multiobjective investment planning under uncertainty

OMEGA, The Int. J1 of Mlmt SoL, Vol. 3, No. 4. 1975

Multiobjective Investment Planning Under Uncertainty

DA CAPLIN World Institute, Jerusalem

JSH KORNBLUTH Hebrew University and World Institute, Jerusalem

(Receleed d~y 1974; in reviled form $ ~ y 1975)

In this paper we consider the relevance of various planning methods and decision criteria to multiobjective investment plmanins under uncertainty. Assuming that a natural reaction to uncertainty is to operate so as to leave open as many good options as possible (as opposed to maximizing subjective expected utility) we argue that the planning process should concentrate on analyzing the effects of the initial decision, and that for this exercise the ~ methods of mixed integer programming are inappropriate. We demonstrate how the technique of dynamic programming can be extended to take aocount of multiple objectives and use dynamic programming as a framework in which we analyze the robtatnem of an initial decision in the face of various types of uncertainty. In so doing we also analyze the risks involved in both the planning and decision making functions.

INTRODUCTION

"NOTHING in progression can rest on its original plan, We may as well think of rocking a grown man in the cradle of an infant."--Edmund Burke, 1729-1797.

In v i r tua l ly all inves tment p lann ing in government and indus t ry there a re three aspects to be cons ide red :

1. the cur ren t inves tment decis ion is the first o f a series o f decis ions which form a long- te rm p l an ;

2. there is uncer ta in ty a b o u t the future, and consequent ly the p lan will be up- da t ed to t ake accoun t o f the deve lopmen t s in the economic env i ronment and the changes in fu ture expec ta t ions ; a n d

3. such fu ture inves tments will be m a d e in the l ight o f the prevai l ing c i rcum- s tances (and expecta t ions) and the degree o f manoeuverab i l i t y afforded by the previous inves tment p r o g r a m m e .

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The extent to which an investment decision taken today will constrain the possible future courses of action is of particular importance when one is dealing with integrated complexes in which:

(a) the capacity available of one component of the system may have a significant effect on both the present and future performance of other parts of the system; and

(b) there may be a variety of conflicting performance measures associated with the total complex.

Standard approaches to planning under risk tend to rely on the optimization of one criterion (the expected value or utility of the total project) or on the presentation of the expected distributions of key parameters as obtained by simulations [2, pp. 551-559, 6, 7].

However, such methods appear to meet with little acceptance in practice [11]. In many cases even the simple subjective estimates of future conditions prove to be totally inaccurate [1 ] and provide tenuous data for a probabilistic analysis of future plans [8]. In most cases the methods place far too much emphasis on the total plan and far too little emphasis on the current decision and its immediate results.

Rosenhead et al [5] introduced the concept of the robustness of the initial investment decision. Robustness tries to measure the flexibility of the initial decisions in the sequence of capital investment decisions, by measuring the proportion of 'good' end-states which can be reached from the initial decision. The emphasis is thus shifted from identifying a single optimal strategy to a consideration offamilie~ of strategies. Attention is focused on the desire (often observed in actual business decision-taking) to keep one's options open for as long as possible. Although robustness is based on deterministic information, the approach pre- serves flexibility in the knowledge that the future state of the environment is uncertain, and that data and techniques may not be adequate to allow subjective probability distributions to be assigned to the parameters with any sense of realism. In this paper we explore some of the consequences of this approach to the determination of capital investment strategies in the light of recent work in programming and planning with multiple objectives (e.g. [9, 13, 16]).

A characteristic of most long-range planning problems is that the decision- maker and the planner (or planning team) are at different levels of the organiza- tional hierarchy. As a consequence there is imperfect communication between them and the planner is always subject to the risk that he supplies an inadequate or inappropriate set of plans for the decision-maker's consideration. We will attempt to define planning measures which quantify the quality of the planning process, and will relate these measures to various situations of uncertainty using appropriate measures of the 'robustness' of the original decision. We will illu- strate the thesis with an example of a multi-objective programming problem.

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In contrast to standard planning methods under risk and uncertainty which attempt to condense the various aspects of the problem into one composite function, the methods suggested below enable one to preserve a reasonable rich- ness of variety throughout all the steps of the planning process, up to the point at which the initial decision is made.

PLANNING WITH MULTIPLE OBJECTIVES

A major study of multi-objective models by Johnsen [9] shows that firms operate in an environment which is measured in terms of several performance measures, and presents a series of models which can be adapted to suit this type of behaviour. From the planner's point of view, the existence of several performance measures is a considerable complication. If there is only one performance measure (or objective function) the planner can use this for selecting the best plan. This might be the plan which maximizes the objective or brings the objective as near as possible to some predetermined goal. If the firm is in a multiobjective environment the planner's task is more complicated.

