Competitive analysis for online choice of foreign production strategy

Abstract

Motivated by a sequential choice of foreign production strategies (CFPSs) in an actual decision for entering the lesser-known foreign markets, this article proposes two online models to analyze an optimal online strategy where the objective is to minimize the cost to supply the total demand of the foreign market. A basic online model for the CFPS problem with exporting (EXP) and wholly owned subsidiary (WOS) is developed to show how the transport/tariff cost and the entry/exit cost affect the switching timing and the competitive ratio of the online strategy. We investigate the online model where the firm can choose the joint ventures (JVs) besides EXP and WOS. Our results show that online strategies may not necessarily transition from EXP to JV mode. The decision-makers need to determine whether to transition from EXP to JV mode based on the cost characteristics among the three modes of EXP, JV, and WOS.

Introduction

With the globalization of the world's economy, the competitions facing the firm in the domestic market become increasingly fierce due to the multinational corporation's (MNC’s) entry. In order to offset the MNC's competition in the domestic market, companies must enter the international market and participate in global competition. The first important problem for domestic enterprises is to choose the foreign entry mode that determines how the company organizes its production activities to serve the foreign market. This is also known as the choice of foreign production strategy problem (CFPS problem) on global operations management. ¹ According to the degree of control over the foreign production activities that each strategy affords, the alternatives that a firm can choose are exporting (EXP), licensing (LG), and foreign direct investment (FDI). The FDI can be divided into joint ventures (JVs) with local partners and wholly owned subsidiary (WOS) in the foreign country. The EXP and WOS are two extreme cases where the EXP cannot influence the foreign production activities, while the WOS can control these activities completely. The LG and JV are the eclectic strategies between EXP and WOS, where the degree of control depends on the firm's monopolistic advantage and the share of the firm in the foreign enterprise.

Meeting overseas demand by a company is a long-term process that involves many sub-cycles. The input sequence of overseas market demand appears one by one. Due to the unfamiliarity of company decision-makers with the overseas market, they only know the current sub-cycle demand of overseas markets, and do not know the future sequential demand of the entire process. In order to enter the overseas market, companies must choose an entry mode to meet the input sequence of overseas market demand in the current period. The decision-making of this problem is a typical online problem.

In the analysis of decision making under uncertainty, the traditional probabilistic analysis is developed with the goal of optimizing the expected behaviors with respect to probability distributions. In this article, we develop an online formulation for the CFPS problem in which a company has to choose a mode to meet sequential demand in order to enter overseas markets without knowing the entire process demand in the future. The objective function is to minimize costs while meeting the entire sequence of demand in overseas markets. In an online model, a sequence of events $\sigma ={\sigma }_1,{\sigma }_2,\dots$ appears over time and have to be dealt with immediately. The decision-maker designs an online strategy in response to an event without knowing what events the future holds, and the performance of the online strategy is compared with the performance of an optimal offline strategy that has knowledge of the entire future and operates optimally. Given an input sequence $\sigma$ , let $C_A(\sigma )$ denote the cost incurred by online algorithm A in serving $\sigma$ , and $C_A(\sigma )$ denote the minimum cost using the optimal offline algorithm. For a minimization problem, we call online algorithm $Ar-$ competitive, if $C_A(\sigma )\le r\times C_{\mathrm{OPT}}(\sigma )$ for all input sequences $\sigma$ . The factor r is called the competitive ratio of online algorithm A. This technique of evaluating an online strategy by comparing its performance to the optimal offline strategy is called competitive analysis and was first used by Sleator and Tarjan. ² Later, it was widely applied to operations management in uncertain environments. For example, online scheduling problem, ³ online routing problem, ⁴ and online inventory problem. ⁵

Online algorithm is one of the important methods for solving uncertain decision-making problems. This method is mainly used to deal with decision-makers in long-term uncertain environments who lack accurate probability information of historical data and unknown events, and whose problem processing cannot wait for decision-makers to obtain sufficient information and must make decisions in the current period. Compared to traditional probability models, online optimization methods can ensure that the gap between the performance of the decision solution and the optimal solution is always within a certain range. With the increase of internal and external uncertainties, the deviation between the average optimal value obtained by applying probability models and the actual situation is becoming larger and larger. Decision makers are increasingly willing to choose online optimization solutions with robust performance to mitigate decision-making risks. This is especially true in starting point of the CFPS problem, where it is very difficult to characterize the uncertain demand, not only because the entrant has little information, but also because the total demand consists of the sequential sub-period demand.

Related work

Choice of foreign entry mode

The research on the significant impact of CFPS problems began with the study of FDI problems. The problem assumes that MNCs preparing to enter foreign markets have monopoly advantages or patents, and decision-makers choose between three modes of EXP, LG, and FDI based on the cost function of different entry modes. Buckley and Casson ⁶ regarded the choice of foreign entry modes as a function of transportation/tariff costs, fixed costs of FDI, and foreign market demand, and studied the optimal entry modes under different market demands. Eicher and Kang ⁷ considered the impact of the level of competition in overseas markets on the optimal entry mode, and analyzed the effects of different levels of competition and tariff policies on the welfare of local enterprises. Appiah-Kubi et al. ⁸ studied the impact of tax incentives on African countries’ absorption of FDI and found that direct investment responds to lower corporate income tax. Andreu et al. ⁹ analyzed the influence of family character on the choice of foreign entry mode for family businesses. Schmitzet al. ¹⁰ studied how digital enterprises choose their overseas entry modes and which specific background modes are preferred. The results show that in situations where institutions are weak, acquisitions and greenfield projects are incomplete, JVs and digital entry are more desirable.

