Fixed versus flexible production systems: A real options analysis

Available online at www.sciencedirect.com

European Journal of Operational Research 188 (2008) 169–184 www.elsevier.com/locate/ejor

Decision Support

Fixed versus flexible production systems: A real options analysis Dalila B.M.M. Fontes

*

LIACC, Faculdade de Economia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-464 Porto, Portugal Received 23 February 2006; accepted 4 April 2007 Available online 19 April 2007

Abstract In this work, we address investment decisions in production systems by using real options. As is standard in literature, the stochastic variable is assumed to be normally distributed and then approximated by a binomial distribution, resulting in a binomial lattice. The methodology establishes a discrete-valued lattice of possible future values of the underlying stochastic variable (demand in our case) and then, computes the project value. We have developed and implemented stochastic dynamic programming models both for fixed and flexible capacity systems. In the former case, we consider three standard options: the option to postpone investment, the option to abandon investment, and the option to temporarily shut-down production. For the latter case, we introduce the option of corrective action, in terms of production capacity, that the management can take during the project by considering the existence of one of the following: (i) a capacity expansion option; (ii) a capacity contraction option; or (iii) an option considering both expansion and contraction. The full flexible capacity model, where both the contraction and expansion options exist, leads, as expected, to a better project predicted value and thus, investment policy. However, we have also found that the capacity strategy obtained from the flexible capacity model, when applied to specific demand data series, often does not lead to a better investment decision. This might seem surprising, at first, but it can be explained by the inaccuracy of the binomial model. The binomial model tends to undervalue future decreases in the stochastic variable (demand), while at the same time tending to overvalue an increase in future demand values. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Fixed and flexible capacity investments; Real options; Stochastic dynamic programming; Binomial lattice; Investment analysis

1. Introduction Several types of flexibility can be considered: a good survey on manufacturing flexibility is that of Sethi and Sethi (1990). Here, we are concerned with the so-called ‘‘volume’’ flexibility. That is, the ability to operate with profit at different output levels. In addition, we also consider the possibility of changing production capacity. In this work, we consider an investment problem where we must decide upon the optimal production capacity to be in place for a production system. Two possible types of system are considered: (i) a fixed production capacity system and (ii) a flexible production capacity system. In the first type, we must determine which is the *

Tel.: +351 225 571 100; fax: +351 225 505 050. E-mail address: [email protected]

0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.04.001

170

D.B.M.M. Fontes / European Journal of Operational Research 188 (2008) 169–184

optimal fixed production capacity. Reaction to demand fluctuations can only be achieved through optimal production output choice. For the flexible system, we must choose the optimal initial production capacity, as well as the optimal capacity changing strategy. In this case, the system can be adapted to changes in demand not only through the produced quantity but also through production capacity contraction and/or expansion. Furthermore, in both cases we can shut-down the production facility temporarily and also abandon the project by getting the residual value. A possible way of measuring flexibility is to estimate its value, which can be compared with the cost of acquiring it. Some authors advocate that its value can be given by Net Present Value (NPV) calculations, while others believe that flexibility can be best given by options, or ‘‘real’’ options to separate them from the more familiar financial options. The NPV tends to systematically undervalue investments, since it assumes that future cash-flows follow a constant pattern and can be accurately predicted. Another disadvantage is that it is linear and static in nature. Furthermore, the NPV analysis typically considers projects to be irreversible, i.e. it does not allow for corrective actions to be taken, and only immediate accept-or-reject decisions are allowed (Dixit and Pindyck, 1994). Thus, it ignores the flexibility to time project initiation optimally. NPV says that an investment should be made whenever the expected discounted future cash-flows match investment costs. Real options theory, on the other hand, requires expected discounted future cash-flows to be significantly above the investment costs. This happens since making the investment implies the loss of the waiting option and hence, it represents an opportunity cost. Therefore, when there is an option element in an investment, traditional discounted cash-flow methods may result in wrong project valuation and hence inadequate decisions, see e.g. Pindyck (1991) and Dixit and Pindyck (1994). In this work, we present models to value investment decisions based on real options. In the problem considered we incorporate partial reversibility by letting the firm reverse its capital investment at a cost, both fully or partially. For this problem we consider two types of system: a fixed production capacity system and a flexible production capacity system. Regarding the fixed capacity system, the following three options are simultaneously considered: defer investment, temporarily shut-down, and abandon project. For the flexible production capacity system, in addition to the above three options, we also consider the option of corrective action in terms of production capacity – capacity switching option. Three different flexible capacity options are to be considered in this work, corresponding to the different capacity corrective actions that may be taken. Thus three different versions will be considered (i) switch to a lower capacity level – contraction; (ii) switch to an upper capacity level – expansion; or (iii) switch either to a lower or to an upper capacity level – both contraction and expansion. As in most approaches in the literature, the stochastic variable (demand) is assumed to be normally distributed and then approximated by a binomial distribution, resulting in a binomial lattice. Dynamic programming models are developed to value the investment decision numerically. The project value is computed on the binomial lattice by backward induction. We compute the predicted investment project value, as well as the associated capacity strategy, considering both a fixed capacity system and a flexible capacity system. We also test the capacity strategies, obtained by solving the aforementioned models, on several data sets. As will be shown later, the project value obtained for the fixed capacity systems is often better than that obtained for the flexible capacity system, when the previously devised strategies are implemented. 2. The real options approach The term ‘‘real options’’ was coined by Myers (1977). It referred to the application of option pricing theory to the valuation of non-financial or ‘‘real’’ investments with learning and flexibility, such as multi-stage R&D and manufacturing plant expansion (Myers, 1977). This type of approach is based on three factors, namely: the existence of uncertainty about future cash-flows; investment irreversibility, at least partially, that is, money cannot be fully recovered once an investment decision has been made; and the timing of project initiation. A large and rapidly growing literature on investment under uncertainty interprets the firm as consisting, either wholly or in part, of a portfolio of options,1 and uses options-based models and option pricing techniques to 1 An option is the right, but not the obligation, to take some action in the future under specified terms. A call option gives the holder the right to buy a stock on a specified future date (maturity) at a specified price (exercise or strike price). This option will be exercised (used) if the stock price on that date exceeds the exercise price.

