- Email: [email protected]
Generating alternative solutions for dynamic programming-based planning problems
Socto-Econ. Plunn 9;. Vol. 24. No. I. pp. 27-Y. Prmtcd III Greal Bntain. All nghts reserved
1990
0038-0121 90 43.00 + 0.00 Copyright C 1990 Pcrgamon Press plc
Generating Alternative Solutions for Dynamic Programming-Based Planning Problems BRIAN W. BAETZ Department
of Civil Engineering and Engineering Mechanics, Hamilton, Ontario, Canada LgS 4L8
McMaster
University.
University,
Durham.
ERIC I. PAS Department
of Civil and Environmental Engineering, NC 27706. U.S.A.
Duke
ALAN W. NEEBE Graduate
School of Business Administration. The University NC 27599. U.S.A.
(Received
Ocroher
I989
;
of North
received fiw publication
Junuary
Carolina,
Chapel
Hill.
1990)
Abstract-An approach is developed for generating alternative near-optimal solutions for dynamic programming-based planning problems. The proposed methodology improves on an existing approach in three respects. First, computational efficiencies are achieved by recomputing the optimal policy for only a subset of the total number of stages. Second, the approach can be easily structured such that there is increased variation in alternatives within the first stages. Third, the maximum allowable difference in objective function value between alternative solutions can be explicitly stated by the user. A hypothetical shortest route problem is used to illustrate the characteristics of the proposed approach. Results are also presented for a facility capacity planning problem in the municipal solid waste managemenl area.
INTRODUCTION The planning and development of systems to fulfill the wide range of human needs is often subject to a variety of objectives. Multiobjective programming is a general technique for optimizing development activities subject to a number of different objectives and associated constraints (21. An important limiting requirement for successful multiobjective analysis is the proper quantification of decision-maker preferences for various and often conflicting objectives. For complex planning problems, this requirement is often difficult to achieve. Another approach, the modeling to generate alternatives (MGA) approach, recognizes the difficulty in developing estimates of decision-maker preferences. MGA attempts to provide an optimal solution and several near-optimal alternative solutions to a planning problem, subject to a single objective. Preferably, the alternative solutions are close to the optimal solution with respect to objective function value, but considerably different from the optimal solution in terms of system characteristics (e.g. facility location, technology, capacity and utilization). A decision-maker can then review the optimal and near-optimal alternatives and internalize the tradeoffs between the differences in the objective function value and the differing system characteristics. Previous applications of the MGA approach have been in the areas of water resources, land use planning, wastewater treatment, and agricultural economics [3]. Another fertile application area involves municipal solid waste management. Here, high development costs and increasing public 27
BRIAN W. BAETZ e! al.
28
demand for sound siting and operating policies emphasize the desirability of generating alternative solutions for decision-maker consideration. The need for alternative solutions is underlined by the presence of uncertainty and many non-quanti~able cost factors relating to monitoring and site remediation, and the possibility that public opposition may eliminate the optima1 solution from further consideration. In this paper, we develop an approach that improves upon an existing dynamic programmingbased technique for the generation of alternative near-optimal solutions. The structure of the approach is discussed, and improvements relative to the existing approach are highlighted, using a hypothetical problem. The MGA approach is applied to the capacity planning of municipal solid waste management facilities, a public sector planning problem of significant concern to many North American and European municipalities.
MODELING
APPROACH
An MGA approach for planning problems using dynamic programming as the solution technique is outlined in this section. The stages of the dynamic program are assumed to represent the time intervals of a planning horizon, but the approach is equally applicable to problems having stages that are not time-related. The state space of the dynamic program represents some selected aspect of the system at a particular point in the time horizon. For example, capacity planning for a municipal solid waste management system (incorporating waste-to-energy and landfitl facilities) can be modeled using dynamic programming with a muIti-dimensional state space representing the facifity capacities throughout the planning period [I]. Our proposed solution generation algorithm is shown in Fig. 1. Initially, the dynamic program generates an optimal solution for the entire time horizon using a backwards recursion. Thus, a sequence of dynamic programming tables is calculated, starting at the end of the time horizon and working backwards to the first stage, Then, the optimal solution is traced, starting at the first stage and working through the state space to the end. To generate the first alternative solution, the user identifies some penalty stage i and some cost penalty P to be added to the total cost for the optimal state at stage i. If P is very large, the etfcct is to rule out from further consideration all solutions which pass through this currently optimal state at stage i. The dynamic programming tables then need to be recalculated only for stages I through (i - I). The alternative solution is obtained by tracing through the new set of tables. Note that this alternative solution will have an objective function value within P of the previously optimal value. If no such alternative solution exists, the procedure would then return with the same solution. Finally, should another alternative solution be desired, the user would select a new penalty cost, P, add it to the total cost for the optimal states at stage i for all previously generated solutions, and the process would repeat.