The FDI can be divided into JVs with local partners and WOS in the foreign country. Anderson and Gatignon ¹¹ studied the impact of bargaining behavior among shareholders and cultural and institutional differences between countries on the choice of equity structure in overseas enterprises based on transaction cost theory. Gome-Casseres ¹² found that when local partners complement the company's capabilities and market transactions are more costly than equity cooperation, MNCs tend to form JVs with local enterprises.

CFPS problem under uncertainty

Uncertainty is a key factor affecting a company's choice of foreign entry mode. Dixit ¹³ investigated the entry and exit decisions under uncertainty and discovered that the entry and exit cost would produce “hysteresis” in response to uncertainty. Kouvelis et al. ¹ developed a stochastic dynamic programming formulation for the CFPS problem with switching cost, in which the firm can adjust the ownership structure of the foreign enterprise dynamically in response to real exchange rate fluctuations. This model also discussed the impact of the switching cost on “hysteresis phenomenon.” Goldberg and Kolstad ¹⁴ studied the choice of foreign entry mode with uncertain exchange rates and demand. The results show that when there is a non-negative correlation between demand and exchange rate fluctuations, MNCs are more willing to choose the entry mode of FDI. When the fluctuation of exchange rates increases, and the correlation with demand fluctuations increases, the company will increase its equity investment in overseas enterprises. Pennings and Sleuwaegen ¹⁵ considered timing and foreign entry mode simultaneously under uncertainty about future profits. Wooster et al. ¹⁶ developed a real option model of foreign entry mode choices under environmental uncertainty. Li and Xiong ¹⁷ investigated the impact of the host country's environmental uncertainty and technological capability on the choice of entry mode simultaneously.

Online leasing problem

Another important issue related to the online CFPS problem is the well-known ski-rental problem in computer science. ¹⁸ The players design an online strategy that is to rent a sled until the rental cost accumulates to almost equal the sled price, and then make a one-time payment to purchase the sled. This online strategy can ensure that the sled usage cost does not exceed twice the optimal strategy cost. El-Yaniv et al. ¹⁹ presented a randomized online algorithm for the ski-rental problem; their work showed that the competitive ratio in the contiguous case is exactly equal to the competitive ratio in the non-contiguous case. Al-Binali ²⁰ introduced a risk-reward framework for this problem and presented the complete set of strategies that respect the risk tolerance level. Lotker et al. ²¹ developed a randomized online algorithm for the multi-slope ski rental problem that all strategies cannot be rejected. Dai et al. ²² studied an online financial leasing problem in which the lessee can buy ownership of equipment after the lease agreement ends. Zhang et al. ²³ considered the deterministic and randomized online strategies for depreciable equipment with opportunity cost. Xin et al. ²⁴ introduced an online leasing problem of durable equipment and analyzed the impact of transaction costs on online strategies. Feng and Chu ²⁵ investigated an online leasing problem in which the price of the required equipment is assumed to fluctuate over time and lie in a predetermined range.

In this article, we design an online algorithm to preserve flexibility in investing in the foreign market, which first selects EXP to supply the demand of the foreign market, then determines whether to adopt the JV mode and the timing switching to JV, and finally switches to WOS mode. Our results show that the optimal switching timings depend on the entry/exit cost, transport/tariff cost caused by market factors, and the reimbursement influenced by the firm's advantages over the local partner. The optimal online algorithm can guarantee maximal cost of implementing global objective in the worst case. Obviously, this approach is not appropriate for solving all CFPS problems under uncertain demand, but we do believe that it is appropriate for the initial stage of CFPS problems when the entrant has little information or doesn't have the experience of international expansion. Once the decision-maker has sufficient data and experience, choosing the production strategy with the goal of optimizing the expected behaviors could then be more appropriate.

Basic online model

Online CFPS problem description and assumptions

Consider a company that operates only domestically, facing increasing competition from MNCs in the domestic market. In order to offset the MNC's competitive pressure in the domestic market, companies have to enter overseas markets to participate in competition. Due to a lack of overseas market experience, company decision-makers in the early stages of entering the overseas market only know the current stage of overseas market demand, without knowing the demand in the later stages and how long the demand will last (how long the company can survive in the overseas market). Decision makers must choose a mode between EXP and WOS to meet the future demand of overseas markets without information about the demand. The decision-making process for this problem is a multi-stage dynamic online problem.

Motivated by the above description, there are main premises in the online model as following:

Due to the fact that MNCs can continuously increase competitive pressure on the domestic market by gaining profits from foreign markets, in order to offset the pressure from MNCs, companies must enter foreign markets as early as possible to participate in global competition.

The total demand of the foreign market, in the long-term strategic process, is the main element to determine production strategy, which consists of sequential sub-period demand. In the initial stage of entering foreign markets, the company does not have sufficient information to predict future market demand in foreign markets, and decision-makers only know the demand of the next sub-period, has little information to know the total demand or probability distribution of the total demand.

The decision-maker needs to choose between EXP and WOS at a discrete sub-period. Once the decision-maker chooses the EXP to enter foreign market, the transport/tariff cost for per unit goods must be paid, and avoids paying transport/tariff cost if choosing WOS.