D.B.M.M. Fontes / European Journal of Operational Research 188 (2008) 169–184

171

study the investment decision. This approach also recognizes the sequential nature of investment decisions as a key feature. The options studied include the option to temporarily shut-down (McDonald and Siegel, 1985; Brennan and Schwartz, 1985), the option to continue or discontinue a planned series of investments (Majd and Pindyck, 1987), the option to defer investment (McDonald and Siegel, 1986), the option to abandon the project earlier (Myers and Majd, 1990), and the option to increment capacity (Bollen, 1999; Pindyck, 1988; Kandel and Pearson, 2002), amongst others. Managerial flexibility has been valued by option pricing for almost two decades and during this time different kinds of real options have been treated. Kulatilaka (1988) uses a stochastic dynamic programming model to value the options in a flexible production process and incorporates the effects of switching costs. Andreou (1990) values process flexibility in different configurations of dedicated and flexible equipment when demand for two products is uncertain. Triantis and Hodder (1990) value process flexibility, in a given fixed capacity equipment as a complex option. The profit margins of different products are assumed to be stochastic and dependent on the quantity produced. The latter effect is a result of allowing for downward sloping demand curves. In their model there are no switching costs. Capacity constraints are considered and the model allows the firm to temporarily shut-down and restart operation. He and Pindyck (1992) examine investments in flexible production capacity. Here, the capacity choice problem is considered, i.e. whether to buy flexible or non-flexible equipment and how much capacity with respect to the fact that investment is irreversible. As in Tannous (1996), demand is uncertain but in this case differs, via a demand shift parameter depending on whether market is perfectly competitive or not. Kamrad and Ernst (1995) model multi-product manufacturing with stochastic input price, output yield uncertainty and capacity constraints to value multi-product production agreements. During one period, only one product type is produced with respect to the inventory available. Tannous (1996) values volume flexible, when demand is uncertain, in order to find the optimal level of investment. The characteristic of the flexible equipment is that the production rate can be adjusted to respond to fluctuations in demand. The effect of having inventory available is also considered. Bollen (1999) values the option to switch between production capacities. The demand stochastic process is governed by a stochastic product life cycle which is modelled by using a regime switching process. 3. Problem description and formulation Following the approach outlined in Dixit and Pindyck (1994) and Trigeorgis (1996), the opportunity of adopting a specific value of production capacity can be viewed as representing a real option to the firm. Furthermore, for the flexible project, the capacity decisions can be cast as a sequence of embedded decisions since the current capacity decision has implications on future decisions. Our starting point is the irreversible investment model by Pindyck (1988), which is a flexible and tractable example of the options-based models, and can be readily generalized to allow for partial reversibility. In this model, a monopolist faces a demand function, typically assumed to be a function of price Pt, that shifts stochastically with a fixed slope k, towards and away from the origin over time as Qt ¼ ht  kP t ;

ð1Þ

where Qt is the industry output and ht models the dynamics of demand. Of course, for the case of monopoly, which is our case, the demand function given by Eq. (1) is also the demand curve faced by the firm. (In financial options it is standard to assume that the underlying security is traded in a perfectly competitive market. However, many real asset markets are monopolistic or oligopolistic, rather than perfectly competitive.) The standard assumption is that total variable production costs, for fixed capacity systems, are a quadratic function of quantity produced, see for example (Pindyck, 1988; Trigeorgis, 1996) CðQÞ ¼ c1 Q þ

c2 2 Q: 2

ð2Þ

Here, although we must distinguish between fixed capacity systems and flexible capacity systems, the fixed and flexible project values must be comparable. Therefore, for both projects the capacity must also be included in the cost function and thus, in the operating profit.

172

D.B.M.M. Fontes / European Journal of Operational Research 188 (2008) 169–184

Therefore, the total production cost is given by CðQt ; MÞ ¼ c1 Qt þ

c2 2 Q þ c3 M; 2M t

 where M ¼

m0 ;

for fixed capacity;

mt1 ;

for flexible capacity:

ð3Þ

c2 Qt are, respectively, the fixed and variable coefficients of the marginal cost funcFor these projects c1 and 2M c2 tion, while c3 M represents the overhead costs. It should be noticed that c1, 2M , and c3 M are constant for the fixed capacity project, since the capacity is constant throughout the whole project life. However, for the flexible capacity project the latter two values vary with the available capacity. The operating profit p of period t, given the demand Qt in period t and the production capacity available M (since the beginning m0 or installed in the previous period mt1 ), is then computed as p ¼ P ðQt Þ  Qt  CðQt ; MÞ and is given by     ht 1 c2 þ  c1 Q t  ð4Þ pðht ; MÞ ¼ Q2t  c3 M: k 2M k

(Note that for the flexible capacity project we are assuming the capacity in place in period t to be the capacity chosen in the previous period. However, this is not a limitation since the reasoning and formulae given can be applied with any number of installation periods.) The firm maximizes operating profit over produced quantity and hence, the optimum quantity Qt , which is op obtained by solving oQ ¼ 0, is given by Eq. (5). Furthermore, the quantity to be produced in each period Qt is t bounded from above by the production capacity and from below by zero, thus it is as given in Eq. (6): ht  kc1 ; ð5Þ 2 þ kc2 =M Qt ðht Þ ¼ maxð0; minðQt ðht Þ; MÞÞ; ð6Þ  m0 ; for fixed capacity; where M= Therefore, the optimal operating profit p* is computed as in Eq. (4) mt1 ; for flexible capacity: for the optimal production quantity Qt, which is given by Eq. (6). For the flexible capacity project we also consider partial reversibility, which is incorporated into the model by letting the firm reverse its capital investment at a cost. The ability to partially reverse the capital investment is modelled through capacity sellout, which allows for recovering a fraction of the purchase price for each unit sold. More specifically, following the work by Bollen (1999), we use Sðm1 ; m2 Þ to represent the additional or recovered investment associated with changing capacity level from m1 to m2: 8 if m2 ¼ m1 ; > < 0; Qt ðht ; MÞ ¼

