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2 44 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 44, NO. 1, FEBRUARY 1997 a numerical analysis issue. Thus, it does not raise the question of distorting a decision that involves continuous variables to fit (discrete) modeling tools. Similarly, with continuous decision variables, standard continuous mathematical optimization methods can be used to determine the optimum at each decision point in the model. As with continuous random variables, the solution procedure for continuous decision variables may require discretization, but this is once again only a solution issue, rather than being part of the problem s formulation. Furthermore, continuous mathematical optimization solution methods can be much faster than the exhaustive search approach that is often necessary when a discrete sequential decision model is solved. Hence, while this may seem somewhat paradoxical, it may be possible to solve larger decision models with continuous decision variables than with discrete decision variables. Suppose, for example, that a decision problem includes five continuous decision variables, and it is desired to determine optimal values for each of these accurate to within 5%. With a discrete approximation, this requires using 19 different levels for each variable, and hence different decision strategies must be analyzed. On the other hand, if the method of bisection [8] can be used with a continuous model to search for the optimal value of each continuous variable, it is only necessary to consider five levels for each variable to obtain a solution that is accurate to within 5%, or a total of different strategies, which is almost a factor of 800 fewer. The relative computational advantage of the method of bisection grows rapidly as the desired solution accuracy increases. II. BACKGROUND AND FORMULATION Our approach to formulating decisions with continuous variables is a generalization of Kirkwood s approach [16] and [17] to formulating decisions with discrete variables. With his approach, information about the path which leads from the root node of a sequential decision model (decision tree) to a particular interior node or endpoint is summarized in a state s. Each node or endpoint is identified by an unique node index i and a node type, which can be decision, chance, endpoint, or auxiliary. Each decision or chance node has branches, where is an integer, and each branch is identified by an unique integer branch index j, with. An auxiliary node is analogous to a deterministic node in an influence diagram and can be used to clarify the structure of a decision analysis model. (Auxiliary nodes are equivalent to decision or chance nodes with one branch.) Each node (but not endpoint) has a next node function, which yields a valid node index, and a branch value function. Each chance node has a branch probability function. Finally, each node has a node variable, which specifies each algebraic variable in the decision analysis model. With this notation, the expected utility [ ] for node given state is described by (see (1) at the bottom of the page) where is an utility function, and is the state formed by adding to the node variable and its associated value as determined by. For example, suppose that is {COST : 500}, where COST : 500 means that node variable COST has a value of 500. If the next node variable in the path is REVENUE and is 800, then this new information is summarized in as {REVENUE : 800, COST : 500}. For a decision node, a preferred alternative is a that maximizes. A number of other approaches have recently been developed for formulating and solving large-scale discrete sequential decision models [10], [26], [7], [25]. Kirkwood s approach emphasizes the use of algebraic variables and functions more strongly than these other approaches. Because of this emphasis on variables and functions, it is straightforward to generalize Kirkwood s approach to handle continuous decision and random variables. The formulation can be generalized to include decisions with continuous decision and random variables by recognizing that the index at each node in (1) needs to be converted to a continuous variable, which we will call. When this is done, (1) can be modified into (see (2) at the bottom of the page) where is the feasible region for decision variable, is the range of possible values for random variable, assigns a value to node variable, and is the cumulative distribution function for random variable given state. (Note that the function at endpoint nodes, and the and functions at auxiliary nodes, do not depend on.) As the notation indicates, all of these functions can depend on state. A preferred alternative at decision node is an that maximizes. decision chance endpoint auxiliary decision chance endpoint auxiliary (1) (2)

3 STONEBRAKER AND KIRKWOOD: FORMULATING AND SOLVING SEQUENTIAL DECISION ANALYSIS MODELS 45 Fig. 1. Influence diagram for the Synfuels decision problem. The discrete formulation in (1) can be viewed as a special case of (2) where consists of the integers 1, 2,, and the probability distribution function at chance node is discrete over the integers 1, 2,,. For many practical decisions, the feasible region will consist of a closed interval, where and will be called the lower bound function and upper bound function, respectively. For example, consider a three-period resource allocation problem where the budgets to be allocated for successive periods are,, and, respectively, subject to a total budget constraint. Then the feasible regions of the three decision variables are and therefore for (3) Fig. 2. Tree diagram for Cost However, before turning to that application, we consider the solution of (2). Also, for many practical decision problems with continuous decision variables, the node variable for each decision node will be equal to the decision variable, and thus when decision. The illustrative application below to R&D planning shows that a formulation in terms of (2) can be compact and, furthermore, that such a formulation can show the quantitative nature of the interrelationships among the decision and random variables in a decision more clearly than a discrete formulation. (4) III. SOLUTION APPROACHES Kirkwood [16] presents a solution procedure for (1), and he notes that the recursive nature of the equation (that is, the function EU is defined in terms of itself) leads to a particularly simple solution algorithm when programmed in a computer language which supports recursive function calls. While the solution algorithm presented in [16] is brute force in the sense that it traverses all paths through the decision tree corresponding to (1), empirical results show that it is sufficiently fast to realistically analyze discrete decision models with up to a few million endpoints using widely available personal computers.