COMPARISON
OF PROPOSED APPROACH TO EXISTING GENERATION APPROACH
ALTERNATIVE
The proposed approach contains improvements over a technique to generate alternative solutions utilizing dynamic programming originally proposed by Hopkins et al, [4]. The existing approach applies a penalty cost to all active links of the optimal solution, and then resolves the entire dynamic program with the transformed set of link costs. The proposed alternative approach contains improvements which relate to computational efficiency, variation of alternatives in the near term, and the explicit establishment of cost differences between alternative solutions. These features are discussed below, and are illustrated with respect to the hypothetical shortest route problem shown in Fig. 2. The features of the approach are described for this example problem without loss of generafity, since any dynamic programming problem may be viewed as a shortest route problem in an acyclic network. Link costs and minimum costs to proceed from a particular point to the end of the time horizon are shown, as is the optimal path with an objective function value of 105 units. Three iterations of the alternative generation approach are presented in Fig. 2. At each iteration, a P value of IO units was added to the total cost of the optimal states at penalty stage t = 2 for all previously generated solutions.
Generating alternative soiutions for dynamic programming-baa
It
Et(o8i
problems
29
tw the
cost fur 5ptfnmf St8ta fft
Spacu
Pensttg
lssef Stags f
(and aff stale
spsce
Penalty stage t and Penalty Cast P me User-Specffferi Fig. 1. Flow chart of solution generation approach.
The benefits of the proposed approach
Cimpar uliod
are as follows.
eficieff cy
instead of recalculating the optimal policy for the entire time horizon, as is done in the approach of [4]. the procedure shown in Fig. 1 recalculates the optimat policy only for the first stage up to the penalty stage. Depending upon the range of feasible state space points at each stage, this win generally result in ~ropo~iona~ computational savings For each iteration. As shown in Fig, 2, after the penalty cost is added to the optimal state level at the penalty stage in any iteration, only five nodes are considered to determine the near-optimal alternative solution, This five node requi~ment compares with thirteen nodes for solving the entire time horizon, yieiding an approximately 60% savings in computation. The corresponding financial savings could be appreciable for a real-life problem,
Intuitively, it may appear that the computational efficiency is achieved at the expense of reduced variation in the alternative generated solutions. Actually, the opposite is true, since the establishment of the penalty stage near the start of the time horizon for a backwards recursion will cause increased variation in the alternative solutions in the earlier time periods. This is often preferred by a decision-maker who is most interested in alternative strategies for near-term consideration and development, white evaIuating the tong-term consequences of the near-term options.