Sðm1 ; m2 Þ ¼

> :

s1 c4 ðm2  m1 Þ þ s3 ; if m2 > m1 ; s2 c4 ðm1  m2 Þ þ s3 ; otherwise:

ð7Þ

In cost function (7), s1 and s2 are percentages of the initial capacity cost c4 and s3 is a fixed switching capacity cost. It should be noticed that, although in the example solved in this work we set s1 = 1 and 0 < s2 < 1, each could assume any positive or negative value. Moreover, the switching costs could also be time dependent. We assume that markets are dynamically complete, implying that there exists a risk-neutral probability or equivalent martingale measure such that the value of the firm is given by the expected discounted value of its profits less the investments in capacity. The assumption that markets are dynamically complete amounts to assuming that stochastic changes in demand are spanned by existing assets, or that markets are sufficiently complete to ensure that the firm’s decision to invest does not change the opportunities available to investors. 4. Solution methodology For the fixed capacity project we must decide on the production capacity, but only once as it remains at the same level throughout the whole project life, and on production quantities. To solve the flexible capacity problem it is necessary to find the optimal sequence of capacity choices, namely: invest in additional capacity; sell out excessive capacity; keep exactly the same capacity, as well as the optimal production output in each period

D.B.M.M. Fontes / European Journal of Operational Research 188 (2008) 169–184

173

given the capacity decision previously made. The two decision types must be addressed simultaneously, since the existence of switching costs implies that a capacity decision made in a period alters future switching costs and future profits and thus, future switching decisions. Therefore, the project value must be determined simultaneously with the optimal production capacity policy. The solution approach we use is to discretize the problem and set up a discrete-valued lattice for which a dynamic programming model is derived and then solved by backward induction. The uncertainty of the underlying risky asset, demand, is modelled through the use of the standard binomial lattice, as explained in the next section. To develop the dynamic programming model, we must go through the decision process defining the decisions, as well as the variables involved (decision and state variables), see Sections 4.2 and 4.3, for the fixed and the flexible projects, respectively. The decision variables are associated with the possible strategic decisions on production capacity and production quantity that managers may undertake during the investment process, while the state variables are related to time. In these sections, we also present the dynamic programming models and explain how to solve them by backward induction. The optimal value of the investment project and the corresponding optimal production quantity in each period are given by the solution to the proposed models. The flexible capacity model also provides the optimal capacity strategy. 4.1. The binomial lattice The analysis performed in this work makes use of the multiplicative binomial model of Cox et al. (1979), the standard tool for option pricing in discrete time. Thus, the underlying stochastic variable is assumed to be governed by a geometric diffusion, which implies that there is only one constant growth/decay rate. If this is then assumed, a natural way of obtaining a valued-lattice for the stochastic variable (demand) is to discretize it through a standard binomial lattice, see Fig. 1. A node of value hit can lead to two nodes with their values being given by hktþ1 ¼ uhit and hjtþ1 ¼ dhit with probability qu and qd ¼ 1  qu , respectively. The probability of reaching each of these nodes is the usual equivalent martingale measure used in the binomial option pricing model of Cox et al. (1979): qu ¼

ð1 þ rf Þ  d ud

and qd ¼ 1  qu ;

ð8Þ pffiffiffiffiffi pffiffiffiffiffi where rf is the risk free rate over the interval Dt, u ¼ expðr DtÞ, and d ¼ expðr DtÞ. It should be noticed that, as Dt ! 0 the parameters of the multiplicative binomial process converge to the geometric Brownian motion.

Fig. 1. A lattice discretizing demand.

174

D.B.M.M. Fontes / European Journal of Operational Research 188 (2008) 169–184

The advantages of this approach are that (i) it can be used to price options other than European, like American and path dependent options; (ii) it does not depend on the investor subject probabilities of an upward/ downward price movement, since it uses risk-neutral probabilities; and (iii) it is simple to understand and implement. The binomial model breaks down the investment horizon into n (potentially very large) time intervals or steps. The lattice is then developed in a forward movement by finding the demand values. At each step the price moves up or down by an amount obtained from its volatility, which is assumed to be constant and known, and the time step length. Thus, the lattice provides a representation of all possible demand values throughout the whole project life. 4.2. The decision process – fixed capacity Let us first consider the fixed capacity investment, since in this case the investment problem, and thus the dynamic programming model, are less complex. We should recall that, for the fixed capacity investment we have three simultaneous options: defer investment, temporarily shut-down, and abandonment. The defer option exists only at the beginning before any money has been invested. After such a decision has been made, we can only decide upon the quantity to be produced or to abandon the project. In each period the firm must decide whether it remains in operation or the project is abandoned. In the latter case, the firm receives an abandonment value, which is either the project selling price R or the capacity scrap value Sðm; 0Þ, whichever is largest. The temporarily shut-down option is a reversible option that is only taken if we are better off not producing, which in our problem is done implicitly. Such a decision is reached whenever the optimal quantity to be produced is determined to be zero. Note that, in such cases, a cost (fixed cost) is incurred. If the project is not abandoned, then we must decide on the quantity to be produced, which is given by maximizing the operating profit and hence, the optimal quantity Qt to be produced at period t, as given by Eq. (6). Therefore, at each period the project value is dependent on the level of demand and on the fixed production capacity and is obtained as the maximum between the sum of the optimal current period’s profit with the optimal continuation value and the abandonment value, which is received if the project is abandoned. The continuation value is given by the discounted expected future profits. The optimal project value in period t given the demand ht and the available fixed production capacity m0 is then given by ( maxfR; Sðm0 ; 0Þg; if abandonment; f ðht ; m0 ; tÞ ¼ max maxfpðh ; m Þg þ E½f ðhtþ1 ;m0 ;tþ1Þ ; otherwise; ð9Þ Qt