4 46 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 44, NO. 1, FEBRUARY 1997 Fig. 5. Tree diagram for Foreign Fig. 3. Tree diagram for Cost Fig. 6. Tree diagram for Foreign Fig. 4. Tree diagram for Cartel 1985 and Cartel Developing an analogous solution procedure for (2) requires that an integration procedure be implemented for the chance nodes and that a maximization procedure be implemented for the decision nodes. The integration of probability density functions has been considered by Keefer [14], Smith [27], Merkhofer [21], Keefer and Bodily [15], and Miller and Rice [22] among others. Keefer [14] and Keefer and Bodily [15] show that in many cases accurate expected utilities and expected values can be obtained using a three-point discrete approximation consisting of the 0.05, 0.50, and 0.95 fractiles of the continuous distribution. Interested readers are referred to these papers for further details. The most common case in decision analysis practice is where the feasible region for each decision variable consists of an interval specified by lower and upper bound functions as discussed above. In general, the problem resulting from the analysis at each decision node is a nonlinear single-variable optimization problem subject to lower and upper bound constraints. In cases where there is a single local optimum at an interior point, the golden-section search algorithm [12] will efficiently determine the optimum. The Fibonacci search algorithm is also sometimes used. However, the golden-section search is often selected over the Fibonacci search because the total number of search intervals for the Fibonacci must be chosen in advance, whereas the golden-section search does not require this [8], which simplifies implementation. Other iterative approaches that use calculus, such as the Taylor series algorithm [8], are not practical to solve (2) because they require a closed-form specification of the objective function in order to differentiate it. In situations where there are multiple local optima or the optimum occurs at a boundary, it is always possible to use a grid search solution procedure where the feasible region is sectioned into a grid of equally spaced points, and these points are then exhaustively searched to find the optimal point [12]. By using a fine enough grid, any desired degree of accuracy

5 STONEBRAKER AND KIRKWOOD: FORMULATING AND SOLVING SEQUENTIAL DECISION ANALYSIS MODELS 47 Fig. 7. Foreign 1995 when Cartel 1995 is strong. can be obtained, but of course, considerable computation may be required if a fine grid is used. We present empirical results related to the use of the goldensection search and grid search in the context of an illustrative application below. See [29] for further details of these solution procedures. Note that the specific solution procedure used does not impact the decision problem formulation, as given in (2). That is, the formulation of the decision model can be done using the notation of (2) regardless of the specific procedure used to solve this equation. IV. ILLUSTRATIVE APPLICATION In his January 1975 State of the Union Address, President Ford called for the accelerated development of U.S. energy technology and resources so as to achieve a lowcost alternative to imported oil. The Interagency Task Force on Synthetic Fuel (Synfuel) Commercialization was formed to evaluate alternatives and make recommendations to the President. The Decision Analysis Group at Stanford Research Institute was selected to assist the Task Force [31], [30]. This Group developed a large-scale discrete sequential decision analysis model (a endpoint decision tree) to aid the Task Force. At the time of their modeling effort, no general purpose algebraic formulation method existed for decision analysis models, either discrete or continuous. Thus an ad hoc description was presented which covered 33 pages in an appendix to [30], including a four-page, step-by-step calculation of the net benefit for a sample path through the endpoint decision tree. A. Original Discrete Model Formulation The decision problem was formulated as a sequential resource allocation problem, with an initial decision in 1975 as to whether the United States should have a synfuels development program and, if so, at what funding. A subsequent decision in 1985 would determine if the current synfuels production capacity should be expanded and, if so, to what level of funding. These two decision variables were intrinsically continuous, as were a number of the random variables in the model (for example, the cost to produce the synfuel). However, following standard decision analysis practice, all of these continuous variables were approximated with discrete variables. Because of the complexity of the resulting discrete model, the analysis of the initial decision in the synfuels problem was limited to considering only four alternatives, namely: produce billion barrels of synfuel per year (Bbbl/yr) ( no program ), produce Bbbl/yr ( informational program ), produce Bbbl/yr ( medium program ), and produce Bbbl/yr ( maximum program ). The expansion decision in 1985 was approximated with six discrete levels, i.e., produce 0.000, 0.365, 0.730, 1.095, 1.460, and Bbbl/yr.