The decision-maker can switch from EXP to WOS, but cannot switch from WOS to EXP because the cost per unit of EXP is more than that of WOS. The time of implementing a mode switch is assumed insignificant (Kouvelis et al., 2001).

The investment size of new plant in foreign country is assumed to be constant, which consists of the price of production facilities and the cost of developing a distribution network. The price of the used plant is the difference between the investment size of the new plant and the aggregated depreciations.

The salvage value of the used plant is always more than the exit cost of selling it within the lifecycle, so the decision-maker can sell the used plant when he no longer needs it. The new decision starts if the lifecycle is over.

The production function is uniform between EXP and WOS, which means that each production strategy has the same production cost. The total cost per unit of EXP includes the production cost and the transport/tariff cost, and the total cost per unit of the WOS equals the production cost.

The entry/exit cost of EXP is assumed insignificant because the EXP mode doesn't develop new facilities in foreign country.

As the firm chooses the WOS mode to enter a foreign market, there exists an entry cost resulting from the technology transfer and the market barriers of entering host country.

In this research, the following set of notations is used in the model. Table 1 summarizes parameters, variables, and details between parameters.

Table 1.

Notations of parameters and variables.

$D$	The total demand of the foreign market.
$d_t$	The demand of sub-period $t$ ( $t=1,\dots ,T$ ). The decision-maker knows the sub-period demand $d_t$ at time t, but T is not known.
$Q$	The aggregated quantity of supplying foreign market. We may use the assumption with the normalization $d_t=1$ , thus the discrete sub-period t that the decision-maker make choice can be represented by the parameter Q.
$B$	The investment size of new plant.
$b$	The depreciation cost per unit.
$B_t$	The price of the plant at time $t$ ( $t=1,\dots ,T$ ). We have $B_t=B-bD$ , which is a reasonable inference.
$c$	The production cost per unit.
$h$	The transport/ tariff cost per unit. The transportation and tariff costs between two countries are greater than zero.
$k$	The entry cost of WOS.
$s$	The exit cost of selling foreign plant with WOS mode. The entry/exit cost of EXP is zero.
$C_{\mathrm{ON}}$	The cost of the online algorithm achieving the global strategy.
$C_{\mathrm{OPT}}$	The cost of the optimal offline algorithm.

Optimal algorithm in the basic online model

Consider the following algorithm.

Algorithm-Q^W = (We use the switching timing Q^W between EXP and WOS to denote an online algorithm). The firm chooses EXP to supply foreign demand until $Q^W=[(k+s)/h]-1$ and then chooses WOS mode.

Compared to the traditional economic models where the decision-maker chooses an optimal entry mode in terms of a given total demand or the probabilistic distribution of total demand, the main idea of the online algorithm is to delay the tremendous investment of establishing a foreign plant by paying a small cost (transport/tariff cost) and preserve flexibility of global strategy when the decision-maker don't know the foreign markets.

We first prove that Algorithm-Q^W is an optimum algorithm for online CFPS problems where the decision-maker can choose EXP to serve foreign market at least one unit before employing WOS.

Lemma 1.

If $D\le Q^W$ , the cost of optimal offline algorithm is $C_{\mathrm{OPT}}=(c+h)D$ ; if $D>Q^W$ , the cost of optimal offline algorithm is $C_{\mathrm{OPT}}=cD+k+s$ .

Proof.

For a given total demand, the cost function of EXP is

$$C_{\mathrm{EXP}}(D)=(c+h)D$$

(1)

The cost function of the WOS mode can be formulated as the following:

$$C_{\mathrm{WOS}}(D)=k+B+cD-bD-B_t+s$$

(2)

The first term (k) of the right-hand side above is the entry cost of WOS, the second term (B) is the investment size of setting up the foreign plant with WOS, the third term (cD) is production cost supply total demand, the fourth item (bD) is the investment depreciation recovered from the total production, the fifth item (B_t) is the residual value of the enterprise after completing the total output, $B_t=B-bD$ and the last term is the exit cost of selling foreign plant with WOS mode after achieving the total production. Replacing $B_t=B-bD$ into equation (2) and rearranging terms, we have

$$C_{\mathrm{WOS}}(D)=cD+k+s$$

(3)

There exist $c+h>c$ and $k+s>0$ , we have that the EXP, WOS are in ascending order of fixed costs and descending order of variable costs. Therefore, the total cost of EXP is equal to or less than that of WOS when $D\le Q^W$ , and the total cost of EXP is greater than that of WOS if $D>Q^W$ . □

Theorem 1.

Algorithm-Q^W has competitive ratio $C_{\mathrm{ON}}/C_{\mathrm{OPT}}=1+[h(k+s-h)/(c+h)(k+s)]<2$ .

Proof.

Consider the following two cases:

Case 1. $D\le Q^W$ . This means that Algorithm-Q^W doesn't switch from EXP to WOS. Hence, $C_{\mathrm{ON}}=C_{\mathrm{OPT}}$ in this case and the competitive ratio is 1.

