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A successful supply chain management (SCM) should aim to maximize the net present value of joint profits along the supply chain. However, the volatile market conditions cause the future cash flows along the supply chain more difficult to anticipate. To obtain higher supply chain value, the supplier and the retailer should cooperatively determine the optimal entry time. This paper proposes a two-stage dynamic optimization model by using a real option approach and then performs the sensitivity analyses for the option value and the investment threshold. The impacts of some critical factors, including the growth rate and the volatility of demand shock, sunk cost, and relating operational costs (cost rates, fix costs, holding costs of inventory, and shipping costs), are investigated.

In an inflationary world today, the declining demands are seriously hampering the profitability of companies and forcing them to compete for the diminishing profits. However, the proper coordination in the marketing channel could generate more joint profits along the supply chain. The extra profits can be then distributed among the players of the supply chain. Many previous researches have focused on the issues of channel coordination and profits sharing. Giannooccaro and Pontrandolfo (2004) classified the supply chain into two categories: centralized and decentralized. Firstly, the centralized supply chain involves the environment under which a unique decision maker controls the pricing policy due to its dominance and bargaining power along the supply chain. It can be also regarded as a vertical integrated system. Secondly, the decentralized system allows the players in a supply chain to make their own pricing policies separately and independently. However, the local maximization could not necessary be global maximum. Yong-Wu Zhou and Shanlin Yang (2008) demonstrated that the joint profits generated by the players in a supply chain when they make the pricing decisions coordinately are 1.3 and 2.3 times the profits, which are generated when pricing decisions are made independently, in the context of 2-echelon and 3-echelon systems, respectively. Biehl et al. (2006) also examined the effectiveness of joint decision making with buyer–supplier relationships in manufacturing and indicated that cooperative decision making contributes to the efficiency of SCM and to their competitiveness in a market. To facilitate the coordination along the supply chain, some mechanisms can be developed through quantity discount, price discount, and profit-sharing contract, etc. All the mechanisms are used to distribute the extra profits among the players in a supply chain favorably and fairly, and to ensure that all of them will gain more profits when they make decisions jointly than they do separately. Giannooccaro and Pontrandolfo (2004) used the revenue sharing contract in their model to coordinate the supply chain, but they assumed that the market demand is independent of the retailing price, while Zhou and Yang assumed the deterministic price-sensitive demand. Stefano Battiston et al. (2007) investigated the impacts of interactions among firms which are connected by production and credit ties, on the bankruptcy prorogation due to supply failures. All the profit-sharing mechanisms are designed to distribute extra profits among the players in a supply chain favorably and fairly. To measure the effectiveness and desirability of profit sharing mechanism, Giannooccaro and Pontrandolfo (2009) proposed a desirability index by calculating a ratio of profit after coordination relative to profit before coordination for each player in a supply chain. The higher the ratio, the more incremental profit the player gets, and the more the player is satisfied with profit sharing mechanism. Even though many researches emphasize the coordination issues, but they discussed in different settings. Xiao et al. (2005) proposed a SCM model with one manufacturer and two retailers, and found that the contractual arrangements can enhance the coordination among them, while Sethi et al. (2005) formulated a SCM with two suppliers and a single retailer. Similarly, we aim to explore the cooperative decision making for the joint profits in a single-supplier and single-retailer system. There exist two echelons in the supply chain: the first level belongs to the business market with a single supplier, while the second level belongs to the consumer market with a single distributer. The supplier delivers the goods to the distributer, who will then sell the products directly to the final consumers. The optimal delivery rates of the supplier and the distributer are jointly determined through the maximization of the total profits in the supply chain, while the optimal entry time is determined by the rule that the stochastic supply chain value should be higher than the investment threshold (value threshold or demand shock threshold), which