There are several possible approaches open to him:

(a) He can attach a priori weights to the various objectives and treat the problem as one with a single objective function;

(b) He can set up a hierarchy of objectives and proceed to optimize the objective function with the highest priority;

(c) He can look for efficient undominated solutions;

(d) He can adopt a satisficing approach by defining a series of bounds which an acceptable solution must satisfy, one for each performance measure, and can 'solve' the problem by searching for a solution (or set of solutions) which satisfies all the planning requirements.

The first approach, the construction and use of a utility function, is discussed by Johnsen and found to be operationally unsatisfactory [9, p. 433]. In general it is extremely diffcult for the decision-maker to supply the necessary weights (which may be functions of the values assumed by the performance measures and the state variables) and the resulting function (if constructed) is only valid for one point in time. The approach has the additional disadvantage that the construction of a single utility function tends to obscure the multidimensional nature of the problem at the very outset. The second approach, that of an ordered hierarchy of objectives, is only applicable when there is one over-riding objective and a series of subsidiary ones.

o 314-o 425


It is the third and fourth approaches which are of interest in our analysis since it is these approaches which preserve the essential multi-dimensional nature of the problem for as long as possible and do not require explicit information regarding the decision-making process. Using these approaches, the planner can work with the multiple objectives right up to the point at which the decision- maker decides between the final set of alternatives. However, it must be borne in mind that the use of multiple criteria and the preservation of the multidimensional nature of the problem complicate the planning process.

UNCERTAINTY AND PLANNING RISK

As we have already noted, planning and decision-making take place at different levels of the organization and there is generally an incomplete flow of information between the decision-maker and the planner. (Typically, the decision-maker fails to describe the goals and objectives in sufficient detail, and the planner fails to emphasize the assumptions and constraints associated with the various proposals.) Since both are operating in an uncertain world, both are subject to the risk of making a wrong decision---or of not making a correct one. These two types of error are analogous to the errors in hypothesis testing (see [I0, Vol. 2, p. 164]), i.e.

1. wrongly rejecting a good strategy, and

2. wrongly accepting a bad strategy.

In the face of such uncertainty, it is illogical for the planner to propose a strategy which will be fixed for all future decisions, irrespective of the actual developments of the environment. The natural response to such uncertainty is to specify the initial decisions which must be taken nos,--with due consideration of possible updating of the strategy in the future. This change of emphasis from planning the total strategy to selecting the initial decision should also imply a change of emphasis from criteria which aggregate performance over time (e.g. average discounted values) towards criteria which measure the degree of 'risk' involved with a particular course of action. Since we would also argue that it is impossible to obtain a practical measure o! the risk associated with any particular strategy, it is essential that we develop other measures (either substitute or surrogate) which will be of use to both the planner and the decision-maker, and for this purpose we will turn our attention to strategies which 'leave open as many future options as possible', i.e. to measures of the robustness of the initial decisions.

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PROGRAMMING AND PLANNING

The two major normative tools available for the analysis of the type of problems outlined above are MIP (mixed integer programming) and DP (dynamic programming) [14]. a

Mathematical programming The MIP approach has a number of disadvantages:

(i) It gives just one optimal plan, and one initial set of decisions for the first sub-period. It is difficult to estimate the various results of different initial decisions taken during the first sub-period;

(ii) Although the values of the dual variables are obtained (as in standard LP) their use is restricted to continuous variables. They represent the shadow costs of departing from the optimal solutions assuming that the values of the integer variables remain constant. To obtain the opportunity loss associated with an integer variable one must re-run the model;

(iii) As soon as we proceed to mixed-integer programming problems with multiple objective functions, the method is inapplicable. Although in the case of continuous linear programming we can work with a number of objective functions and thus find a set of efficient or undominated solutions (see [13]), to our knowledge no fast algorithms exist for finding efficient solutions with multiple objectives in MIP. In order to find the set of efficient plans using mathematical programming, we would have to enumerate all feasible plans---a lengthy and time-consuming process.

The main attention of the MIP model is focused on the allocation of opera- tional resources throughout the entire planning period, whilst the initial investment decisions are obtained only as a by-product of these calculations. Yet it is these initial investment decisions which ought to form the main area of enquiry; in particular, sensitivity analysis of the 'optimality' of the initial investment decision with regard to changes in the assumptions of the model and its environment is of prime importance.

