Theorem 5.Theorem 5.
The optimal option pricew0∗ of RFO satisfies the following formula:(30)N∗+w0∗+w1∗−b11−ψzw1∗−pi−b2+1−ψzK−N∂N∗∂w0∗=0,l2w0∗=0,l2≥0.Proof.
The Kuhn–Tucker conditions of formula (28) are constructed as follows:(31)∂ϑ∂w0=∂Π∂w0+l2=0,ϑ=Π+l2w0,l2≥0,l2w0=0.
Next, take the first-order partial derivative with respect tow0 for the Lagrangian function ϑ as follows:(32)∂Π∂w0=N∗+w0∂N∗∂w0+∂N∗∂w0w1−b1∫w1∞hpidpi−∂N∗∂w0pi−b2+∫K−N∞fD2dD2=N∗+w0+w1−b11−ψzw1−pi−b2+1−ψzK−N∂N∗∂w0.
Then, substituting formula (32) into formula (31), formula (30) is obtained.
Therefore, Theorem5 is proved.


### 4.3. Results and Discussion

The optimal pricing decision of RFO can be divided into four cases as follows:(1)
w0∗>0,w1∗=0In this case, the amount of RFO provided by railway transportation enterprises is equal to the contract market capacity. In other words, the contract signed with the contract customer is a fixed volume contract.(2)
w0∗=0,w1∗>0In this case, the railway transportation enterprise does not provide any RFO, and all transport capacities are sold in the spot market. In other words, the railway transportation enterprise accepts all transport capacities reservations without charging any fees.(3)
w0∗>0,w1∗>0In this case, the railway transportation enterprise provides both RFO and the spot market capacity, and the contract signed with the contract customer can be regarded as RFO.(4)
w0∗>0,w1∗>0The solution is a theoretical local optimal solution without any practical economic significance.According to the above analysis, when the railway transportation enterprise writes RFO, the option strike pricew1 should be no more than the long-term preparation cost b1. A reasonable option price w0 can make up for the opportunity cost of the RFO order quantity that cannot be sold in the spot market and also avoid the risk that the marginal cost is higher than the option strike price. Meanwhile, the contract customer’s RFO purchase amount N is negatively correlated with the strike price w1 and the option price w0. In addition, the strike price w1 is more sensitive to the impact on the expected supply chain utility than the option price w0. This is mainly because if only the total amount that the contract customer has to add up to the option strike price w1 is less than the spot freight rate pi at that point, the contract customer will exercise his RFO, and of course, the contract customer will abandon his RFO in reverse. Therefore, the railway transportation enterprise should pay more attention to the strike price w1 in terms of maximizing system utility and achieving Pareto optimal.


## 4.1. Optimal Option Strike Price of RFO

Under the circumstance of contract market coexisting with spot market, the article should not focus on the decentralized decision-making between contract customers and railway transport enterprises, but the equilibrium of supply chain system. Therefore, according to the principle of supply chain utility maximization and formula (13), a programming optimization model is established as follows:(23)Γ=maxw1EUq1+q2−piED1+Epi−w1+D1∧N∗−b1q1+Epi−b2+K−N∗∧D2−KC,s.tw1≥0.According to the proof of Theorem2, formula (23) can be rewritten as follows:(24)Γ=∫0∂U/∂NUDpi−piDpidhpi+∫∂U/∂N∞UN−piNdhpi+∫w1∞pi−w1Ndhpi+∫w1∞pi−b2K−Ndhpi−∫w1∞b1Ndhpi−KC=∫0∂U/∂NUDpi−piDpidhpi+∫∂U/∂N∞UN−piNdhpi+∫w1∞pi−w2K+b2−b1−w1dhpi−KC.Using Kuhn–Tucker conditions to solve formula (23), the optimal option strike price w1∗ of RFO can be obtained, as described in Theorem 4:Theorem 4.
The optimal option strike pricew1∗ of RFO satisfies the following formula:(25)b2K+b2N∗−piK−b1N∗−w1∗N∗hw1∗+b2−b1−w1∗1−ψzw1−pi1−ψzU‘HpiU″N∗−N∗1−ψzw1=0,l1w1∗=0,l1≥0.Proof.
The Kuhn–Tucker conditions of formula (23) are constructed as follows:(26)∂ϑ∂w1=∂Γ∂w1+l1=0,ϑ=Γ+l1w1,l1≥0,l1w1=0.
Next, take the first-order partial derivative with respect tow1 for the Lagrangian function ϑ as follows:(27)∂Γ∂w1=∂∂w1pi−b2K∫w1∞hpidpi+b2−b1−w1N∗∫w1∞hpidpi−∂N∗∂w1pi∫∂U/∂N1∞hpidpi+l1=b2−piKhw1−N∗∫w1∞hpidpi+b2−b1−w1N∗hw1+∂N∗∂w1b2−b1−w1∫w1∞hpidpi−∂N∗∂w1pi∫∂U/∂N1∞hpidpi+l1=b2K+b2N∗−piK−b1N∗−w1N∗hw1+b2−b1−w11−ψzw1−pi1−ψzU′∂N∗∂w1−N∗1−ψzw1,in which zw1=w1−μD2/σD2 and ψzw1=∫−∞w1ftdt.
Then, substituting formulas (22) and (27) into formula (26), formula (25) is obtained.
Therefore, Theorem4 is proved.