t

0

1þrf

where R is the project selling price and Sðm0 ; 0Þ is the capacity scrap value, which is computed as given in Eq. (7). If this is a new investment, the optimal project net expected discounted value is given by   f ðh1 ; m0 ; 1Þ max  Iðm0 Þ ; ð10Þ 06m0 6mmax 1 þ rf where Iðm0 Þ ¼ m0 c4 is the required initial investment to install production capacity m0. (It should be noticed that if the capacity value m0 that maximizes Eq. (10) is zero, then the project is not worth to undertaking.) Otherwise, if the project already exists with capacity m0, then the optimal project net expected discounted value is given by f ðh1 ; m0 ; 1Þ : 1 þ rf 4.3. The decision process – flexible capacity Let us recall that, for the flexible capacity investment, in addition to the options of temporarily shutdown, project abandonment, and defer investment, we also can take corrective actions in terms of production

D.B.M.M. Fontes / European Journal of Operational Research 188 (2008) 169–184

175

capacity. In each period a corrective action may be taken and it can be to contract capacity, or to expand capacity, or to both contract or expand capacity, depending on the options being considered. As explained before, the temporarily shut-down option is only taken if we are better off not producing, this happens whenever the optimal quantity to be produced is determined to be zero. The abandonment option corresponds to giving up the project by receiving an abandonment value, which is the maximum between the project selling price and the capacity scrap value. The defer investment or disinvestment option can now be taken throughout the whole project life and it corresponds to remaining in operation at the same capacity level. The options to contract capacity, to expand capacity, or being able to both contract and expand capacity are quite similar since all correspond to remaining in operation potentially at a different production capacity level and thus, involve switching costs. In this way, we only have to explicitly address the abandonment option and the capacity switching option, which may involve a capacity increase, a capacity decrease, or remaining at the same capacity level. In each period the firm must take two decisions, one regarding the quantity to be produced Qt and another regarding the production capacity mt that will be in place from the following period, if the project is not abandoned. The decision on the quantity to be produced is given by maximizing the operating profit and hence, the optimal quantity Qt to be produced in period t is as shown in Eq. (6). The decision about the production capacity is related to the future periods profit since the chosen capacity is to be available from the following period. We should recall that an increase in installed capacity requires an investment, while a sellout of capacity leads to a partial recovery of investment, as given in Eq. (7). Therefore, in each period the project value is dependent on the level of demand and production capacity and is given either by the abandonment value if the project is abandoned or obtained by maximizing the sum of the optimal current period’s profit with the optimal continuation value for each possible capacity value. The latter value is given by the discounted expected future profits net of switching costs. The decision to be taken is the one that maximizes project value gðht ; mt1 ; tÞ, where S is the switching cost function given in Eq. (7), and it is to be chosen from amongst the following: 1. Abandon project maxfR; Sðmt1 ; 0Þg: 2. Defer (dis)investment maxfpðht ; mt1 Þg þ Qt

E½gðhtþ1 ; mt1 ; t þ 1Þ : 1 þ rf

3. Expand production capacity maxfpðht ; mt1 Þg þ Qt

max

mt1
4. Contract production capacity maxfpðht ; mt1 Þg þ max Qt



06mt


 E½gðhtþ1 ; mt ; t þ 1Þ þ Sðmt1 ; mt Þ : 1 þ rf

 E½gðhtþ1 ; mt ; t þ 1Þ þ Sðmt1 ; mt Þ : 1 þ rf

The temporarily shut-down option is not explicitly addressed here as such a decision is reached whenever the optimal quantity to be produced is computed to be zero. If all options are considered simultaneously, then it can be seen that capacity contraction and expansion can be handled simultaneously, since they only differ in the possible values that mt can take. Furthermore, it should be noticed that the option to defer (dis)investment corresponds to a capacity switch to the same value. Therefore, these three options can all be gathered in a capacity switching option, as given below in Eq. (11). This way, we only have to explicitly address the capacity switching and project abandonment options:   E½gðhtþ1 ; mt ; t þ 1Þ maxfpðht ; mt1 Þg þ max þ Sðmt1 ; mt Þ : ð11Þ Qt 06mt 6M max 1 þ rf

176

D.B.M.M. Fontes / European Journal of Operational Research 188 (2008) 169–184

The optimal project value gðht ; mt1 ; tÞ in period t given the demand ht and available production capacity mt1 is then given by 8 maxfR; Sðmt1 ; 0Þg; > > n o < E½gðhtþ1 ;mt ;tþ1Þ þ Sðm ; m Þ ; if abandonment; ð12Þ max maxfpðht ; mt1 Þg þ max t1 t 1þr f Qt 06mt 6M max > > : otherwise: If this is a new investment, the optimal project net expected discounted value is then given by   gðh1 ; m0 ; 1Þ max  Iðm0 Þ ; 06mt 6M max 1 þ rf

ð13Þ

where Iðm0 Þ ¼ m0 c4 is the required initial investment to install production capacity m0. (It should be noted that if the m0 value that maximizes Eq. (13) is zero, then the project, at the moment, is not worth undertaking.) Otherwise, if the project already exists with capacity m0, then the optimal project net expected discounted value is given by gðh1 ; m0 ; 1Þ : 1 þ rf 4.4. Boundary conditions As said before, and in order to allow for earlier exercise, the valuation procedure begins at the last stage and works backward to initial time. At the final period t = T, for each demand value and each possible capacity value, the project value is given either by the final operating profit plus the discounted capacity scrap value or the abandonment value, whichever is the largest. Thus, in the last period the project value for the fixed capacity project and the flexible capacity project is given by Eqs. (14) and (15), respectively ( maxfR; Sðm0 ; 0Þg; if abandonment; f ðhT ; m0 ; T Þ ¼ max maxfpðh ; m Þg þ Sðm0 ;0Þ ; otherwise; ð14Þ T