Single and multiple objective dynamic programming Unlike mixed integer programming, the DP approach can be extended to the

case of multiple objectives. As we have already implied above, the case of multiple objectives can be covered by defining an undominated or "'efficient" solution instead of an "optimal" one. The formal definition is as follows: If we assume that there are q objective functions f l (x ) . . . fq (x ) , then a feasible solu-

z We exclude search techniques and simulation methods from the present discussion. The comments of the remaining sections of the paper apply equally to these techniques.

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tion x* is efficient if and only/f there exists no other feasible solution y such that:

f~ ty) ;~ f~ (x*) i =, I . . . q

with

f~ Cv) > fj (x*) for some j

i.e. a solution is efficient if it is undominated by all other solutions. With these definitions we can easily extend the principle of optimality, so as

to take account of the multiple objective situations thus:

"A path F/through the network is efficient if and only if for all points i , j on the path, the sub-path F/t~ is an efficient path between i and j "

and can use this extension of DP to solve the multiple-objective problem. The following example illustrates this approach.

EXAMPLE OF DP WITH MULTIPLE OBJECTIVES

Let us assume that we are planning the development of an integrated complex comprising three major components, A, B, and C with two performance measures--NPV (net present value) and total employment. A particular configuration of the complex w/ll be identified by stating the capacity levels of each of the three components, say (Lt, L2, L3)" each index L~ refers to the level of capacity of component i in the current configuration, e.g. Lj = 1 implies that A has size 40,000 units etc.

We assume that A has three possible sizes and that B and C have two possible sizes each, and that the complex must contain each of the components A, B, and C. There are twelve possible configurations. The capital costs of erecting facili- ties are described by the movement from index 0 to index 1, 2 or 3, and are shown in Table 1. We will also assume that it is impossible to reduce the capacity of a facility once it is installed; thus it is only possible to move to configurations with a higher index, not a lower one, e.g. the move (I, 2, 3) -* (2, 3, 3) is feasible, whereas (1, 2, 3) -* (1, 2, 2) is not. (Other restrictions in the build up of the complex can easily be included.)

These capital costs are calculated as of the present date, and might include estimates of inflation. We assume that the planning will be carried out by making five possible investments--and we therefore divide the total planning period into five sub-periods of length 2, 3, 5, 5 and 10 yr respectively. (This division of time is an arbitrary assumption made for ease of exposition.) On the

2 It is convenient to arrange the indices such that over time we tend to move from a lower index to a higher one. A movement f rom one configurat ion to another will be represented by moving from one vector (say (1, 2, 3)) to another (say (2, 3, 3)).

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basis o f cer ta in ty a b o u t es t imates o f fu ture d e m a n d (or by assuming means) the

net annua l ope ra t ing profi ts for each o f the conf igura t ions are calculated. These are shown in Tab le 2.

T~LE 1. C~rrAt. COSTS OF INrn~ AND INCREMENTAL CONSTRUCTIONS (M $)

Capital costs when Capacity constructed from

Component Alternatives (units p.a.) 0 1 2

A 1 40 ,000 80 - - - - 2 60,000 110 40 3 100,000 140 70 40

B 1 200 ,000 120 - - 2 400,000 200 120

C 1 500 60 2 700 90 35

m

m

TAat~ 2. NET ANNUAL O l ~ o rzortTs (M $)

Period Configuration 1 2 3 4 5

(1, 1, 1) 32 53 68 68 68 (1,1, 2) 31 52 73 73 73 (1,2,1) 14 33 57 65 85 (1, 2, 2) 12 33 61 70 90 (2,1,1) 32 52 74 81 81 (2,1, 2) 27 48 75 82 82 (2,2,1) 9 27 57 73 93 (2" 2, 2) 6 26 61 77 96 ( 3 , 1 , 1 ) 21 40 61 72 84 (3, 1, 2) 18 38 63 75 86 (3, 2, 1) --3 15 43 63 94 (3, 2 , 2 ) - - 6 13 47 67 98

Us ing s t anda rd d i scount ing p rocedures (and a d i scount rate o f 1 0 ~ ) , the figures o f Tab le 2 are conver ted into the N P V o f opera t ing profi ts as shown in Table 3.

W e assume tha t the pro jec t is being sponsored (in par t ) by a government agency interested in the general increase o f emp loymen t in the a r e a : W e therefore include the to ta l e m p l o y m e n t over the length o f the project as our second

3 At present the Israeli Government is interested in a policy which provides new employment opportunities in development areas, hence we have included the maximization of a function of employment as one of the performance measures. The comparison of employment with NPV then provides the useful statistic of the incremental cost of employment opportunities.