30
Generating alternative solutions for dynamic programming-based
problems
31
Variation of alternatives in the near-term may also be experienced with the existing dynamic programming-based approach [4], but not in all cases. Re-optimization with penalty costs added to all of the active links may lead to an alternative solution that varies from the previous optimal solution only in terms of the links in later time periods. For systems with fixed starting conditions, the penalty stage for the proposed approach must be set at least one stage away from the start of the time horizon. For systems where the starting condition is a decision variable, the penalty stage may be at the start of the time horizon or at any later stage. For the example problem, the penalty stage was set at the halfway mark of the time horizon, with the iterations yielding a range of alternative solutions that differ considerably in their initial characteristics. One potential disadvantage of the proposed approach is that it eliminates from consideration all solutions that are at the optimal state level of the penalty stage in previously generated solutions. Alternatives with considerable variation from previously generated solutions may actually pass through the same state at the penalty stage. The likelihood of this occurrence diminishes as the user-specified penalty stage is established nearer the start of the time horizon, since the paths of the alternative solutions will then have fewer opportunities to diverge from previously generated paths and then rejoin these paths at the penalty stage. The variation in alternative solutions in the first stages may be less critical for dynamic programming problems with stages which are not time-oriented. However, variation in the decision variables associated with the first stage up to the penalty stage will be obtained with the technique; these decision variables should be appropriately sequenced in order to generate differences in those choices where significant variation is desired. Establishment
of objective jiinction
value differences between solutions
The following discussion relates to the optimal solution and the first near-optimal alternative solution to be generated, but applies equally to any later iterations where an alternative solution has been generated and another alternative solution is about to be generated. Hopkins et al. [4] do not allow for the direct specification of the maximum objective value difference between the optimal solution and the alternative solution to be generated. This is due to the fact that penalties are attached to the optimal activity links, and it is thus not possible to determine which of these links will be selected for the alternative solution. If, as in our approach, a penalty P is attached to the optimal state space level at the penalty stage, the alternative solution will have an objective value within P of the optimal solution, or the alternative solution will be the same as the optimal solution (with the objective function value equal to the optimal value plus P). For example, in any of the iterations shown in Fig. 2, a penalty cost of less than five units would have resulted in the alternative solution being identical to the previously generated solution (since no alternative solution exists inside a five unit range of the previous solution for any iteration). The ability to set the maximum objective value difference can be used to terminate the alternative solution generation process if the alternative solution is equal to the previously generated solution. The penalty can be made dependent upon the number of alternative solutions previously generated. For example, one logical approach is to let P decrease as further alternative solutions are generated. Thus, after generating a number of alternative solutions, the objective value difference for remaining solutions would be made smaller, thereby decreasing the likelihood that a different alternative solution would be found within the adjusted penalty. The advantages of the proposed approach are thus its computational efficiency, the variation of alternatives in the early stages, and the capability of establishing an upper bound on the objective value difference between two successively generated alternatives. These advantages have been illustrated through the hypothetical shortest route problem, and may be similarly experienced in problems of greater size and increasing complexity. It is noteworthy that both the proposed and existing approaches are heuristics in that they do not guarantee the actual “second best” solution. Their purpose is, instead, to generate, in a computationally efficient manner, solutions that are close to the optimal in terms of objective function value but which are substantially different in terms of system characteristics.
BiuN W. EAETZet ai.
32
APPLICATION
OF THE GENERATION
OF ALTERNATIVES
APPROACH
We now apply the MGA approach shown in Fig. 1 to the facility capacity planning problem [l) within the area of municipal solid waste management, The purpose is to illustrate the approach’s potential for dealing with real-life problems. The stages here correspond to time intervals. while the state space represents the capacities of landfill and waste-to-energy facilities. The objective is to minimize all facility development costs plus operating costs less any energy revenues (all appropriately discounted). Decision variables at the start of each time period are the expansion capacities for both facility types. The optimal allocation of waste at each stage is determined through an embedded trans~rtation problem within the dynamic program. Parameters for typical solid waste management conditions were input into the optimization program, and an optimal solution and alternative near-optimal solutions were generated. The penalty cost was set to a very large value, and the penalty stage was set at i = 2. Thus, the optimal policies were recalculated for the first two stages (6 years) of a 30 year planning horizon. The time requirements for the optimization runs were as shown in Table 1. Each alternative solution was generated in approximately 15% of the original optimum solution time for this particular application. The resulting alternative expansion patterns for the waste-to-energy facility and landfill facility are shown in Fig. 3. These optima1 and near-optimal alternatives are characterized in Table 2 in terms of total discounted cost, front-end financing requirements, excess waste-to-energy capacity, and landfill capacity requirements. For a more general problem, regulatory r~uirements, demand center proximity, and technological reliability may also be important elements to be considered in examining tradeoffs between alternatives. Excess waste-to-energy capacity is a positive characteristic since excess capacity is a marketable commodity to outside municipalities, while landfill capacity requirement is a negative characteristic due to the potential impacts of landfill development upon the natural and social environments. In terms of total discounted cost, the optimal solution has the lowest front-end ~nancing levels, but has low excess waste-to-energy capacity and the highest landfill capacity requirements. The first near-optimal alternative has higher cost and increased financing levels, with lower excess waste-to-energy capacity and similar landfill development. The third and fourth alternatives also have higher cost and financing levels, but offer greater excess waste-to-energy capacity while both have decreased landfill development levels. Assuming that financial constraints are not limiting, the third near-optimal alternative is attractive from the standpoint of excess waste-to-energy capacity and landfill requirements, with a total cost within 12.5% of the optimal solution. The information summarized in Table 2 would allow decision-makers to evaluate and utilize tradeoffs between alternative options to meet future waste disposal demand on the basis of overall cost, financing requirements, waste-to-energy capacity levels, and landfitl siting requirements.