6 48 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 44, NO. 1, FEBRUARY 1997 TABLE I CAPACITY EXPANSION COST FACTORS [30] These decisions are represented in the Fig. 1 influence diagram as Program and Expand. The Program decision is made before the Expand decision, hence the arrow from the former to the latter. In addition to these two decisions, there are a number of uncertain variables. For example, there are four uncertainties associated with the year 1985, namely: cost, cartel, foreign, and embargo; and five uncertainties for 1995: cost, cartel, foreign, embargo, and demand (where demand represents the structure of the market demand, that is, a parameter value for the demand curve). (The years 1985 and 1995 were used to typify the decades of the 1980 s and 1990 s, respectively. Each uncertainty associated with the year 1985 will be resolved prior to the Expand decision.) Cost 1985 and Cost 1995 represent the uncertainty in the cost to produce synfuel in the years 1985 and 1995, respectively. While these are continuous variables, they are approximated as discrete variables. Cost 1985 is independent of Program, and the probability distribution assumed for Cost 1985 is shown in Fig. 2. Cost 1995 depends on Program, and the assumptions made about the dependence are shown in Fig. 3. As shown in Fig. 3, the cost to produce a barrel of synthetic fuel in 1995 decreases as the production of such fuel increases. Cartel 1985 and Cartel 1995 represent the state (strong or weak) of the OPEC cartel in 1985 and 1995, respectively. Cartel 1995 depends on the outcome of Cartel 1985 as shown in Fig. 4. For example, if the cartel is strong in 1985, then it is more likely that the cartel in 1995 will be strong than if the cartel is weak in Foreign 1985 and Foreign 1995 represent the price of foreign oil in 1985 and 1995, respectively. Foreign 1985 depends on the state of the cartel in 1985 as shown in Fig. 5. For example, if the cartel is strong in 1985, then the price of foreign oil in 1985 will be higher than with a weak cartel. Foreign 1995 depends on the state of the cartel in 1995 and the cost of synthetic fuel in 1995 as shown in Figs. 6 and 7. For example, if the cartel is strong in 1995, then the price of foreign oil in 1995 will be higher than if the cartel is weak. Equilibrium 1985 and Equilibrium 1995 are deterministic or auxiliary variables that represent the market equilibrium prices of oil for 1985 and 1995, respectively. As shown in Fig. 1, Equilibrium 1985 depends on Program and Foreign 1985; and Equilibrium 1995 depends on Program, Expand, Foreign Fig. 8. Tree diagram for Demand , and Demand As specified in [30], Equilibrium 1985 and Equilibrium 1995 are the minimum of the marketclearing price of synfuel and the price of foreign oil in the years 1985 and 1995, respectively. The market-clearing price is determined from the following demand curve: where is the price of synfuel, is the amount of synfuel demanded, and,, and are parameter values for the demand curve. The parameter values used for the 1985 demand curve are,, and. The parameter values used for the 1995 demand curve are and, whereas the parameter for the demand curve for 1995 (Demand 1995) is uncertain, and the assumed probability distribution is shown in Fig. 8. Embargo 1985 and Embargo 1995 represent the likelihood of an embargo and its associated loss in 1985 and 1995, respectively. The probability of an embargo was judged to be 0.10 in 1985 and 0.05 in The duration of an embargo was assumed to be five months. As derived in [30], the economic effect of an embargo is a loss of consumer surplus, as given by embargo loss (6) where is the amount of foreign oil and synfuel demanded at the market equilibrium price ; is the amount of synfuel demanded; and and are parameter values for the demand curve. (5)

7 STONEBRAKER AND KIRKWOOD: FORMULATING AND SOLVING SEQUENTIAL DECISION ANALYSIS MODELS 49 TABLE II CONTINUOUS-VARIABLE DECISION ANALYSIS MODEL FORMULATION OF THE SYNFUELS DECISION PROBLEM Consumer surplus is the difference between the value of a good (i.e., energy) to a consumer and the amount paid for it. Given a demand curve, such as (5), and a market equilibrium price, the consumer surplus is determined from the area below the demand curve and above the market equilibrium price. Consumer surplus, as derived in [30], is given by consumer surplus (7) where,, and are parameter values for the demand curve; and is the market equilibrium price. The supply curve used in [30] is the product of Cost 1995 and the appropriate capacity expansion cost factor shown in Table I. Producer surplus is the difference between the amount received by a producer for a good and the cost to produce it. Given a supply curve and a market equilibrium price, producer surplus is determined from the signed sum of the area above market equilibrium price and below the supply curve (thus including possible negative contributions to producer surplus), and the area below the market equilibrium price and above the supply curve (thus including possible positive contributions to producer surplus). The deterministic variable Cost Factor in Fig. 1 is a contribution to producer surplus and is determined by the area under the supply curve. As derived in [30], producer surplus for 1985 and 1995, respectively, are determined by and (8) and (9) at the bottom of the page. Consumer surplus, producer surplus, embargo loss, and environmental and socioeconomic cost specify the objective function, Net Benefit. The environmental and socioeconomic cost of producing synfuel is \$0.40 per barrel [30]. The net ben producer surplus (Equilibrium Cost 1985) Program (8) 1995 producer surplus (Equilibrium Cost 1995) Program Equilibrium 1995 (Program + Expand) Cost Factor (9)