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Caplin, Kornbluth--Muitiobjective Investment Planning

TAeLe 3. NPV or OPERATING PROFITS PER PERIOD (EXCLUDING CAPITAL ~ ' M ~ ¢ r ) (M S)

Period Configuration 1 2 3 4 5

(1, 1, 1) 56 108 161 100 100 (1, 1, 2) 53 108 ]72 107 107 (1, 2, 1) 24 68 133 95 125 (I, 2, 2) 21 67 144 102 132 (2, 1, |) 56 107 174 118 119 (2, 1, 2) 46 98 176 119 120 (2, 2, 1) 15 56 135 106 136 (2,2,2) 11 54 144 112 142 (3, 1, 1) 37 83 143 105 123 (3, 1, 2) 31 78 149 109 127 (3, 2, 1) --6 30 101 92 139 (3, 2, 2) --10 28 110 98 144

objective. ( F o r s implici ty we assume tha t there is no t ime preference in this object ive a l though it is qui te s imple to include a t ime preference, a mul t ip l ier effect or a con t inu i ty const ra int . )

The e m p l o y m e n t figures pe r conf igurat ion per per iod are presented in Table 4.

TASTE 4. TOTAL EMPLOYMENT PIER PIglUOD (MAN YEARS X 10 2)

Period Configuration I 2 3 4 5

(1, 1, 1) 61 96-3 1 6 3 . 5 163"5 325 (1, 1, 2) 43.4 78 130-5 130-5 390 (1, 2, 1) 77 123 191 191 560 (1, 2, 2) 64 102 185 185 390 (2, 1, 1) 68 101"4 1 7 2 . 5 172.5 350 (2, 1, 2) 55 81-6 141 141 290 (2, 2, 1) 90 150 285 285 640 (2, 2, 2) 71 106-5 195 195 440 (3, I, 1) 69 108 185 185 390 (3, 1, 2) 57 90 155 155 325 (3, 2, 1) 94 159 325 325 780 (3, 2, 2) 72 114 225 225 580

The calculations The object ive o f our calcula t ions at present is to compu te all the feasible

efficient ways of sett ing up the complex over t ime, in the hope tha t an analysis o f their associa ted initial investment decisions will help us to solve the present dec is ion-making p rob lem. F o r our analysis we define a feasible investment strategy as one which moves th rough the poss ib le conf igurat ions over t ime in

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such a way that the index of capacity is increased or left unaltered. There are 711 such strategies.

Let us consider the strategies in which one of the configurations (1, I, 1), (1, 1, 2) or (I, 2, 1) is in operation at the end of the first time period. It can be seen from Table 1 that construction costs of the components of the configuration (1, 1, 1) at the beginning of the first time period are 80, 120 and 60 respectively. (Since these are expenditures at the initial decision point, they need not be discounted.) The discounted operating profit of the configuration (1, 1, 1) during the first time period is 56, thus, at the end of the first time period, (1, 1, 1) has a net discounted profit of --204 (=56--80--120--60) , and an employment of 61. It can only be set up by constructing the configuration (1, 1, 1) at time 0 at a cost of 260. Similarly, configuration (1, I, 2) has a net discounted profit of --237 (=53- -80- -120- -90) and an employment of 43.4, whilst configuration (1, 2, 1) has a net discounted profit of --316 (=24--80--200--60) and an employment of 77.

We now move to the next period and, for each configuration, we ask the question " I f we were to be in this configuration, in this period, which are the best configurations to have come from ?" From Table I we know that the costs of expanding configuration (1, I, 1) to (1, I, 2) or (1, 2, 1) are 35 and 120 (m $) respectively. If these expansions are undertaken at the beginning of the second period, they imply discounted costs of 29 and 99 respectively.

For strategies having the configuration (1, 1, 2) in period two, there are two ways of starting the second period:

(a) via configuration (1, 1, 1) in period one with additional discounted investment of 29 at the end of period one; or

(b) via configuration (1, 1, 2) in period one with no additional investment.

Route (a) would imply that we begin period two with a NPV of --233 ( = --204--29) and an employment of 61, whereas route (b) would imply a starting point of --237 and 43.4. Clearly, route (a) is the more efficient route. It implies a greater employment and a smaller loss. This procedure is illustrated in Fig. 1.