SUMMARY An approach has been developed to generate alternative near-optimal solutions to planning problems in which dynamic programming is used as the solution method. The proposed technique improves on an existing approach to generate alternatives utilizing dynamic programming with respect to computational efficiency, variation of alternatives in the near-term, and the control of the maximum objective value differential between alternative solutions. The technique can be used to efficiently generate near-optima1 alternatives that differ considerably from the optimal solution Table
i.
Computationa requirements for mukiplc soiutions
Computation time (CPU seconds: IBM 3081) :I: 64 71
Number of rolutions generated Optimal Optimal Optimal Optimal
only + I alternative + 2 akmativrs + 3 altcmative
Generating
alternative
solutions for dynamic
programming-based
problems
* /,////,/ : / /
///
// ///
llu
i-
Demand Optlmsl Pettern
0
3
6
i-o
l-l
l-2
L
Pattern
#I
Near-Optlmal
Pattern
#2
Near-Optlmal
Pattern
03
i-4
l-3
penalty
Near-Optimal
i-5
JO
TIME (years)
i-6
STlEES
stage (a)
LEGEND
0
3
6
-
Optlmal Pattern
WH n n
Near-Optimal
Pattern
*I
0 0 0 0
Near-Optlmal
Pattern
82
NH R/Y
Near-Optlmal
Pattern
#3
II
16
23
30
TIME (years)
l-4
l-5
l-6
STACES
W Fig. 3. (a) Alternative solutions for waste management capacity capacity. (b) Alternative solutions for waste management capacity depletion patterns.
planning planning
example: waste-to-energy example: landfill capacity
33
BRIANW. BAETZer al.
34
Table 2. Characteristics of optimal and near-optimal alternatives for waste management applicatmn Pattern Consideration Total discounted cost (MS) Front-end financing requirements (MS) Capability to provide waste-to-energy capacity to other regions (excess capacity in 1000 ton units) Landfill capacity requirements (acres)
Opttmal 70.2 49 Low (350) 220
Near-optimal 72.7 61 Minimal (70) 210
I
Near-optimal 2
Near optimal 3
74. I 69 Low to moderate (470) 135
79.0 80 High (1350) 130
in their near-term characteristics. Such alternative solutions may be of great interest to decisionmakers faced with difficult and controversial choices. Acknowledgemenrs-This vork has been supported by the Natural Sciences and Engineering Research Council of Canada. the Department of Civil and Environmental Engineering. Duke University, and the Business Foundation of North Carolina.
REFERENCES I. B. W. Bactz (1988) Capacity planning for production facilities and alternative facilities with consumable capacity: application to waste management systems. Unpublished Ph.D. dissertation, Department of Civil and Environmental Engineering. Duke University, Durham. N.C. 2. J. L. Cohort. Mulriobjective Programming and Planning. Academic Press, New York (1978). 3. J. S. Gidley and M. F. Bari. Modelling to generate alternatives. Am. Sot. Cit. Engrs Warer Forum ‘86, 1366-1373 (1986). 4. L. D. Hopkins, E. D. Brill and B. Wong. Generating alternative solutions for dynamic programming models of water resources problems. War. Resow. Res. 18(4), 782-790 (1982).