can be derived in this framework. According to the definition of Dixit and Pindyck, 1994a and Dixit and Pindyck, 1994b, investment threshold is a cutoff value, greater than which termination of real option (undertaking investment) is optimal, less than which continuation of real option (waiting) is optimal, and at which termination and continuation is equally optimal. Based on the above definition, optimal entry time is defined as the time when undertaking an investment is optimal. In other words, when corporate value is greater than investment threshold, it is the optimal entry time to undertake the investment. As to the investment strategy to be discussed in this framework, it is the strategy of investment timing, defined as a rule to determine the optimal time to invest. Compared with the price trigger in the research of Sigbjorn Sodal (2006) on the entry and exit decisions for the investment projects, the value threshold in this model is similar to the price trigger. This paper extends the “Capacity Choice” model by Pindyck (1988), who employs the contingent claim approach to propose that the market value of a firm can be maximized through the optimal choice of the production capacity, which depends deeply on how the stochastic demand evolves. Instead, we intend to propose an optimization model by using the stochastic dynamic programming approach to valuate a dynamic supply chain. Moreover, Pindyck (1991) indicated in the framework of “Investment Timing” that the value threshold is chosen to maximize the net payoff of an investment, which once undertaken incurs the irreversible sunk cost. The common decision rule for those papers focusing on the entry strategies is that the investment will not be undertaken until the project value stochastically evolves higher than the value threshold. The first passage time when the value of the underlying real asset is above the threshold is referred to as the optimal entry time. Under the assumption that the uncertainty of market demands follows a geometric Brownian motion (GBM), the delivery rates for the supplier and the retailer can be simultaneously determined in maximizing the supply chain value by a real option approach. Many discussions on the real option model in continuous time could be found in the researches by Brennan and Schwartz (1985), McDonald and Siegel (1986), Pindyck, 1988 and Pindyck, 1991, Dixit (1989) while the lattice method in discrete time is used to valuate the real option by Brandao and Dyer (2005), who derive the numerical solutions by a decision tree tool. Most of them focus on the revenue-related uncertainty, while some others focus on the technological uncertainty, e.g. Grenadier and Weiss (1977), Farzin et al. (1998), Doraszelski (2004). Moreover, Pauli Murto (2007) investigates the interactions between the technological and the revenue-related uncertainties. The main reason why the scholars valuate a real asset (a project or firm value) by the real option approach is that the traditional NPV method underestimates the value of the real asset because of the neglect of the uncertainty and the manufacturing flexibility (Pindyck, 1991 and Triantis and Hodder, 1990). They focused on the effects of the price volatility on the output rate and found that the increase of the price volatility induces the increase of the output rate. The other effects of the price volatility could be found in the works of Van Wijnbergen (1985), Ingersoll and Ross (1992). In this model, the demand uncertainty, instead of the price uncertainty, is introduced. The price elasticity of demand and the discount rate are also considered in the derivation of the value threshold for the supply chain investment. Sensitivity analyses of the value threshold and the option value to the demand uncertainty and elasticity are also performed to provide readers with more insights into this framework. This paper is organized as follows: Section 1 addresses the necessity and benefits of implementing a coordinated supply chain and introduces the recent researches on the maximization of joint profits. A real option approach is also introduced to explore and the effects of the price (demand) uncertainty and the interest rate on the output rate for a manufacturing company. Section 2 presents a two-echelon supply chain model under the assumption of the stochastically evolved demand uncertainty in a single-supplier/single-retailer system. The dynamic maximization in joint profits generated from the supplier and the retailer is proposed and the analytical solutions are obtained. In Section 3, we use the empirical data to test the fit of a GBM process for several different products and then estimate the parameters used in this framework. Section 4 performs sensitivity analyses of value thresholds and option values. The analytical results are also depicted in figures and the implications are discussed. Section 5 concludes this paper and recommends the possible further studies.