Case 2. $D>Q^W$ . In this case, the cost of the Algorithm-Q^W can be formulated as the following:

$$C_{\mathrm{ON}}=(c+h)Q^W+k+c(D-Q^W)+s$$

(4)

The first term $(c+h)Q^W$ of the right-hand side above is the cost of choosing EXP mode to supply the demand (Q^W) of the foreign market. Similar explanations to equations (2) and (3), the last three terms are the cost of employing WOS mode to supply surplus demand ( $D-Q^W$ ) of the foreign market. Replacing $Q^W=[(k+s)/h]-1$ into (4) and rearranging terms, we have

$$C_{\mathrm{ON}}=cD+2(k+s)-h$$

(5)

By Lemma 1, the cost of the offline algorithm is $C_{\mathrm{OPT}}=cD+k+s$ . Thus, we have a competitive ratio

$$r={\displaystyle \frac{C_{\mathrm{ON}}}{C_{\mathrm{OPT}}}={\displaystyle \frac{cD+2(k+s)-h}{cD+k+s}}}$$

(6)

Differentiating r with respect to D, we have $\mathrm{\partial }r/\mathrm{\partial }D<0$ . Therefore, the competitive ratio decreases as the total demand increases. In the worst case, to make the competitive ratio as large as possible, the offline algorithm will choose the total demand $D=Q^W+1$ . After replacing $D=Q^W+1$ into equation (6), we get .□

Theorem 2.

Algorithm-Q^W is the optimal, which obtains the best possible competitive ratio.

Proof.

Consider an arbitrary online algorithm where $Q\ne Q^W$ . Note that for an offline adversary, in order to make the competitive ratio as large as possible, he still chooses the total demand $D=Q+1$ . We consider the following two cases:

Case 1. $Q>Q^W$ . By the proof of Theorem 1, we have

$$r={\displaystyle \frac{C_{\mathrm{ON}}}{C_{\mathrm{OPT}}}={\displaystyle \frac{(c+h)Q+c+k+s}{c(Q+1)+k+s}={\displaystyle \frac{c+h}{c}-{\displaystyle \frac{h(c+k+s)}{c(cQ+c+k+s)}}}}}$$

(7)

Differentiating r with respect to Q, we have $\mathrm{\partial }r/\mathrm{\partial }Q>0$ . Therefore, the competitive ratio increases as the value of Q increases, and we have that the competitive ratio of Algorithm-Q is more than that of Algorithm-Q^W for $Q>Q^W$ .

Case 2. $Q<Q^W$ . By Lemma 1 and the fact that the offline algorithm always chooses $D=Q+1$ , we have that $C_{\mathrm{ON}}=(c+h)Q+c+k+s$ and $C_{\mathrm{OPT}}=(c+h)$ $(Q+1)$ . Thus, we have

$$r={\displaystyle \frac{C_{\mathrm{ON}}}{C_{\mathrm{OPT}}}=1+{\displaystyle \frac{k+s}{(c+h)Q+c+h}}}$$

(8)

Differentiating with respect to Q, we have $\mathrm{\partial }r/\mathrm{\partial }Q<0$ . Therefore, we have that the competitive ratio of Algorithm-Q is more than that of Algorithm-Q^W for $Q<Q^W$ . □

Structural result of competitive ratio and managerial insight

An interesting finding is that in our model, the competitive ratio, which measures the performance of an online algorithm, is determined by the entry/exit cost and the transport/tariff cost. In this sub-section, we prove the structural properties of the competitive ratio function and analyze how the changes in the transport/tariff cost and the entry/exit cost affect the competitive ratio of the optimal online strategy.

Theorem 3.

Algorithm-Q^W has the maximal value of the competitive ratio when the transport/tariff cost is $h*=\sqrt{c(c+k+s)}-c$ , the competitive ratio increases as the entry/exit cost increases, but is at most $1+h/(c+h)$ .

Proof.

We first prove that Algorithm-Q^W has the maximal value of the competitive ratio when the transport/tariff cost is $h*=\sqrt{c(c+k+s)}-c$ . Assuming that the entry/exit cost is a known constant, we consider the competitive ratio function of Algorithm-Q^W with respect to h.

$$r(h)=1+{\displaystyle \frac{h(k+s-h)}{(c+h)(k+s)}=1+{\displaystyle \frac{h}{c+h}-{\displaystyle \frac{h}{k+s}+{\displaystyle \frac{c}{k+s}-{\displaystyle \frac{c^2}{(c+h)(k+s)}}}}}}$$

(9)

This is a global maximum, since we have ${\mathrm{\partial }}^{2}r(h)/\mathrm{\partial }{h}^2<0$ from equation (4). Thus, we have that there exists a value of h maximizing the competitive ratio. In order to determine the value of h in the worst case, we differentiate $r(h)$ with respect to h and set this amount equal to 0. This gives

$${\displaystyle \frac{c}{{(c+h*)}^2}-{\displaystyle \frac{1}{k+s}+{\displaystyle \frac{c^2}{(k+s){(c+h*)}^2}=0}}}$$

Therefore, we have $h*=\sqrt{c(c+k+s)}-c$ .