Considering strategies having the configuration (1, 2, 1) in period two, we have two ways of starting the second period:

(a) via configuration (1, 1, I) in period one with additional discounted investment of 99 at the end of period one; or

(b) via configuration (l, 2, 1) in period one with no additional investment.

In this case both routes are efficient. The route via (1, 1, 1) gives greater profits and lower employment, whilst the route via (1, 2, 1) gives lower profits but greater employment. We therefore carry forward both alternatives.

431

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Caplin, Kornbluth--Muhiobjective Investment Planning

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Proceeding in this manner we calculate the efficient investment strategies for each configuration up to the end of the fifth period. There are 121 such efficient strategies leading to the twelve possible terminal configurations. Some of these strategies are dominated by others, i.e. some strategies although efficient for a particular configuration are dominated by other strategies leading to other configurations. The envelope of undominated strategies for all twelve configurations is shown in Fig. 2; the associated strategies are tabulated in Table 5. In this simple planning exercise, the planner could present the decision-maker with the results of Table ~--from which the decision-maker would select the most 'desirable' initial configuration. The decision-maker's choice would not have to be justified on any other grounds other than subjective ones, without specifying any of the implied weights or preferences.

From this example we can see that the multiple-objective dynamic programming approach is capable of generating the entire e.~cient era,elope associated with the planning problem, i.e. the set of efficient or undominated terminal states that can be reached from an initial investment decision during the first sub-period.

The programme used for these calculations required less than 15 seconds on a CDC 6400 computer, and it is quite feasible to extend the consideration to more

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Omega, Vol. 3, No. 4

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I I I f t I I f I ! O~ 9o0 to00 i lo0 t20o 130o HOO I ~ t6oo

Cumulative labour X 102

FIG. 2. The set of solutions at the planning horizon and the efficient boundary (details of numbered strategies are given in Table 5).

than two performance measures. (Clearly the presentation of the results will be less elegant than in the two dimensional case.) The methodology remains unchanged.

We may conclude that the multiple objective dynamic programming approach has three major advantages:

(i) it automatically generates all the efficient solutions to the planning problem.

TAet.E 5. E F t ' l C l ~ INV~TMENT STRATEGIES AND THEIR VALUES

Path Total Period 1 2 3 4 5 Cumulative employment

Strategy profit (X 100)

I (I,I,I) (I, 1, I) (2,1,1) (2,1, I) (2,1,1) 290 853 2 (1,1,17 (1,1,1) (2,1,1) (2,1, I) (3,1,1) 284 893 3 (1,1,1) (1,1,1) (2,1,1) (2,1,1) (2,2,1) 278 903 4 (1,1,1) (1,1,1) (2,1,17 (2,1,1) (3,2,17 272 1283 5 (2,1,1) (2,1,1) (2,1,1) (2,1,1) (3,2,1) 266 1296 6 (2,1,1) (2,1,1) (3,1,1) (3,2,1) (3,2,1) 176 1459 7 (I,I,I) (I,I,I) (2,2,1) (2, 2,17 (3, 2,1) 175 1507 8 (1,1.17 (1.1,1) (2, 2,1) (3,201) (3, 2,1) 156 1547 9 (2,1,1) (2,1,17 (2,2,1) (3,2,1) (3,2,1) 149 1559

I0 (2,1,1) (2,2,1) (2,2,1) (2,2,1) (3,2,1) 93 1568 11 (I,1,1) (I ,2,1) (2,2,1) (3,2,1) (3,2,1) 91 1574 12 (1,2,1) (1,2,1) (2,2,1) (3.2,17 (3,2,1) 78 1590 13 (2,1,1) (2,2,1) (2,2,1) (3,2,1) (3.2,1) 74 1608 14 (1,2,1) (2, 2,1) (2,2,1) (3, 2,1) (3, 2,1) 58 1617 15 (2,2,1) (2,2,1) (2,2,1) (3,2,1) (3,2,1) 52 1630

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Caplin, Kornbluth--Muitiobjective Investment Planning

(ii) the tabulation of the efficient strategies can be used as a starting point for the analysis of the effects of the initial decision, and

(iii) one can obtain estimates of the opportunity costs associated with each decision by comparing the set of outcomes with those of the efficient envelope.

Despite these advantages the straightforward application of dynamic programming (with single or multiple objectives) will not necessarily solve the planner's problem. There may be many efficient strategies associated with any real system--and it is unlikely that the planner will feel happy in presenting the decision maker with a large number of possible choices. Some further screening will be required, during which it is important to remember that:

(i) it is the initial decision which is of most importance at the moment; and

(ii) even if the problem has been analyzed from an assumption of certainty, this assumption is unlikely to be justified in practice.