QT

( gðhT ; mT 1 ; T Þ ¼ max

0

1þrf

maxfR; SðmT 1 ; 0Þg; maxfpðhT ; mT 1 Þg þ QT

if abandonment; SðmT 1 ;0Þ ; 1þrf

otherwise;

ð15Þ

where m0 is the value of the optimal or existing capacity for the fixed capacity project and mT 1 the value of the optimal capacity available at the last stage (chosen at stage T  1) for the flexible capacity project. 4.5. Expected continuation project values The implementation of the dynamic programming recursion, given by Eq. (9) or (12), on a standard binomial lattice computes expected value of future profits as E½f ðhtþ1 ; m; t þ 1Þ ¼ qu f ðuht ; m; t þ 1Þ þ qd f ðdht ; m; t þ 1Þ

ð16Þ

E½gðhtþ1 ; mt ; t þ 1Þ ¼ qu gðuht ; mt ; t þ 1Þ þ qd gðdht ; mt ; t þ 1Þ;

ð17Þ

or where qu, qd, u, and d are as explained in Section 4.1. 4.6. The dynamic programming algorithm In this section, we give a detailed explanation of how the Dynamic Programming (DP) models have been computationally implemented. Only one model implementation is discussed, since the other can be implemented in a similar manner. For the sake of simplicity we have chosen to discuss the fixed capacity model. The implementation is divided into four parts as in Fig. 2. A detailed explanation of each of them follows.

D.B.M.M. Fontes / European Journal of Operational Research 188 (2008) 169–184

177

Fig. 2. Flowchart of the DP algorithm.

The first module consists of inputting the values of the parameters associated with selling price, production and switching costs, maximum capacity and capacity step, and risk free rate. It also computes the demand values given the initial demand and the binomial model parameters. The initialization module sets all states as not computed and then initializes all terminal states as given by Eq. (14). In this module, we also compute the optimal quantity to be produced and corresponding profit for each possible combination of demand value and available capacity. The switching costs are also computed here. The computed project value module is the central module of the DP model implementation and thus it is described in greater detail. The optimal project value for a given capacity and initial demand is obtained by calling a recursive function Compute(h1, m, 1), which implements recursion (9). The advantage of the recursive implementation is that only existing states are computed, that is, not all demand values are considered in each period but only the ones resulting from the binomial evolution. Compute (h, m, t) If state ðh; m; tÞ has already been computed then return f ðh; m; tÞ aux1 ¼ pComputeðuh;m;tþ1Þþð1pÞComputeðdh;m;tþ1Þ þ p ðh; mÞ 1þr aux2 ¼ maxfR; Sðm; 0Þg If aux1 > aux2 then f ðh; m; tÞ ¼ aux1 decðh; m; tÞ ¼ 1 else f ðh; m; tÞ ¼ aux2 decðh; m; tÞ ¼ 2 Return f ðh; m; tÞ This function is then repeated for all possible values of m and the maximum project value net of investment costs is chosen as the optimal investment project value, as in Eq. (10). In order to retrieve the solution strategy, which is that of the decisions taken during the project life time, in this case wether to continue project or to abandon project (for the flexible case we would also get the capacity

178

D.B.M.M. Fontes / European Journal of Operational Research 188 (2008) 169–184

values), we have developed another module. This last module is also recursive and backtracks through the information stored, the decision made in this case, during the states computation. This information is to be used when testing the devised strategies, see Section 5.3. 4.7. An illustrative example Consider a four period problem where k = 1, h1 = 1, u = 1.114 (d = 1/u), p = 0.51, 1 + r = 1.008, no selling opportunity exists (R = 0), c1 = 0.1, c2 = 0.5, c3 = 0.1, c4 = 2, S2 = 0.85, and S3 = -0.05. Let us consider that the capacity can vary between 0 and 1.5 with 0.5 increments. The demand evolves over time as given in Fig. 3. Using Eq. (7), for this specific example we have 8 0; if m ¼ 0; > > > < 0:80; if m ¼ 0:5; ð18Þ Sðm; 0Þ ¼ > 1:65; if m ¼ 1; > > : 2:50; if m ¼ 1:5: The optimal profit p ðht ; mÞ for a given demand value and available capacity is computed using Eqs. (4)–(6). For this example, we have obtained the values given in Table 1. Obviously, when there is no capacity available (m = 0) there is no project, either because it has not been undertaken or because it has been abandoned, thus all Q and p* values are zero. As explained earlier we start with the terminal states, i.e. states at period 4, which are initialized as given by Eq. (14). For example, states (1.383, 0.5, 4) and (1.114, 0.5, 4) are computed as follows:   0:8 f ð1:383; 0:5; 4Þ ¼ max 0:8; 0:224 þ ¼ 1:018; 1:008   0:8 ¼ 0:915: f ð1:114; 0:5; 4Þ ¼ max 0:8; 0:1214 þ 1:008 For the remaining periods, states are computed as given by Eq. (9), e.g. state (1.241, 0.5, 3) is computed as   0:51  f ð1:383; 0:5; 4Þ þ ð1  0:51Þ  f ð1:114; 0:5; 4Þ f ð1:241; 0:5; 3Þ ¼ max 0:8; 0:167 þ ¼ 1:127: 1:008 The corresponding decision is to continue. In Table 2, we list all possible states and for each one of them we give the optimal state value and corresponding decisions.