Next, we prove that the competitive ratio increases as the entry/exit cost increases, but is at most $1+h/(c+h)$ . Assume that the transport/tariff cost is constant, we consider the competitive ratio function with respect to the entry/exit cost.

$$r(k+s)=1+{\displaystyle \frac{h(k+s-h)}{(c+h)(k+s)}=1+{\displaystyle \frac{h}{c+h}-{\displaystyle \frac{h^2}{(c+h)(k+s)}}}}$$

(11)

Differentiating r with respect to $(k+s)$ , we have $\mathrm{\partial }r/\mathrm{\partial }(k+s)>0$ from equation (11). Therefore, the competitive ratio increases as the value of $(k+s)$ increases. Therefore, we have $C_{\mathrm{ON}}/C_{\mathrm{OPT}}<1+$ $h/(c+h)$ . □

Online CFPS problem with JV mode

In this section, we consider an online CFPS problem where the firms can accept or reject the eclectic modes such as JV between EXP and WOS. The cost per unit of the JV mode consists of the production cost and the reimbursement to the local partner. Without loss of generality, we assume that the firm can switch mode from JV to WOS by purchasing the share from the local partner, but the switch incurs switchover cost. We assume that the firm can choose EXP to meet foreign demand at least one-unit product before switching from EXP to JV, and employ JV to supply at least one-unit product before switching from JV to WOS. Let $k_J$ be the entry cost of JV mode, $s_J$ is the exit cost of selling the share of the firm in foreign enterprise; v is the reimbursement per unit to the local partner; $k_W$ is the entry cost of WOS mode, $s_W$ is the exit cost of selling the foreign enterprise with WOS; ${\delta }_{JW}$ is switching cost from JV to WOS mode, the cost of switching between two modes is equal to the difference in entry cost between the two modes $({\delta }_{JW}={\lambda }_W-{\lambda }_J)$ ; the entry/exit cost of EXP is 0, so the switching cost from EXP to JV mode is equal to the entry cost of JV mode $({\delta }_{EJ}={\lambda }_J)$ , similarly ${\delta }_{EW}={\lambda }_W$ ; $Q^{EJ}=(k_J+s_J)/(h-v)-1$ , $Q^{JW}=(k_W+s_W-k_J-s_J)/v-1$ and $Q^{EW}=(k_W+s_W)/$ $h-1$ . We begin with the following assumptions and a useful lemma.

Assumption 1.

There exists a $h>v>0$ .

The reimbursement per unit to the local partner in JV mode is greater than zero. Otherwise, local partners are unwilling to establish JVs with the company. The reimbursement per unit to the local partner in JV mode is less than the transportation and tariff costs per unit of product. Otherwise, the company is unwilling to establish JVs with local partners.

Assumption 2.

There exists a $c+h<c+v+$ $k_J+s_J<c+k_W+s_W$ .

This is a reasonable assumption because the cost of using EXP mode to supply one-unit product must be less than that of employing JV to supply one-unit product. Otherwise, the decision-maker doesn't choose EXP but chooses JV in the first period. Therefore, we have $c+h<c+v+k_J+s_J$ . Similarly, we have $c+v+k_J+s_J<c+k_W+s_W$ .

Lemma 2.

If $Q^{EJ}>Q^{JW}$ , there exists a $Q^{EJ}>Q^{EW}>Q^{JW}$ ; if $Q^{EJ}\le Q^{JW}$ , there exists a $Q^{EJ}\le Q^{EW}\le Q^{JW}$ .

Proof.

We first prove the lemma for $Q^{EJ}>Q^{JW}$ . Consider the case $Q^{EW}-Q^{JW}$ .

$$\begin{array}{rl}{Q}^{EW}-Q^{JW}=& {\displaystyle \frac{k_W+s_W}{h}-{\displaystyle \frac{k_W+s_W-k_J-s_J}{v}}}\\ =& {\displaystyle \frac{h(k_J+s_J)-(h-v)(k_W+s_W)}{hv}}\end{array}$$

(12)

Since $Q^{EJ}>Q^{JW}$ , there exists $Q^{EJ}-Q^{JW}>0$ .

$$Q^{EJ}-Q^{JW}={\displaystyle \frac{k_J+s_J}{h-v}-{\displaystyle \frac{k_W+s_W-k_J-s_J}{v}}}$$

$$={\displaystyle \frac{h(k_J+s_J)-(h-v)(k_W+s_W)}{hv}>0}$$

(13)

Therefore, we have $Q^{EW}>Q^{JW}$ . Similarly, we have $Q^{EJ}>Q^{EW}$ .

The proof of the case $Q^{EJ}\le Q^{JW}$ is similar to the proof of the case $Q^{EJ}>Q^{JW}$ . □

Lemma 3.

For $Q^{EJ}\le Q^{JW}$ , if $Q^{EJ}\ge D\ge 0$ , the cost of the optimal offline algorithm is $C_{\mathrm{OPT}}=(c+h)D$ ; if $Q^{JW}\ge D\ge Q^{EJ}$ , $C_{\mathrm{OPT}}=(c+v)D+k_J+s_J$ ; if $D>Q^{JW}$ , $C_{\mathrm{OPT}}=cD+k_W+s_W$ . For $Q^{EJ}>Q^{JW}$ , if $D\le Q^{EW}$ , the cost of the offline algorithm is $C_{\mathrm{OPT}}=(c+h)D$ ; if $D>Q^{EW}$ , $C_{OPT}=cD+k_W+s_W$ .

Proof.

We first prove the lemma for $Q^{EJ}\le Q^{JW}$ . For a given total demand, the cost function of EXP is $C_{\mathrm{EXP}}(D)=(c+h)D$ , the cost function of JV is $C_{\mathrm{JV}}(D)=(c+v)D+k_J+s_J$ , and the cost function of WOS is $C_{\mathrm{WOS}}(D)=cD+k_W+s_W$ .

