Valuing Operational Flexibility in Manufacturing Systems Using Real Options

Harriet Black Nembhard Leyuan Shi

Department of Industrial Engineering University of Wisconsin-Madison Madison, WI 53706-1572, U.S.A.

Mehmet Aktan

Department of Industrial Engineering Atatrk University

Erzurum, 25240, TURKEY

Abstract In this paper, we look across the manufacturing enterprise to consider opportunities to improve value by switching among suppliers, plants, and markets. We extend this model to address the reality that these switches may often take a period of transition time, rather than occur instantaneously. We value the real options associated with this flexible approach using both a lattice and Monte Carlo simulation. This valuation gives decision makers a way to choose the appropriate manufacturing enterprise strategy based on an integrated view of the market dynamics. Overall, the manufacturing enterprise maximizes its expected, discounted profit through effective supply chain network decisions.

Keywords Real Options, Monte Carlo simulation, supply chain

1. Introduction Increased competition in the global market has caused organizations to realize that the most competitive way of survival is high value. This can often be achieved through increased flexibility. We present a real options model for a company that wants to maximize its profit by increasing flexibility in an environment of uncertain exchange rates. We specifically consider the operational decisions to increase flexibility through selection of suppliers, allocation of production among plants, and choice of market regions. We use the options approach to find the value of such flexibility during a specified length of time, considering future uncertain exchange rates between the home country and other countries where operations can be located. The optimal policy maximizes the total expected discounted global profit. This valuation gives decision-makers a way to choose the appropriate outsourcing strategy based on an integrated view of the market dynamics.

We present the problem formulation in Section 2. In Section 3, we present the valuation of the problem with a lattice model. In Section 4, we propose an advanced model where the decisions can be implemented after a transition time has been observed. In Section 5, we value the problem with a Monte Carlo simulation approach. Conclusions are given in Section 6.

2. Problem Formulation We consider a company that can supply from a number of global suppliers in foreign countries, can make production in different plants, and can market its products in several market regions. The time horizon for the problem is divided into T time intervals. In each time interval, the company selects an option from a group of N manufacturing options. A particular global manufacturing strategy option O in period t is defined in terms of available sources of supply, plant

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capacities, market regions, and open supply linkages within the firms global supply chain network. Switching between the manufacturing options is costly because of the costs of opening and closing plants as well as opening and closing supply linkages. We formulate the option valuation problem as a stochastic dynamic program, similar to the formulations of Huchzermeier [6] and Huchzermeier and Cohen [7]. In each state, the company maximizes its value by selecting a manufacturing option Ot from the set of manufacturing options t, given that the company selected option O

*t-1 in the

preceding period and the current exchange rate scenario between the home country and foreign countries is ekt, where k is the number of possible jumps for the exchange rate scenario in each time period. Let be the discount factor and succ(ekt) be the set of the successor states of exchange rate scenario ekt. The value V of the company in period t and state (ekt, O

*t-1) is defined as follows:

Vt(ekt, O*t-1) = max

ttO WPt(ekt, O

*t-1, Ot) + EVt+1( 1+te , Ot)

= maxttO W

Pt(ekt, O*t-1, Ot) +

+ )(1, kttk esucce'kkp . Vt+1( 1, + tke , Ot) (1)

VT( 1, -Tke , O*T-1) = max

TTO W PT( 1, -Tke , O

*T-1, OT).

Jump probability from state k to state k is denoted by 'kkp in Equation (1). Single-period profit function Pt accounts for both the switching costs between global manufacturing strategy options and the global profit determined by optimizing the single-period subproblem formulation SPt.

Pt(ekt, O*t-1, Ot) = [(O

*t-1, Ot) + SPt(ekt, Ot)] (2)

We assume that the exchange rate ei between the home country and the supplier country i follows geometric Brownian motion as

iiiti

ti dzdte

desm +=

,

, (3)

where mi is the drift of the exchange rate changes for foreign country i, si is the volatility of the exchange rate for foreign country i, dz is a standard Wiener disturbance term.