Both these points can be taken into account by appropriate definitions of 'robustness'.

ROBUSTNESS

Rosenhead, et al [5, 12] suggest the use of 'robustness' as a measure of the desirability of the first of a series of planning decisions. They describe the robustness of an initial decision dt as: "the number of 'good' end states for expected external conditions which remain as open options. . ." or " . . . the ratio of that number to the number of good end states considered".

We can interpret this measure as follows: An initial decision which has a high measure of robustness will leave a large number of good strategies open for future adoption. Any one of these can be selected at the next decision point in the light of revised information about the environment. This interpretation of robustness is true except where the measure is inflated by the inclusion of a large number of strategies which are essentially the same (i.e. strategies which differ by trivial diversions from each other but lead from the same initial decision to the same ultimate end point) or where different outcomes have different probabilities of materializing.

The measure suggested by Rosenhead et a / i s intended for situations of uncertainty [12, p. 315] but the actual mention of uncertainty is omitted. Al- though the definition has been amplified [15] the specific inclusion of the probabilistic nature of the system is of little advantage: "uncertainty rather than risk is the defining characteristic of strategic problems" [l 2].

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In general, the methods based on probability calculations tend to reduce the variety of the set of plans being considered by combining factors into uni- dimensional expected values, whereas the methods based on robustness attempt to preserve as much variety as possible (whilst bearing in mind the scale of the planning exercise and the effort that can be utilized).

We will now analyze the use of robustness as a planning criterion (and as a surrogate measure of risk) in the situations of uncertainty in the dichotomized roles of planning and decision making, and will show the relevance of such a criterion when planning in the three situations of uncertainty about:

(i) the trade-offs between objectives;

(ii) the states of the environment in the future and the response of the organization to such future states; and

(iii) the appropriateness of the model space and the subspace chosen for investigation.

In (i) we will assume that both the states of the future and the appropriateness of the model space are known with certainty; in (ii) we will assume that only the appropriateness of the model space is known with certainty. In (iii) we will relax the assumptions completely. This progressive analysis of the risks involved with the project is one which planners adopt (or should adopt) when confronted with planning problems involving uncertainty.

AN ANALYSIS OF PLANNING UNDER UNCERTAINTY

(1) Uncertainty about the trade-offs between objectives Let us first assume that the future states of the environment are known with

complete certainty (or with completely quantifiable risk and probability functions) but that the planner is uncertain about the relative trade-offs that might be made by the decision-maker (either because of a lack of communication between them or a lack of definition on the part of the decision-maker).

We can argue as follows:

(i) Assuming that the decision-maker and the planner have utility functions {which can always remain unspecified) which are increasing functions with respect to attainments in each of the objectives, the planner can concentrate his attention on obtaining the set of efficient final configurations, and can neglect all non-efficient strategies. (This has been the case in our example.) The adoption of a strategy which does not lead to an efficient terminal configuration will come under the category of a type 2 error (i.e. the inclu-

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sion of a strategy which is a "bad" strategy). In any realistic problem there may be a large number of efficient strategies;

(ii) If we further assume that there is no dominant objective, it may be possible to cut clown the number of acceptable efficient strategies by limiting the ratio between selected sets of attainments, and/or the acceptable trade-offs between objectives. In our example we might exclude strategies 11-15 on the grounds that the ratio between NPV and employment is unacceptable. We might also reject strategies 1-5 where there is considerable gain in employment for small percentage loss in N'PV, and strategies 9-10 for the opposite reason. This heuristic argument is a reinterpretation of Geoffrion's concept of "proper efficiency" [4] in which the rate of trade-off between acceptable efficient solutions is bounded by some limiting value (the decision-making body should find little difficulty in providing the planner with such a value);

(iii) Having reduced the number of efficient strategies, the planner can turn his attention to analyzing the results of the alternative decisions. For each of these initial decisions he can calculate a measure of robustness and can present the various alternatives to the decision maker for final consideration. It should be noted that in situations where the future environment is assumed to be known with certainty, the measures of robustness are likely to be inflated by the inclusion of strategies which derive from the same initial decision but lead, via different paths, to terminal configurations whose measured performance differs 'little' in all criteria. From the decision- maker's point of view such strategies are identical.