8 50 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 44, NO. 1, FEBRUARY 1997 TABLE III MEAN AND STANDARD DEVIATION FUNCTIONS FOR CONTINUOUS RANDOM VARIABLES IN THE SYNFUELS DECISION MODEL TABLE IV PROBABILITY DISTRIBUTIONS FOR DISCRETE RANDOM VARIABLES IN THE SYNFUELS DECISION MODEL efit for 1985 is the sum of consumer surplus in 1985, producer surplus in 1985, embargo loss in 1985, and environmental and socioeconomic cost in Similarly, the net benefit for 1995 is the sum of consumer surplus in 1995, producer surplus in 1995, embargo loss in 1995, and environmental and socioeconomic cost in The total discounted net benefit is measured in constant 1975 dollars, as determined by multiplying the net benefits for 1985 and 1995 by the appropriate discount factors and summing. (A 10% discount rate is used in [30] and results in discount factors of 4.2 for 1985 and 1.62 for 1995.) B. Continuous Variable Model Formulation As mentioned above, several of the variables in the synfuels decision problem are intrinsically continuous. These include the decisions Program and Expand, as well as the random variables Cost 1985, Foreign 1985, Cost 1995, Foreign 1995, and Demand As shown above, the original formulation approximated the possible levels for these variables with a small number of discrete levels. To illustrate the usefulness of a continuous-variable decision analysis formulation, such a formulation is presented for the synfuels decision problem in Tables II V. Table II contains the basic structure of the decision model using the notation of (2). The two continuous decision variables Program and Expand have node indexes one and seven, and Table II shows that the feasible regions for these variables are specified by lower and upper bounds, which are simply fixed values. The continuous random variables Cost 1985, Foreign 1985, Cost 1995, Foreign 1995, and Demand 1995 have node indexes 2, 4, 8, 10, and 11, respectively. To illustrate the continuous formulation approach for these random variables, normal distributions specified by N[, ], where is the mean and is the standard deviation, are assumed for the five continuous random variables. (Since no continuous distributions were specified in the original analysis, these should be considered only illustrative.) The functional forms for the means and standard deviations of these normal distributions are specified in Table III. The remaining (discrete) random variables are Cartel 1985, Embargo 1985, Cartel 1995, and Embargo 1995, with node indexes 3, 6, 9, and 13, respectively. The probability distributions for these random variables are given in Table IV, and some of the entries in this table are further defined in Table V.

9 STONEBRAKER AND KIRKWOOD: FORMULATING AND SOLVING SEQUENTIAL DECISION ANALYSIS MODELS 51 TABLE V SUPPLEMENTAL FUNCTIONS IN THE SYNFUELS DECISION MODEL There are three auxiliary variables Equilibrium 1985, Equilibrium 1995, and Cost Factor, with node indexes 5, 12, and 14, respectively. The specific functions which determine the values of these auxiliary variables are given in Table V. Finally, node index 15 is the endpoint for the decision model, and the function which determines the endpoint values for the model is given in Table V. The formulation in Tables II V is significantly more compact than the original 33-page formulation shown in [30], but a greater advantage is that it shows the structure of the decision problem more clearly. We quickly see that the two decision variables can lie anywhere within a bounded interval, and the specific quantitative nature of the interdependencies among the decision and random variables is more apparent. For example, the mean and standard deviation for Cost 1995 both depend linearly on Program. In addition, Embargo 1985 is a function of Program and Equilibrium 1985, and Embargo 1995 is a function of Program, Expand, Demand 1995, and Equilibrium 1995, with the specific functional relationships shown in the table.

11 STONEBRAKER AND KIRKWOOD: FORMULATING AND SOLVING SEQUENTIAL DECISION ANALYSIS MODELS 53 Craig W. Kirkwood received the S.B., S.M., E.E., and Ph.D. degrees from the Massachusetts Institute of Technology, Cambridge, MA. He is currently a Professor at Arizona State University, Tempe, and has served on the faculties of the University of Colorado and the University of Michigan. He worked with the decision analysis group at Woodward Clyde Consultants, San Francisco. His research interests include large scale decision analysis models and decision analysis using partial or approximate information. Dr. Kirkwood is a member of the Institute for Operations Research and the Management Sciences and has served on the Advisory Panel for the National Science Foundation s Decision, Risk, and Management Science Program.

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