By Assumptions 1 and 2, there exist $c+h>c+v>c$ and $0<k_J+s_J<k_W+s_W$ , we have that the EXP, JV, and WOS are in ascending order of fixed costs and descending order of variable costs. Consider the case for $Q^{JW}\ge D\ge Q^{EJ},$ we have $C_{\mathrm{JV}}(D)-C_{\mathrm{EXP}}(D)<0$ when $D\ge Q^{EJ}$ and $C_{\mathrm{JV}}(D)-C_{\mathrm{WOS}}(D)<0$ when $D\le Q^{JW}$ . Therefore, the cost of the optimal offline algorithm is $C_{\mathrm{OPT}}=(c+v)D+k_J+s_J$ when $Q^{JW}\ge D\ge Q^{EJ}$ . Similarly, $C_{\mathrm{OPT}}=(c+h)D$ when $Q^{EJ}\ge D\ge 0$ , $C_{\mathrm{OPT}}=cD+k_W+s_W$ when $D>Q^{JW}$ .

The proof of the case $Q^{EJ}>Q^{JW}$ is similar to the proof of the case $Q^{EJ}\le Q^{JW}$ . □

Consider the following two algorithms:

Algorithm- Q ^EJ − Q^JW.

The decision-maker chooses EXP to supply the demand of the foreign market until $Q^{EJ}$ and then, switches from EXP to JV, and switches from JV to WOS after $Q^{JW}$ .

Algorithm-Q^EW.

The decision-maker chooses EXP to serve foreign market until $Q^{EW}$ , and then ignores the JV mode and switches directly from EXP to WOS.

First, we show that Algorithm- Q ^EJ − Q^JW is the optimum online algorithm for case $Q^{EJ}\le Q^{JW}$ .

Assumption 3.

There exists a $c+h+k_J+c+v+s_J<k_J+2(c+v)+s_J$ .

The cost of using EXP mode to supply one-unit product, and switching from EXP to JV and then serving the foreign market one-unit product must be less than the cost of choosing directly JV mode to supply two-unit products. Otherwise, the decision-maker doesn't choose EXP and then switches from EXP to JV, but chooses directly JV on the first product unit.

Assumption 4.

There exists a $c+h<k_J+c+v+s_J$ .

The cost of employing EXP mode to supply one-unit product must be less than the cost of choosing EXP to enter foreign market and then switching from EXP to JV, and using JV to serve foreign market one-unit product. Otherwise, the decision-maker ignores EXP and switches immediately from EXP to JV after choosing EXP to enter the foreign market.

Theorem 3.

For the case $Q^{EJ}\le Q^{JW}$ , Algorithm- Q ^EJ − Q^JW has competitive ratio

$$\begin{array}{rl}& C_{\mathrm{ON}}/C_{\mathrm{OPT}}\\ & \le max\left\{\begin{array}{c}1+{\displaystyle \frac{(h-v)(k_J+s_J-h+v)}{(c+h)(k_J+s_J)},}\\ 1+{\displaystyle \frac{v(k_W+s_W-h)}{c(k_W+s_W-k_J-s_J)+v(k_W+s_W)}}\end{array}\right\}<2\end{array}$$

and this result is best-possible.

The theorem follows from following four propositions. Proposition 1 shows that for case $D\le Q^{EJ}\le Q^{JW}$ , Algorithm- Q ^EJ  −  Q ^JW has a competitive ratio of 1. Proposition 2 shows that for case $Q^{EJ}\le D\le Q^{JW}$ , Algorithm-Q^EJ − Q^JW has competitive ratio , and $Q^{EJ}$ is the optimal timing switching from EXP to JV. Proposition 3 shows that for case $Q^{JW}<D$ , Algorithm-Q^EJ− Q^JW has competitive ratio , and $Q^{JW}$ is the optimal timing switching from JV to WOS. Proposition 4 shows that for case $Q^{EJ}\le Q^{JW}$ , the competitive ratio of Algorithm- Q ^EJ  −  Q ^JW is less than that of the arbitrary algorithm switching directly from EXP to WOS.

Proposition 1.

Under Assumptions 3 and 4, Algorithm-Q^EJ − Q^JW has a competitive ratio of 1 for case $D\le Q^{EJ}\le Q^{JW}$ .

Proof.

Consider the case $0\le D\le Q^{EJ}$ . This means that Algorithm- Q ^EJ  −  Q ^JW doesn't switch to JV. Hence, $C_{\mathrm{ON}}=C_{\mathrm{OPT}}=(c+h)D$ in this case and the competitive ratio is 1. □

Proposition 2.

Under Assumptions 3 and 4, for case $Q^{EJ}\le D\le Q^{JW}$ , Algorithm-Q^EJ − Q^JW has competitive ratio , and $Q^{EJ}$ is the optimal timing switching from EXP to JV.

The proof of Proposition 2 is similar to the proof of Theorems 1 and 2.

Assumption 5.

There exists a $c+h+k_J+c+v$ $+{\delta }_{JW}+c+s_W<3c+k_W+s_W$ .

Assumption 5 is similar to Assumption 3.

Proposition 3.

Under Assumption 5, for case $Q^{JW}<D$ , Algorithm- Q ^EJ  −  Q ^JW has competitive ratio , and $Q^{JW}$ is the optimal timing switching from JV to WOS.

The proof of Proposition 3 is similar to the proof of Theorems 1 and 2.