Measure of robustness for Case 1 We still have to consider the possible measures of robustness for such situa-

tions of uncertainty about the future trade-offs between objectives. Four measures can be suggested, in addition to the number of good solutions associated with a particular decision. These are:

(a) The measure of robustness defined by Rosenhead et al, i.e. the proportion of efficient strategies (or terminal configurations) which can be attained by implementing a particular decision. In the case of certainty with respect to the environment, good strategies are those which are e~cient. The measure is:

r , - - -~- f f ) (I)

where = n(~t) is the number of efficient states that ran be reached by implementing the initial decision dr, and n(,¢~) is the total number of efficient solutions to the problem, S being the set of long range plans;

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(b) A measure of the flexibility of the remaining decisions after a particular decision dt i.e., the ratio

n($,) t, s (2)

nf$,)

where n(S~) is the total number of strategies deriving from the initial decision d~. The measure r~ represents the 'chance' that an efficient solution derives from a decision d~, whereas the measure tt represents the 'chance' that, all other things being equal, a subsequent path will lead to an efficient solution;

(c) A measure of robustness which reflects the span of efficient possibilities that can be attained after implementing decision dr; or

(d) Rt a measure of robustness given by say the sum of squares between acceptable points reached as the result of the initial decision dr.

These measures fall into two distinct groups: (a) and (b) measure the numerical flexibility that can be associated with a particular decision. Measure (a) is an a priori measure "what proportion of good strategies are associated with a particular decision". Measure (b) is more of an a posteriori measure: " i f I accept decision dl what proportion of the resulting paths axe 'good' ". Measure (b) could be used as a probabilistic assessment of the ability of the decision-maker to choose the right strategy. A high measure of robustness reduces the possibility that a decision-maker might inadvertently choose a wrong second decision after taking a particular first decision d,.

The second group (c) and (d) attempt to quantify the span of possibilities resulting from an initial decision du and thus measure the ability of the decision- maker to opt for radically different strategies at a date beyond the first decision point. In a sense it can be argued that the decision-maker is covering himself against future changes in the trade offs by using (c) and (d) rather than (a) and (b). In any case these four measures are not mutually exclusive and could well be used simultaneously in any planning exercise.

(2) Uncertainty about future states and responses If the planner or decision-maker is uncertain about the future states of the

environment (or the way in which the system will respond to such states), the planning process becomes considerably more complex. As we have seen the 'classical' methods [7] attempt to build up a probabilistic picture of the future in order to provide the planner and decision-maker with estimates of mean expected returns from alternative strategies, standard deviations, measures of risk etc. If we assume that the planner or decision-maker is unwilling or unable to assign probabilities or utilities to uncertain events in the future, such methods become inapplicable.

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One approach to planning under uncertainty about future events is that of planning with respect to a series of alternative scenarios. It is tacitly assumed that a representative (and relatively small) number of scenarios can be selected so as to represent all possible developments in the future---and all plans are reviewed with respect to each of these scenarios in turn. In such planning exercise, robustness can be used in two complementary ways,

(i) to measure the robustness of a decision over time with respect to one particular scenario,

(ii) to measure the robustness of a decision at a particular point of time over all possible scenarios.

Using appropriate measures of robustness, the planner can evaluate plans according to the degree to which they leave good options open for future acceptance--both over time and over the scenario set. Unfortunately, the actual planning exercise itself is necessarily lengthened by the need to review plans under different basic assumptions. We cannot even reduce the number of plans to be considered by using the efficiency arguments used in the certainty case. Where the future is uncertain the planner is not justified in concentrating attention on plans which appear to be efficient (at present) nor can he amalgamate or disregard strategies which have the same initial decision and which lead to the same ultimate performance via different paths. The reasons are as follows:

(a) Since the economic climate that eventually rules is never exactly the one used at the time of planning and at the time when the initial decision is made, strategies which initially appear to be efficient (ex-ante) may well turn out not to be so. On the other hand, strategies which were not efficient (ex-ante) may well turn out to be efficient (expost). The planner may commit errors of type A by not including strategies which are good (ex-ante) even though they are not efficient (ex-ante). In order to avoid such errors it is necessary to broaden the scope of the planning exercise to include "good" solutions and not just efficient solutions;

(b) Given that the decision-maker is capable of defining bounds that will define satisfactory (acceptable or good) solutions, there remains a temptation to classify solutions as good (those which just satisfy the relevant criteria) and very good (those which exceed all lower bounds or lie well within the region mapped out by the decision-maker). It is natural, but not necessarily correct, to make the assertion that in the face of uncertainty about the environment, solutions which are very good in the initial analysis will at least turn out to be good (and acceptable) whereas solutions which are just acceptable based on present calculations may not remain so in the light of future development. In view of this change of attitude on the part of the planner (and decision-

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maker) there may be a call to include some weighting of solutions in the numerical measures of robustness. Sensitivity analysis might also be used to estimate the weight that might be attached to a particular solution [3];

(c) The planner cannot use the discernment argument to reduce the number of strategies included in the robustness measure, since this only applies to uncertainty about objectives. Where the future state of the environment is not known with certainty, paths which could have been treated as leading to the same final result must now be treated as quite distinct responses to future events.