This paper employs a real option approach to investigate the impacts of the growth rate and the volatility of demand shock, sunk cost, shipping costs, holding costs, and cost rates of goods sold on the investment threshold, as well as on the option value for a single-supplier/single-retailer system in a two-echelon supply chain setting. In the framework, we assume that cost rate is a proportion of the selling price with fix cost considered. The analytical solutions for threshold and real option value demonstrate that the effect of the fix cost is similar to the effect of the shipping cost. We also assume that the supplier and the retailer agree to cooperatively determine the optimal timing to establish the supply chain for the maximized combined profits. As a result, both of them get more profits jointly than they do separately. The uncertainty of the joint profits comes from the stochastic demand shock in the inversed demand function proposed by Carruth et al. (2000). The demand shock evolves as a geometric Brownian motion process, validated by empirical data acquired from the database of Global Insight. In this model, a retailer has rights to place orders, depending on the realized demand shock and its threshold level, and then receive goods form the supplier. The demand shock threshold can be equivalently transformed into the value threshold (Dixit and Pindyck, 1994a and Dixit and Pindyck, 1994b), which is more economically meaningful than the demand shock threshold. It can be shown that the higher threshold allows a firm to defer the investment because it will take more time to reach the threshold level and then activate the investment activity. As a result, the optimal entry time in a real option approach is later than in the NPV approach, which requires the investors to make decisions immediately. Moreover, the retailer incurs the shipping costs and the holding costs for the inventory, which are charged according to the quantity of the goods delivered from the supplier. To obtain the maximum combined operational profits along the supply chain, the supplier and the retailer should jointly monitor the market demand shock (or the project value) in a coordination way to see whether it reaches the threshold level, at which the investment is undertaken and the goods-delivery begins. In other words, the supplier and the retailer jointly own a call option to invest, which permits them to wait for the optimal investment opportunity. While the demand shock (or the project value) is below the threshold level, the retailer and the supplier keep waiting for a better opportunity. To construct the value dynamics of the supply chain in the context mentioned above, this paper extends the option model proposed by Dixit and Pindyck, 1994a and Dixit and Pindyck, 1994b to embed the derived stochastic value process in this framework. The value process is driven by the market demand uncertainty. In this paper, analytical solutions are derived for the succeeding sensitivity analyses. Based on the analytical results, we conclude that the growth rate of demand shock and the price elasticity have decreasing impacts on the threshold, while the volatility of demand shock has increasing impacts on the threshold at lower level, but decreasing effects at higher level. However, the price elasticity has stronger effects than the volatility. Even though the drift term, the volatility, and the price elasticity have different impacts on the threshold of demand shock, they all give positive impacts on the option value. When we consider the relevant costs along the supply chain, the higher cost rate of goods sold for the supplier raises the threshold, while the higher cost rate for the retailer lowers the threshold. Conversely, the increasing cost rate for the supplier lowers the option value, while the increasing cost rate for the retailer raises the option value. The higher holding costs of inventory and the higher shipping cost for the supplier raises the threshold, while the higher shipping cost for the retailer lowers the threshold. Only the holding cost and the shipping cost for the supplier give decreasing effects of the option value as usual. The shipping cost for the retailer has slight positive impacts on the option value. Based on the thresholds derived in this paper, the supplier and the retailer could closely and jointly monitor the uncertainty of the stochastic demand according to step 3 of the procedures for parameter estimation in Section 3 and then respond quickly to the dynamic demand shock process to see whether it reaches the level of the threshold in Eq. (30), at or above which the supplier and the retailer should immediately deliver the goods to the final customers. However, in a less volatile market, the increasing threshold is obtained in this model to inform that the supplier and the retailer should keep waiting longer for the optimal entry time. But in a more volatile market, the decreasing threshold informs the supplier and the retailer of undertaking the investment earlier. That is, the retailer and the supplier should coordinately pay attention not only to the volatility but to price elasticity when they make any decisions for the supply chain activities. For a better investment decision, they should be also aware of the relevant costs, such as the cost rates of goods sold, the holding and the shipping costs. Therefore, the conclusions drawn upon in this model could be employed to assist the players along the supply chain to make better decisions: the hedging of demand risks and the maximization of the joint profit for a dynamic supply chain. The option discussed in this paper has its implications: the investment opportunity which permits the supplier and the retailer to wait for the time when the maximum supply chain value appears. Moreover, if the demand trend is reversed downwards, the supplier and the retailer could save the losses from their waiting and suspension of the goods deliveries. For those readers who are interested in the topics of the strategies in a supply chain, it is recommended that the assumptions made in this model concerning a single-supplier single-retailer system in the two-echelon supply chain context could be released as the setting in the work by Sethi et al. (2005), who use the game theory to formulate the supply chain with a single retailer and two suppliers, which compete for the retailer's purchases, and concluded by the Nash solutions. Therefore, a more generalized framework can be proposed in further studies to navigate the coordination decisions in the multi-supplier or multi-retailer system.