Journal of Economic Dynamics and Control 10 (1986) 89-91. North-Holland

OPTIMAL TIMING OF CAPACITY EXPANSION

Z. CARVALHAIS

CA PS, Ir~tituto Superior Tecnico, 1090 Lisboa, Portugal

M.H.A. DAVIS

Imperial College, London S W7 2BT, England

Capacity expansion is the process of providing new facilities so as to meet a rising demand for their services. We are mainly interested here in large-scale public utilities such as electricity supply or water resources where the evolution of demand is significantly uncertain within the construction time of a single project. A realistic mathematical model of such a situation must incorporate (a) a stochastic model for the growth in demand and (b) some description of the construction times. The latter may be fixed times, random times, or controlled in some way by the decision-maker.

There is an extensive literature on capacity expansion. An influential early contributor was Manne (1961), while Luss (1982) contains an up-to-date survey~ However, much of the literature assumes either deterministic demand forecasts or instantaneous completion of projects and thus is not applicable to large-scale projects such as those mentioned above. In a recent paper by Davis, Dempster, Sethi and V~rmes (1985) demand is modelled as a point process (either a simple or doubly stochastic Poisson process) and the decision-maker is supposed to control the rate of investment in the current project, completion taking place when total investment equals project cost. Formulated in this way the problem is one of stochastic control theory for piecewise-deterministic Markov processes [Davis (1984)], and by using dynamic programming it is possible to compute optimal policies, at least in simple cases.

In this paper we take an approach, more closely related to impulse control, in which the decision-maker simply decides when to initiate construction of a project, the completion time then being a random variable with known distribution. As in Davis et al. (1985), demand is modelled as a Poisson process. The completion times are supposed to be exponentially or gamma-dis- tributed. There are penalties for overcapacity and for shortage, and we

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90 Z. Carvalhais and M.H.A. Davis, Capaci(v expansion

optimize according to an average cost per unit time criterion. The problem can then be formulated in terms of the 'Generalized Markovian Decision Processes' (MDP) introduced by De L6ve, Federgruen and Tijms (1977).

In a MDP, the evolution of a 'natural process' (a homogeneous strong Markov process) is interrupted by 'interventions' at which the state is moved instantaneously to a new position. A stationary strategy divides the state space into continuation and intervention regions. At each state in the intervention region there is a set of feasible decisions, which may include the null-decision of taking no action. A key assumption of De L6ve et al. (1977) is that there is a non-empty set A 0 such that the null decision is not feasible in A 0 and the natural process eventually reaches A 0 from any starting point. Costs of stationary policies can then be calculated by a renewal-type argument based on the expected times and costs incurred by the natural process between succes- sive returns to A o.

The simplest capacity expansion model in. this framework is as follows: demand is a Poisson process and project completion times are exponentially distributed. The state of the process is (Nt, i t ) w h e r e Nt is the excess capacity and i t is an indicator variable (i t = 1 if a project is under construction and i t = 0 if no project is under construction). Each project supplies a fixed number s of capacity units, so on completion of a project at time t the (right continuous) state jumps from (N t_, 1) to (N , i t) = ( N t_ + s, 0). When a project is initiated the state jumps from (N/_,0) to ( N , i t ) = ( N t _ , l ). The natural process consists of evolution of ( N t, it) from an arbitrary initial state (n, i) to A o = ((0,0)) ( N t stays at zero from the first time zero is reached, i.e., excess demand is rejected). A cost R per unit time is paid while in shortage ( N t = O) and a holding cost h N t per unit time is paid when N t > 0. By using the theory of MDPs we are able to compute explicitly the optimal intervention level at which construction of a new project should be initiated, at least under-certain conditions on the coefficients.

The results can be extended to 7-distributed completion times by using the well-known 'method of stages': a 7(n, ~)-distributed random variable T has the same distribution as T z + T 2 + .. . + T, where 7"1, T 2 . . . . are i.i.d, exponen- tially-distributed random variables with parameter h. If T is the completion time of a project one can thus think of the project as consisting of n exponentially-distributed stages, and a Markov model is obtained by taking as state the triple (Nt, it, Jr) where ( N t, it) are as before and Jt ~ {0,1 . . . . . n - 1} is the number of completed stages of the current project.

Further work will be directed towards control of project size, where one has a range of projects available of differing sizes and costs. The idea of 'economy of scale' can be introduced by supposing that larger projects take longer to complete but supply cheaper capacity. Optimization will then determine whether a large-scale project should be undertaken or a 'quick fix' resorted to.

Z Carvalhais and M.H.A. Davis, Capacity expansion 91

References

Davis, M.H.A., 1984, Piecewise-deterministic Markov processes: A general class of non-diffusion stochastic models, Journal of the Royal Statistical Society B 46, 353-388.

Davis, M.H.A., M.A.H. Dempster, S.P. Sethi and D. Vermes, 1985, Optimal capacity expansion under uncertainty, Advances in Applied Probability, forthcoming.

De L/.we, G., G. Federgruen and H.C. Tijms, 1977, A general Markov decision method I: Model and techniques, Advances in Applied Probability 9, 296-315.

Luss, H., 1982, Operations research and capacity expansion problem: A survey, Operations Research 30, 907-947.

Mann¢, A.S., 1961, Capacity expansion and probabilistic growth, Econometrica, 632-649.