## 4.2. Optimal Option Price of RFO

Considering the optimal option purchase amountN∗ and optimal option strike price w1∗ calculated above, the original utility function independent variable is changed based on the principle of supply chain utility maximization. Therefore, the constraint problem of satisfying the optimal option price w0∗ can be updated as follows:(28)Π=w0N+w1∗q1−b1q1+K−N∗∧D2pi−b2+−KC,s.tw0≥0.According to the proof of Theorem2, formula (28) can be rewritten as follows:(29)Π=w0N+w1−b1∫w1∞Nhpidpi+pi−b2+∫0K−ND2fD2dD2+∫K−N∞K−NfD2dD2−KC.Using Kuhn–Tucker conditions to solve formula (28), the optimal option price w0∗ of RFO can be obtained, as described in Theorem 5.Theorem 5.
The optimal option pricew0∗ of RFO satisfies the following formula:(30)N∗+w0∗+w1∗−b11−ψzw1∗−pi−b2+1−ψzK−N∂N∗∂w0∗=0,l2w0∗=0,l2≥0.Proof.
The Kuhn–Tucker conditions of formula (28) are constructed as follows:(31)∂ϑ∂w0=∂Π∂w0+l2=0,ϑ=Π+l2w0,l2≥0,l2w0=0.
Next, take the first-order partial derivative with respect tow0 for the Lagrangian function ϑ as follows:(32)∂Π∂w0=N∗+w0∂N∗∂w0+∂N∗∂w0w1−b1∫w1∞hpidpi−∂N∗∂w0pi−b2+∫K−N∞fD2dD2=N∗+w0+w1−b11−ψzw1−pi−b2+1−ψzK−N∂N∗∂w0.
Then, substituting formula (32) into formula (31), formula (30) is obtained.
Therefore, Theorem5 is proved.


## 4.3. Results and Discussion

The optimal pricing decision of RFO can be divided into four cases as follows:(1)
w0∗>0,w1∗=0In this case, the amount of RFO provided by railway transportation enterprises is equal to the contract market capacity. In other words, the contract signed with the contract customer is a fixed volume contract.(2)
w0∗=0,w1∗>0In this case, the railway transportation enterprise does not provide any RFO, and all transport capacities are sold in the spot market. In other words, the railway transportation enterprise accepts all transport capacities reservations without charging any fees.(3)
w0∗>0,w1∗>0In this case, the railway transportation enterprise provides both RFO and the spot market capacity, and the contract signed with the contract customer can be regarded as RFO.(4)
w0∗>0,w1∗>0The solution is a theoretical local optimal solution without any practical economic significance.According to the above analysis, when the railway transportation enterprise writes RFO, the option strike pricew1 should be no more than the long-term preparation cost b1. A reasonable option price w0 can make up for the opportunity cost of the RFO order quantity that cannot be sold in the spot market and also avoid the risk that the marginal cost is higher than the option strike price. Meanwhile, the contract customer’s RFO purchase amount N is negatively correlated with the strike price w1 and the option price w0. In addition, the strike price w1 is more sensitive to the impact on the expected supply chain utility than the option price w0. This is mainly because if only the total amount that the contract customer has to add up to the option strike price w1 is less than the spot freight rate pi at that point, the contract customer will exercise his RFO, and of course, the contract customer will abandon his RFO in reverse. Therefore, the railway transportation enterprise should pay more attention to the strike price w1 in terms of maximizing system utility and achieving Pareto optimal.


## 5. Conclusion

For railway freight pricing, this paper introduces the concept of option theory and proposes a new tradable rail freight option (RFO). Different from the previous works, this paper establishes a two-stage Stackelberg game model based on the perspective of maximizing the utility of the supply chain to discuss the optimal pricing strategy of RFO. Considering the coexistence of the contract market and the spot market, this study uses the reverse reasoning method in dynamic programming to solve the optimal combination decision of the contract customer firstly. Then, the optimal pricing decision of RFO is obtained using Kuhn–Tucker conditions and Lagrangian function. The main conclusions of this article are as follows: (1) the option strike pricew1 should no more than the long-term preparation cost b1; (2) the contract customer’s RFO purchase amount N is negatively correlated with the strike price w1 and the option price w0; and (3) the railway transportation enterprise should pay more attention to the option strike price w1 in terms of maximizing system utility and achieving Pareto optimal.



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*Source: 6436729-2020-06-22.xml*