In the light of these comments, it is obvious that in the case of uncertainty about the environment, the planner is forced to include consideration of more strategies than in the deterministic cases and is therefore subject to a greater number of errors of both types.

Measure of robustness in Case 2 In order to take account of the points mentioned above, the use of robustness

in situations of uncertaihty can be amended as follows:

(a) we can include consideration of good solutions as well as efficient solutions; and

(b) we can utilize robustness measures in two dimensions--within each particular scenario, and across the various possible scenarios.

As in the deterministic case there is no unique measure of robustness; the robustness of a decision can be measured by:

(i) as in equation (1), but here n(,~i) is the number of good solutions that can be reached by implementing decision d, either within or across scenarios, and n(~) is the total number of good solutions for the entire problem;

(ii) as in equation (2);

(iii) Rt a measure of the span of the good solutions---either by scatter or range.

Where the future environment is uncertain, both the planner and decision- maker must take adverse results into account. For each strategy there is a chance that it might eventually lead to an irretrievable loss or an irreparable disaster, and strategies which have a high 'chance' of leading to such a loss are to be avoided if at all possible. In order to cope with this aspect of uncertainty, strategies might be analyzed according to the opposite of robustnessqwhich we will refer to as 'debility'. A strategy with a high measure of debility will be defined as one which leads to many bad options (as opposed to robustness which leads to many good ones). Apart from the measure of "the number of bad

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solutions associated with a particular decision" we can suggest at least two other measures of debility:

(a) e, ffi - - ~ ' the proportion of all solutions which are defined as ' bad ' ,

•ffi b($t) the proportion of solutions derived from decision dt which are defined as

'bad'.

where: b(~t) is the number of bad solutions deriving from decision dr, n(S) is the total number of strategies, and n(S~) is the total number of strategies after decision d, has been taken.

Both (a) and (b) measure the numerical debility that can be associated with a particular decision, (a) in an a priori sense, (b) in an a posteriori sense. Any decision with a high measure of debility should be viewed with extreme caution. As with robustness there may be a case for weighting these measures, and they should definitely be considered both within and across scenarios.

(3) Uncertainty about the model space In any exercise of planning or project evaluation there is often considerable

arbitrariness in the way in which the actual model space is formulated or defined. It is always assumed that the planner is sufficiently skilful to be able to define a space which adequately represents the real world---and can thus draw conclu- sions about the real world from the model or abstraction. At best these assumptions are tested by an analysis of the sensitivity of the solutions.

In large-scale problems there is a further difficulty. It may be impossible to undertake a thorough study of the entire model space--for either physical or economic reasons. In this context, robustness may be a useful method of reducing the size of the model space without discarding too many valuable areas of possible study.

For example, the problem described above is a standard dynamic programming problem. As is well known, it is difficult to apply dynamic programming to large scale problems because of the vast computing requirement of the algorithm (even in the case of a single objective function). One method of reducing the size of the network that must be analyzed might be to use Monte Carlo methods to obtain initial estimates of the robustness of each initial decision (or decision element). Only such decisions (or decision elements) which show an initially high robustness would be included in the second stage--i.e, the detailed dynamic programming analysis. The methods would have to be applied over all the envisaged scenarios.

Clearly such two stage methods are likely to lead to an increase in both types of planning error--since the initial inclusion or exclusion of strategies is made

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on a very general basis--but by using appropriate measures of robustness and debility, it should be possible to reduce the size of the planning and decision making problem to manageable yet meaningful proportions.

CONCLUSION

The multidimensional scenario approach to planning suggested in this paper is heuristic and academically unconventionalnand far removed from the classical elegance of decision theory. It requires that the planner and decision- maker confront the uncertain environment with a series of robustness measures: it increases the emphasis placed on the initial decision; it maintains the variety and multidimensional nature of the decision-making problem right up to the point at which decisions are actually made, and utilizes a much more flexible approach than that used in the maximization of expected future utility. As a result, the approach may be much closer to the needs of decision-makers in government and industry. It remains to be seen whether decision-makers can benefit in practice from such an open-ended, lateral approach to forward planning.

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