Fig. 3. Possible demand values.

D.B.M.M. Fontes / European Journal of Operational Research 188 (2008) 169–184

179

Table 1 Optimal production quantity and operating profit for each possible value of demand and available capacity Demand

Available capacity

h

m = 0.5

0.723 0.806 0.898 1.000 1.114 1.241 1.383

m = 1.0

Q

p*

0.208 0.235 0.266 0.300 0.338 0.380 0.426

0.015 0.033 0.056 0.085 0.121 0.167 0.224

m = 1.5

Q

p*

Q

p*

0.249 0.282 0.319 0.360 0.406 0.456 0.513

0.022 0.000 0.027 0.062 0.106 0.160 0.229

0.267 0.303 0.342 0.386 0.435 0.489 0.550

0.067 0.043 0.014 0.024 0.070 0.129 0.202

Table 2 Optimal project value and corresponding decision (C – Continue and A – Abandon) for each period given the demand and available capacity Period

Demand

Available capacity

t

h

m = 0.5

1 2 2 3 3 3 4 4 4 4

1.000 0.898 1.114 0.806 1.000 1.241 0.723 0.898 1.114 1.383

1.137 0.958 1.159 0.856 0.961 1.127 0.808 0.850 0.915 1.018

m = 1.0 C C C C C C C C C C

1.880 1.716 1.945 1.650 1.753 1.952 1.650 1.664 1.743 1.866

m = 1.5 C C C A C C A C C C

2.595 2.500 2.679 2.500 2.529 2.726 2.500 2.500 2.551 2.683

C A C A C C A A C C

The decision corresponding to the optimal project value for m = 0.5 is to continue for all states, since in this case the capacity overhead costs are low and thus it is worth keeping the project going, even for small demand values. For larger capacity values, when demand is low, it is sometimes better to abandon the project since the capacity scrap value is higher. 5. Numerical examples and results In order to test our methodology we have implemented, in MATLAB, the dynamic programming models on the binomial lattice derived from the collected data. As we have considered the project to be a new investment, the initial production capacity also has to be decided, this corresponds to solving Eqs. (10) and (13) for the fixed capacity and full flexible capacity projects, respectively. In these equations, the project value is maximized in relation to the initial capacity value, which can be chosen from 0 to a pre-specified maximum, having increments given by a pre-specified capacity step. Throughout this section we use full flexible to denote the flexible capacity project where all options are considered simultaneously, since other versions of the flexible capacity project are to be considered from Section 5.2 onwards. Both models have been used to find out an optimal project value. For the full flexible capacity project, we also find the corresponding optimal capacity investment policy. The quality of these models is then tested by valuing the policy performance on specific data realization sets. To make the comparison between both projects easier we have considered that the project selling price value is the same as the capacity scrap value, i.e. R ¼ Sðm; 0Þ. 5.1. Problem specification It follows the description of the specific problem solved.

180

D.B.M.M. Fontes / European Journal of Operational Research 188 (2008) 169–184

The values for the parameters associated with price, production costs, capacity and capacity switching costs, production capacity, and risk free rate have been taken from Bollen (1999), and are as follows: Price parameter: The price slope was set to k = 1. Production costs: The parameters for the fixed and flexible production costs were set to c1 = 0.1 and c2 = 0.5, while the capacity overhead cost parameter was set to c3 = 0.1. Capacity and capacity switching costs: The per unit initial capacity cost was set to c4 = 2, a positive value since the total initial capacity cost is deducted from the project value, as in Eqs. (10) and (13). The capacity switching decisions involve a fixed cost, that was set to s3 = 0.05 and a per unit cost s1c4 when capacity is increased or a per unit cost recover s2c4 when capacity is sold. s1 and s2 are percentages of the initial capacity cost and were set to s1 = 1, s2 = 0.85. It should be noted that since s1 and s3 are costs they must be negative as they are being added to the project value, see e.g. Eq. (12). Production capacity values: The production capacity values range from 0 up to 2.5 with capacity step values of step ¼ 0:01; 0:05; 0:1, and 0.5; This means that m can take the following values 0, step, 2  step; . . . ; 2:5. These are also the values that the initial capacity value can be chosen from, since its value is also to be optimized. Annual risk free rate: rf ¼ 10%. We collected monthly sales data for a 48 months period, which has been scaled in order to be of the same magnitude as the demand values used in Bollen (1999). The first 24 periods of the scale data, here and hereafter referred to as model data, are used to set up the Binomial model, while the last 24 periods, together with randomly generated sales data, are used to test the devised strategies. (All data can be obtained from the corresponding author.) The initial demand was set to the average demand over the first 24 periods of the scaled data – model data. The binomial lattice parameters have been computed by using the model data, as given in Section 4.1 to be: u = 1.114, d = 1/u, qu = 0.510, and qd = 1/qu, see Appendix A. 5.2. Obtaining optimal strategies In this section, we report on the results obtained with the models devised for both the fixed and the flexible capacity projects. For each possible value of capacity changing step, we report the model value, i.e. the predicted project value, which is computed as given in Eqs. (10) and (13) and the corresponding initial capacity. In Table 3, we report the results obtained for the fixed capacity project and for the flexible capacity project, considering all options simultaneously. From the results reported it can be seen that the flexible capacity project predicts a higher project value than the fixed capacity project. This does not come as a surprise, since the flexible capacity model generalizes the fixed capacity model, that is, if the best strategy would be to have a fixed capacity throughout the whole planning period then the flexible capacity model would propose such a strategy. Furthermore, the initial capacity to be installed is smaller for the flexible capacity system. This is possible since the flexible system can adapt the capacity value by increasing it and a smaller capacity value leads to savings on overhead costs. Although better values for the predicted project value are usually obtained for smaller capacity steps, when the flexible capacity project is being considered, they do not vary much.