3. A Lattice Model Huchzermeier [6] and Huchzermeier and Cohen [7] propose a lattice model where n-1 exchange rate processes is approximated with an n-nomial lattice. As in other lattice models, they define the size of the movements of exchange rates in the lattice as ui and di, where ui is the one-step increase in exchange rate for country i, and di is the one-step decrease in the exchange rate for country i, i=1n-1. We apply the Huchzermeier-Cohen lattice model in a problem with two uncertain exchange rate processes. Since we have two exchange rate processes in the problem, we will use a trinomial lattice. The trinomial lattice assumes that for any exchange rate scenario ekt = (e1kt, e2kt) at time t, there are three successor scenarios at time t+1, i.e., e1,t+1=(u1.e1kt, u2.e2kt), e2,t+1=(d1.e1kt, u2.e2kt) and e3,t+1=(d1.e1kt, d2.e2kt). The supply chain network for the problem is shown in Figure 1. Each of the two foreign countries is a supplier. There is one production plant in each country. At the same time, each of these two countries is a market region for the final product. Total time horizon of the problem is one year. Total time was divided equally into four time periods, which implies that each period was four months.

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Figure 1. Supply chain network for two suppliers, two plants, and two markets.

Twelve manufacturing strategy options are defined based on possible connections in the supply chain network. These twelve options are shown in Figure 2.

Figure 2. Manufacturing strategy options. Figure 3 shows expected profits for no switching and costly switching. When there is no switching, the same option will be used at each time period. However, when switching is possible, the best option can be selected at each time period considering the switching costs. Then, the option value of flexibility because of being able to switch the manufacturing strategy is the difference between the expected profits of costly switching and no switching. The upper line in Figure 3 shows the expected profits when switching between options is possible, and the lower line shows the expected profits when there is no switching between options. The horizontal axis shows which manufacturing strategy option has been used prior to the first time period in the problem.

Figure 3. Expected profits for no switching and costly switching.

S1

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Supplier MarketPlant

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Option 8Option 7Option 6Option 5

Option 12Option 11Option 10Option 9

4. A Model for Multi-Period System Transitions In the manufacturing strategy option problem in Section 3, we assumed that when a new manufacturing strategy is desired, it is implemented immediately. This means that when the company wants to switch the supplier, all raw material from the new supplier is received instantly; when a switch is desired for the production plants, required amount of production in the new plant is done instantly; and when a switch is desired for the market regions, final product reaches to the new market regions instantly. However, in real manufacturing operations, these assumptions do not truly hold. Switches for suppliers, plants, and market regions need a transition time to be executed. In this section, we propose a solution method that addresses the transition time needed to execute the switching decisions in such problems. In the lattice model of Section 3, we assumed that if the company gives a switch decision at a node, that switch could be implemented at the same node instantly. Now we assume that a switch decision given in a node can be implemented at the successor nodes. We solve the problem where the transition time is one period. In this problem, if a switch decision is given at a node, then the strategies in immediate successor nodes for that node have to be same, because the decision given in a node will be implemented in the next time period. This means that when a decision is given at a state, then that decision will be executed in the next state, whichever the next state happens to be. When using dynamic programming with backward recursion, this requires that the option selections will be same at the nodes that are preceded by the same node. Then, Ot, which is the option selected at time t, will be same at states that are preceded by the same state (see Equation (1)). At the first time period, the option will be the same as the option that was used just before the beginning of the planning horizon. Using this relationship, we modified the structure of the backward recursion of the dynamic program, and obtained the option values of flexibility with the feature of transition time. Comparison of the option value of flexibility with and without the transition time is given in Figure 4. The upper line shows the expected profits when the transition time is zero, i.e., the switching decisions can be implemented instantly. The lower line shows the expected profits when switching is not possible, i.e., when there is not any flexibility. The middle line shows the expected profits when there is a transition time of one period for the implementation of switching decisions.

Figure 4. Expected profits with and without transition time. We see that the expected profits with transition time are less than the profits with no transition time. When the implementation of strategy switches require a transition time, the company must wait until the implementation. Therefore, compared to the case of instant implementation, there is less flexibility when there is transition time. Since the no-switching case has zero flexibility, its expected profits are lowest.