Assumption 6.

There exists a $c+h+k_J+c+v+s_J<c+h+k_W+c+s_W$ .

The cost of employing EXP mode to supply one unit product, and switching from EXP to JV and using JV to serve the foreign market one-unit product must be less than the cost of using EXP mode to supply one unit product, and switching from EXP to WOS and then serving the foreign market one unit product. Otherwise, the decision-maker doesn't switch from EXP to JV, but switches directly from EXP to WOS in the second period.

Proposition 4.

Under Assumption 6, for case $Q^{EJ}\le Q^{JW}$ , the competitive ratio of Algorithm- Q ^EJ  −  Q ^JW is less than that of the arbitrary algorithm switching directly from EXP to WOS.

Proof.

Consider an arbitrary algorithm switching directly from EXP to WOS when the aggregated supply is $Q$ . Given the fact that offline algorithm always chooses $D=Q+1$ after the online decision-maker chooses $Q$ , we consider the following three cases, for $Q\in [0,\mathrm{\infty })$ :

Case 1. $0\le Q<Q^{EJ}$ . The offline algorithm chooses $D=Q+1$ , since Q is a discrete variable, we have $D=Q+1\le Q^{EJ}$ . By Lemma 3, we have that Algorithm- Q ^EJ  −  Q ^JW is the optimal.

Case 2. $Q^{EJ}\le Q<Q^{JW}$ . For an arbitrary algorithm switching directly from EXP to WOS, the cost can be formulated as the following:

$$C_{\mathrm{ON}}^{EW}=(c+h)Q+k_W+c(D-Q)+s_W=(c+h)Q^{EJ}+(c+h)(Q-Q^{EJ})+c(D-Q)+k_W+s_W$$

(14)

The offline algorithm chooses $D=Q+1\le Q^{JW}$ , Algorithm- Q ^EJ  −  Q ^JW cannot switch from JV to WOS, the cost is:

$$\begin{array}{rl}{C}_{\mathrm{ON}}^{EJW}=& (c+h)Q^{EJ}+k_J+(c+v)(D-Q^{EJ})+s_J\\ =& (c+h)Q^{EJ}+(c+v)(Q-Q^{EJ})+(c+v)(D-Q)\\ & +k_J+s_J\end{array}$$

(15)

By Assumptions 3 and 6, we have that there exist $c+h>c+v>c$ and $k_J+v+s_J<k_W+s_W$ . Therefore, we have $C_{\mathrm{ON}}^{EW}>C_{\mathrm{ON}}^{EJW}$ for $Q^{EJ}\le Q\le Q^{JW}$ .

Case 3. For an arbitrary algorithm switching directly from EXP to WOS, the cost is:

$$\begin{array}{rl}{C}_{\mathrm{ON}}^{EW}=& (c+h)Q+k_W+c(D-Q)+s_W\\ =& (c+h)Q^{EJ}+(c+h)(Q^{JW}-Q^{EJ})\\ & +(c+h)(Q-Q^{JW})+c(D-Q)+k_W+s_W\end{array}$$

(16)

The cost of Algorithm- Q ^EJ  −  Q ^JW is

$$\begin{array}{rl}{C}_{\mathrm{ON}}^{EJW}=& (c+h)Q^{EJ}+k_J+(c+v)(Q^{JW}-Q^{EJ})\\ & +{\delta }_{JW}+c(D-Q^{JW})+s_W\\ =& (c+h)Q^{EJ}+(c+v)(Q^{JW}-Q^{EJ})\\ & +c(Q-Q^{JW})+c(D-Q)+k_W+s_W\end{array}$$

(17)

By Assumption 1, we have that there exists $h>v$ . Therefore, we have $C_{\mathrm{ON}}^{EW}>C_{\mathrm{ON}}^{EJW}$ for $Q^{JW}<Q$ .□

Theorem 4.

For the case $Q^{EJ}>Q^{JW}$ , Algorithm-Q^EW has competitive ratio

$$C_{\mathrm{ON}}/C_{\mathrm{OPT}}\le 1+{\displaystyle \frac{h(k_W+s_W-h)}{(c+h)(k_W+s_W)}<2}$$

and $Q^{EW}$ is the optimal timing, switching directly from EXP to WOS.

By Lemma 3, for $Q^{EJ}>Q^{JW}$ , the cost of the offline algorithm is $C_{\mathrm{OPT}}=(c+h)D$ when $D\le Q^{EW}$ ; the cost of the offline algorithm is $C_{\mathrm{OPT}}=cD+k_W+s_W$ when $D>Q^{EW}$ . We can easily have the proof of Theorem 4 that is similar to the proof of Theorems 1 and 2.

Numerical analysis

Effects of the transport/tariff cost and entry/exit cost on the competitive ratio

To discuss the impact of the transport/tariff cost on competitive ratio, we investigated the impact of transport/tariff cost changes on the competitive ratio for three values of the entry/exit costs: $k+s=500$ , 1000 and 1500 when $c=20$ . By Assumption 2, we have that there exists a $k+s>h$ . Therefore, we let the value of h vary from 0 to $k+s$ when adjusting the transport/tariff cost.