Table 3 Results for the fixed and full flexible capacity projects Cap. step

0.01 0.05 0.10 0.50

Fixed

Full flexible

Ratio

Model value

Init. Cap.

Model value

Init. Cap.

FFlex/fix (%)

7.502 7.502 7.502 7.502

0.990 1.000 1.000 1.000

7.953 7.952 7.950 7.914

0.660 0.650 0.700 0.500

106.01 106.00 105.96 105.49

D.B.M.M. Fontes / European Journal of Operational Research 188 (2008) 169–184

181

Table 4 Results for flexible capacity projects Cap. step

Contraction Model value

Init. Cap.

Added value

Expansion Model value

Init. Cap.

Added value

Full flexible Model value

Init. Cap.

Added value

0.01 0.05 0.10 0.50

7.585 7.585 7.584 7.570

1.09 1.10 1.10 1.00

0.083 0.083 0.082 0.068

7.937 7.936 7.934 7.899

0.63 0.65 0.60 0.50

0.435 0.434 0.432 0.397

7.953 7.952 7.950 7.914

0.66 0.65 0.70 0.50

0.451 0.450 0.448 0.412

In Table 4, we also report the predicted project value and the corresponding initial capacity for flexible capacity projects not considering all options simultaneously. These results allow us to find out the value added by the different options. Three versions of the flexible capacity model have been developed. All consider, simultaneously, the options to defer (dis)investment, to shut-down temporarily, and to abandon the project. In addition to these three options, each version also considers the option to: (i) contract production capacity – contraction; (ii) expand production capacity – expansion; and (iii) contract and expand production capacity – full flexible. Again, it can be seen that the predicted project value is not very sensitive to the capacity step value, since although its value varies, the variation is very small. The value added by the possibility of contracting capacity, expanding capacity, or both contracting and expanding capacity has been computed in relation to the fixed capacity project. As can be seen, all options add value to the project. However, the magnitude of these values is quite different. The contraction option is the least valuable, while the option to both contract and expand is the most valuable. Furthermore, the expansion option is almost as valuable as the option to both contract and expand. If the value added by the option to both contract and expand is regarded as the target, then it can be seen that while the value added by the contraction option is about 18% of this value, the value added by the expansion option accounts for over 96% of the total value added. The expansion project and the full flexible one have smaller initial capacity, which allows for overhead cost savings. It should be noticed that the value added by considering the contraction and expansion options simultaneously is smaller than the sum of the values added by the two options when considered in isolation. These results are actually not surprising since the binomial model assumes that demand grows at a constant rate in perpetuity, which intuitively undervalues a decrease in future demand values and overvalues an increase in future demand values, see e.g. Bollen (1999). Usually, real option applications and model valuation stop at this point. Here, however, we go further and test the solution strategies provided by all models developed on the scaled collected data, as well as on sets of randomly generated data. We call this procedure a posteriori analysis. By doing so, we can actually find out how well the devised strategies perform on specific data sets. 5.3. A posteriori analysis: Comparing the performance of the strategy provided The capacity policy devised by each model is applied to the last 24 periods of the scaled collected data and also to five different and randomly generated demand data sets. For each data set five instances have been generated and their characteristics, average demand as a percentage of the average demand model data (used to obtain the binomial model) and standard deviation (StDev), are given in Table 5. In this table, we also report on the characteristics of the remaining 24 periods of scaled collected data that, here and hereafter, is referred to as Set S6. Since we are using five different data instances for each data set, in Tables 6 and 7 we report on the average project value for the fixed and the full flexible projects, respectively, and in Table 8 on their variability, which is measured by the standard deviation. The reported project values refer to the fixed capacity and full flexible (all options available simultaneously) projects. As can be seen, for the first three data sets, for which the average demand is between 80% and 100% of the average demand of the model data, the flexible capacity model provides a better ‘‘real’’ project value.

182

D.B.M.M. Fontes / European Journal of Operational Research 188 (2008) 169–184

Table 5 Average characteristics out of the five instances considered for each data set Set

S1

S2

S3

S4

S5

S6

Demand (%) StDev

79.53 0.40

90.21 0.37

100.88 0.44

109.09 0.47

120.59 0.41

109.38 0.41

Table 6 Average project value for fixed capacity projects Step

Fixed

0.01 0.05 0.10 0.50

S1

S2

S3

S4

S5

S6

2.822 2.808 2.808 2.808

3.876 3.865 3.865 3.865

5.768 5.760 5.760 5.760

7.323 7.316 7.316 7.316

9.134 9.132 9.132 9.132

7.456 7.452 7.452 7.452

Table 7 Average project value for full flexible capacity projects Step

Full flexible

0.01 0.05 0.10 0.50

S1

S2

S3

S4

S5

S6

2.952 2.959 2.920 2.825

3.964 3.970 3.896 4.046

5.796 5.787 5.820 5.717

7.274 7.253 7.343 7.298

8.754 8.715 8.837 8.632

7.195 7.179 7.247 6.729

Table 8 Standard deviation of project value fixed and full flexible capacity projects Step

0.01 0.05 0.10 0.50

Fixed

Full flexible

S1

S2

S3

S4

S5

S1

S2

S3

S4

S5

0.245 0.246 0.246 0.246

0.483 0.482 0.482 0.482

0.564 0.565 0.565 0.565

0.406 0.408 0.408 0.408

0.563 0.564 0.564 0.564

0.236 0.234 0.242 0.222

0.250 0.264 0.223 0.220

0.594 0.589 0.611 0.462

0.497 0.493 0.512 0.421

0.594 0.617 0.632 0.723

However, for sets 4–6, which are associated with a demand increase, the fixed capacity model provides a better ‘‘real’’ value. The project value obtained by the full flexible system varies between 87.96% and 120.67% of the project value obtained by the fixed capacity systems. The fact that the full flexible project is not always better than the fixed capacity project might seem surprising at first. Nevertheless, the explanation is very simple and straight forward. More flexibility means more decision capability, which is only better if the decision agent is in a good position to decide, i.e. has enough information and knowledge about the situation being handled. Since we already know that the binomial model undervalues a decrease in future demand values and overvalues an increase in future demand values, we can say that, at least when considering binomial models, projects with increased flexibility may not perform better, or at least may not always perform better. A possible way of addressing the problem raised here is to use a more accurate model, for instance a Markov model. In another work (Fontes et al., submitted for publication) we have shown the Markov approach to be more reliable and to lead to better decision policies than the binomial model. 6. Conclusions In this work, we address the problem of making investment decisions in a flexible production capacity firm. The problem involves deciding not only the optimal quantity to be produced in every single period but also the