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5. A Monte Carlo Simulation Approach for Multi-Time Period Options and Nonzero Switching Costs Boyle [2] introduced a Monte Carlo simulation method for asset pricing of European options. This approach, however, cannot be used for problems with switching costs since they cannot be treated as a bundle of European options with different expiration dates. Broadie and Glasserman [5] developed a simulation algorithm for estimating the prices of American-style assets. In order to develop valid error bounds on the true option value, they introduced two estimators, one biased high and one biased low, but both asymptotically unbiased as the computational effort increases. These estimators are based on simulated lattices. The simulated trees are parameterized by b, the number of branches per node. State variables are simulated at the finite number of possible decision points, i.e., exercise times. In this section, we propose a Monte Carlo simulation technique that can be used to value real options that have multiple time points to switch between the options and that have nonzero switching costs. In order to value the problem, we will combine dynamic programming with the simulation approach given in Broadie and Glasserman [5]. In the proposed technique, we use a simulation tree with three branches per node. On this simulation tree, we will apply the dynamic programming algorithm that we used for the trinomial lattice. The best decision for each possible previous policy is determined at each node. This means that best decisions are determined at each of the three branches. During the backward recursion of the dynamic programming, we use the mean value of the three branches to select the best decision. Since the simulated values for the state variables are used for calculations, the dynamic programming takes advantage of knowledge of the future to overestimate the option value. Hence, this estimate is biased high. In order to find the low estimator, we use the following approach. At each node, we use the second and third branches to determine a policy, and we apply that policy in branch one. Then, we use the first and third branches to determine a policy, and we apply that policy in branch two. Finally, we use the first and second branches to determine a policy, and we apply that policy in branch three. Then, we use the average of the expected profits from the three branches during the backward recursion of the dynamic program. We will use the average of the high-biased and low-biased simulation estimates to estimate the true option value. Applying this technique on the problem given in Section 3, we obtained the simulation results given in Table 1.

Table 1. Monte Carlo Simulation and Trinomial Lattice Results Prior option 1 2 3 4 5 6 7 8 9 10 11 12 Low -biased estimate 2011.7 2112.8 2097.2 2208.9 2003.0 2107.6 2110.1 2204.2 2100.7 2208.9 2209.8 2299.1 High-biased estimate 2019.4 2128.4 2131.8 2238.1 2040.9 2121.4 2128.4 2223.5 2134.2 2231.6 2229.8 2326.5 Average of estimates 2015.5 2120.6 2114.5 2223.5 2021.9 2114.5 2119.2 2213.8 2117.5 2220.3 2219.8 2312.8 Trinomial lattice result 2015.3 2115.3 2115.3 2215.3 2015.3 2115.3 2115.3 2215.3 2115.3 2215.3 2215.3 2315.3 Difference (%) 0.01 0.25 0.04 0.37 0.33 0.04 0.19 0.06 0.10 0.22 0.20 0.11

The first row in Table 1 shows the policy prior to the beginning of the decision time horizon. The next two rows are the low-biased and high-biased Monte Carlo simulation estimates for the expected profits. The fourth row is the average of these two estimates. The fifth row shows the expected profits that we obtained by using trinomial lattice. The last row shows the percent difference between the expected profits of Monte Carlo simulation and trinomial lattice. It has been shown that lattice methods give close results to the analytical solutions (Nembhard, Shi, and Aktan [9], Boyle [3], Boyle, Evnine, and Gibbs [4], Kamrad and Ritchken [8], Amin and Khanna [1]). We see that the Monte Carlo simulation estimates are close to the trinomial lattice estimates, which implies that the proposed approach yields close results to the analytical solution for this problem. Many manufacturing system models contain more than two state variables. If there are three or more state variables in the problem, lattice techniques become difficult to apply. If we want to use lattice approach in such problems, structure of the lattice gets too complex and the number of possible jumps at each node gets larger. However, the proposed Monte Carlo simulation technique does not require a different tree structure when the number of state variables increases. The same tree structure is used for any number of state variables; the only difference is the

number of random numbers generated. The main advantage of the proposed Monte Carlo simulation approach is its simple application on real options problems with three or more state variables.

6. Conclusions In this paper, we have proposed a technique to value the real options problems in manufacturing enterprises when decisions in the system cannot be implemented immediately. We have presented the application of the proposed technique on a supply chain network with uncertain exchange rates, where costly switching decisions for the suppliers, production plants, and market regions take a transition time to execute. We have also proposed a Monte Carlo simulation technique that can be used to value the real options problems with costly switching decisions. In such problems, there are multiple options. At each time interval, switching between options is possible, and each switch results in a switching cost. We have compared the results of the proposed Monte Carlo simulation with the results of a lattice technique. The comparison has shown that the proposed Monte Carlo simulation technique yields close estimates for the true option value. Real options valuation techniques proposed in this paper give decision makers a way to choose the appropriate manufacturing enterprise strategy based on an integrated view of the market dynamics. Overall, the manufacturing enterprise maximizes its expected, discounted profit through effective supply chain network decisions.

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Quantitative Analysis, 23(1), 1-12. 4. Boyle, P.P., Evnine, J., and Gibbs, S., 1989, Numarical Evaluation of Multivariate Contingent Claims, The

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