Figure 1 shows that there exists unique $h*$ that maximizes the competitive ratio, and the competitive ratio increases as the value of h changes from 0 to $h*$ and decreases as the value of h varies from $h*$ to $k+s$ . The maximal value of the competitive ratio increases as the value of $k+s$ increases from 500 to 1500, but the competitive ratio is always less than $1+h/(c+h)<2$ . Interestingly, we can see that Algorithm-Q^W has the optimal competitive ratio of 1 when the transport/tariff cost is at least 0 or at most $(k+s)$ . This is because the decision-maker will delay the timing switching from EXP to WOS as the transport/tariff cost decreases and then give up the switch when the value of $h$ is close to 0, which is consistent with the optimal offline strategy. On the other hand, the switch will become earlier as the transport/tariff cost increases or the entry/exit cost decreases, and then the decision-maker chooses WOS mode at time 0 when $h=k+s$ . This observation shows that the high transport/tariff cost encourage FDI.

Figure 1.

Effects of the transport/tariff cost and entry/exit cost on the competitive ratio.

As is apparent, the switching timing and competitive ratio of the online algorithm are far from insignificant. Of particular interest is the effect of the entry/exit cost and transport/tariff cost on the competitive ratio, since the decision-maker can measure the performance of the online algorithm by observing changes in the transport/tariff cost and entry/exit cost caused by government policies in host country. Moreover, it is helpful for the host countries because they can forecast the MNC's behaviors when adjusting some policies to varying these parameters.

Effects of JV mode on the competitive ratio

Assume that in the CFPS problem with JV mode, $c=20,h=10$ and the entry/exit cost of EXP mode is 0, $k_J$ =250, $s_J$ =250, $k_W$ =500, $s_W$ =500, ${\delta }_{JW}=250$ . We adjust the value of v in the JV mode so that there are two situations between EXP, JV, and WOS modes: $Q^{EJ}\le Q^{JW}$ and $Q^{EJ}>Q^{JW}$ (Table 2). Then, we analyze the competitive ratio of the online algorithm and determine whether the online algorithm should switch from EXP to JV mode in these two situations (Table 3).

Table 2.

Switch timing of the online CFPS problem with JV mode.

$c$	$h$	$k_J$	$s_J$	${\delta }_{JW}$	$k_W$	$s_W$	$v$	$Q^{EJ}$	$Q^{JW}$	$Q^{EW}$
20	10	250	250	250	500	500	1	55	499	99
20	10	250	250	250	500	500	3	71	166	99
20	10	250	250	250	500	500	5	99	99	99
20	10	250	250	250	500	500	7	166	71	99
20	10	250	250	250	500	500	9	499	55	99

Table 3.

Competitive ratio of the online algorithm.

	Online algorithm	$v$	$Q^{EJ}$	$Q^{JW}$	$Q^{EW}$	$r$
$Q^{EJ}\le Q^{JW}$	Algorithm- Q EJ  −  Q JW	1	55	499	99	1.29
	Algorithm-QEW	1	55	499	99	1.33
	Algorithm- Q EJ  −  Q JW	3	71	166	99	1.23
	Algorithm-QEW	3	71	166	99	1.33
	Algorithm-QEW	5	99	99	99	1.33
$Q^{EJ}>Q^{JW}$	Algorithm-QEW	7	166	71	99	1.33
	Algorithm-QEW	9	499	55	99	1.33

Table 2 shows the changes in the timing of mode transitions from EXP to JV, from JV to WOS, and from EXP to WOS when parameters change between the three modes of EXP, JV, and WOS. When the value of v increases in JV mode， $Q^{EJ}\le Q^{JW}$ becomes $Q^{EJ}>Q^{JW}$ .

Table 3 shows that when $Q^{EJ}\le Q^{JW}$ , the competitive ratio of Algorithm-QEJ −  QJW is smaller than that of Algorithm-Q^EW, indicates that JV mode can improve the performance of online algorithms. When $Q^{EJ}>Q^{JW}$ , the timing of transition from JV to WOS mode occurs earlier than the transition from EXP to JV mode, the JV mode is meaningless for the online CFPS problem, and online algorithms ignore the JV mode and directly switch from EXP to WOS mode.

As shown in the above case, the impact of the JV mode on online strategies is very interesting. Adding JV mode selection in CFPS problem decision-making does not necessarily improve the competitive ratio of online strategies. Decision makers need to determine whether to choose the JV model in CFPS problems based on the conditions of the JV model, that is, the profit distribution between different equity structures and local partners in the JV model, which is influenced by negotiations between decision makers and local partners.

Conclusions

The decision-makers look for an effective strategy to reduce the MNC's threat; a common solution is entering foreign markets: the companies can counteract the competition resulting from the MNC. In this article, we investigate an online CFPS model that captures the company's behavior in the early stages of the CFPS problem under unknown demand. Based on the optimal online strategy, we analyze the impact of the transport/tariff cost and the entry/exit cost on the switching timing and competitive ratio. We also considered the CFPS problems with JV mode, in which the decision-maker has the ability to accept or reject a JV mode. This introduces additional difficulties: how to accept or reject the JV mode between EXP and WOS mode. We design a new online algorithm that is to determine whether the JV model is feasible.

The efficient online strategy that we propose increases the flexibility of entering foreign markets by first choosing EXP mode to delay the decision of FDI, and guarantees maximal cost of implementing global strategy in the worst case. A possible extension of our analysis would be to study how to find average-case performance of online algorithms as well as randomized algorithms. Another extension could include information augmentation. For example, the competence of forecasting demand increases as the length of time the firm has been serving the foreign market increases.