D.B.M.M. Fontes / European Journal of Operational Research 188 (2008) 169–184

183

optimal capacity policy, which determines the production capacity available in every single period, throughout the whole planning horizon. We consider the investments to be at least partially reversible, since capacity sellout allows for partial investment recover. We propose to address this problem by using stochastic dynamic programming implemented on a standard binomial lattice. Our results show that, as expected, an increase in flexibility always leads to an increase in project predicted value. However, although increasing decision flexibility is normally taken as an advantage, it may not always be the case. When the optimal strategies obtained are applied to specific demand data sets, the resulting project values are, in some case, better for the fixed capacity system. This is easily explained since additional flexibility is only an advantage if we have enough good information and knowledge of the decision situation we are faced with. That is, good judgment requires good information to base our decisions upon. The problem here is that the binomial model used – the standard and most used model in real options literature – might not be accurate enough to come up with the best decision policies in all situations. It is well known that the binomial model has constant growth/decay rates at constant probabilities. This implies that the binomial model tends to undervalue a decrease in future demand and to overvalue an increase in future demand. Acknowledgement The financial support of the FCT Project POCI/MAT/61842/2004 is gratefully acknowledged. Appendix A The collected data has been scaled in order to be of the same magnitude as the data used in Bollen (1999). Then and since the binomial is a multiplicative model, the ratios, as a percentage of the previous period value, have been computed. The binomial parameters have then been computed using these ratios (see Table 9).

Table 9 Data used to compute the binomial model parameters Period

Data

Ratios

Period

Data

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

1.1 1.5 1.8 1.1 1.7 1.3 1.5 0.6 1.4 1.7 1.8 1.0 1.3 1.6 1.7 1.1 1.7 1.1 1.9 0.7 1.7 1.7 1.7 1.4

1.00 1.36 1.20 0.61 1.55 0.76 1.15 0.40 2.33 1.21 1.06 0.56 1.30 1.23 1.06 0.65 1.55 0.65 1.73 0.37 2.43 1.00 1.00 0.82

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

1.30 1.80 1.70 1.60 1.90 1.70 2.30 0.90 2.00 2.20 2.00 1.50 1.60 1.70 1.60 1.50 1.50 1.20 1.80 0.50 1.40 1.50 1.10 1.00

184

D.B.M.M. Fontes / European Journal of Operational Research 188 (2008) 169–184

References Andreou, S.A., 1990. A capital budgeting model for product-mix flexibility. Journal of Manufacturing and Operations Management 3, 5– 23. Bollen, N.P.B., 1999. Real options and product life cycles. Management Science 45, 670–684. Brennan, M., Schwartz, E., 1985. Evaluating natural resource investments. Journal of Business 58, 135–157. Cox, J., Ross, S., Rubinstein, M., 1979. Option pricing: A simplified approach. Journal of Financial Economics 12, 229–263. Dixit, A.K., Pindyck, R.S., 1994. Investment Under Uncertainty. Princeton University Press, Princeton, NJ. Fontes, D.B.M.M., Camo˜es, L., Fontes, F.A.C.C., submitted for publication. Real options using Markov chains: An application to production capacity decisions. He, H., Pindyck, R.S., 1992. Investment in flexible production capacity. Journal of Dynamics and Control 16, 575–599. Kamrad, B., Ernst, R., 1995. Multiproduct manufacturing with stochastic input prices and output yield uncertainty. Real Options in Capital Investment: Models, Strategies, and Application. Praeger. Kandel, E., Pearson, N.D., 2002. Option value, uncertainty, and the investment decision. Journal of Financial and Quantitative Analysis 37, 341–374. Kulatilaka, N., 1988. Valuing the flexibility of flexible manufacturing system. IEEE Transaction in Engineering Management 35, 250–257. Majd, S., Pindyck, R., 1987. Time to build, option value, and investment decisions. Journal of Financial Economics 18, 7–27. McDonald, R., Siegel, D., 1985. Investment and the valuation of firms when there is an option to shut down. International Economic Review 26, 331–349. McDonald, R., Siegel, D., 1986. The value of waiting to invest. Quarterly Journal of Economics 101, 707–727. Myers, S.C., 1977. Determinants of corporate borrowing. Journal of Financial Economics 5, 147–175. Myers, S.C., Majd, S., 1990. Abandonment value and project life. Advances in Futures and Options Research 4, 1–21. Pindyck, R., 1988. Irreversible investment, capacity choice and the value of the firm. American Economic Review 78, 969–985. Pindyck, R.S., 1991. Irreversibility, uncertainty, and investment. Journal of Economic Literature 29 (3), 1110–1148. Sethi, A.K., Sethi, S., 1990. Flexibility in manufacturing: A survey. International Journal of Flexible Manufacturing Systems 2, 289–328. Tannous, G.F., 1996. Capital budgeting for volume flexible equipment. Decision Sciences 27, 157–184. Triantis, A., Hodder, J., 1990. Valuing flexibility as a complex option. Journal of Finance 45, 549–565. Trigeorgis, L., 1996. Real Options: Managerial Flexibility and Strategy in Resource Allocation. MIT Press.