1990
0038-0121 90 43.00 + 0.00 Copyright C 1990 Pcrgamon Press plc
Generating Alternative Solutions for Dynamic Programming-Based Planning Problems BRIAN W. BAETZ Department
of Civil Engineering and Engineering Mechanics, Hamilton, Ontario, Canada LgS 4L8
McMaster
University.
University,
Durham.
ERIC I. PAS Department
of Civil and Environmental Engineering, NC 27706. U.S.A.
Duke
ALAN W. NEEBE Graduate
School of Business Administration. The University NC 27599. U.S.A.
(Received
Ocroher
I989
;
of North
received fiw publication
Junuary
Carolina,
Chapel
Hill.
1990)
Abstract-An approach is developed for generating alternative near-optimal solutions for dynamic programming-based planning problems. The proposed methodology improves on an existing approach in three respects. First, computational efficiencies are achieved by recomputing the optimal policy for only a subset of the total number of stages. Second, the approach can be easily structured such that there is increased variation in alternatives within the first stages. Third, the maximum allowable difference in objective function value between alternative solutions can be explicitly stated by the user. A hypothetical shortest route problem is used to illustrate the characteristics of the proposed approach. Results are also presented for a facility capacity planning problem in the municipal solid waste managemenl area.
INTRODUCTION The planning and development of systems to fulfill the wide range of human needs is often subject to a variety of objectives. Multiobjective programming is a general technique for optimizing development activities subject to a number of different objectives and associated constraints (21. An important limiting requirement for successful multiobjective analysis is the proper quantification of decision-maker preferences for various and often conflicting objectives. For complex planning problems, this requirement is often difficult to achieve. Another approach, the modeling to generate alternatives (MGA) approach, recognizes the difficulty in developing estimates of decision-maker preferences. MGA attempts to provide an optimal solution and several near-optimal alternative solutions to a planning problem, subject to a single objective. Preferably, the alternative solutions are close to the optimal solution with respect to objective function value, but considerably different from the optimal solution in terms of system characteristics (e.g. facility location, technology, capacity and utilization). A decision-maker can then review the optimal and near-optimal alternatives and internalize the tradeoffs between the differences in the objective function value and the differing system characteristics. Previous applications of the MGA approach have been in the areas of water resources, land use planning, wastewater treatment, and agricultural economics [3]. Another fertile application area involves municipal solid waste management. Here, high development costs and increasing public 27
BRIAN W. BAETZ e! al.
28
demand for sound siting and operating policies emphasize the desirability of generating alternative solutions for decision-maker consideration. The need for alternative solutions is underlined by the presence of uncertainty and many non-quanti~able cost factors relating to monitoring and site remediation, and the possibility that public opposition may eliminate the optima1 solution from further consideration. In this paper, we develop an approach that improves upon an existing dynamic programmingbased technique for the generation of alternative near-optimal solutions. The structure of the approach is discussed, and improvements relative to the existing approach are highlighted, using a hypothetical problem. The MGA approach is applied to the capacity planning of municipal solid waste management facilities, a public sector planning problem of significant concern to many North American and European municipalities.
MODELING
APPROACH
An MGA approach for planning problems using dynamic programming as the solution technique is outlined in this section. The stages of the dynamic program are assumed to represent the time intervals of a planning horizon, but the approach is equally applicable to problems having stages that are not time-related. The state space of the dynamic program represents some selected aspect of the system at a particular point in the time horizon. For example, capacity planning for a municipal solid waste management system (incorporating waste-to-energy and landfitl facilities) can be modeled using dynamic programming with a muIti-dimensional state space representing the facifity capacities throughout the planning period [I]. Our proposed solution generation algorithm is shown in Fig. 1. Initially, the dynamic program generates an optimal solution for the entire time horizon using a backwards recursion. Thus, a sequence of dynamic programming tables is calculated, starting at the end of the time horizon and working backwards to the first stage, Then, the optimal solution is traced, starting at the first stage and working through the state space to the end. To generate the first alternative solution, the user identifies some penalty stage i and some cost penalty P to be added to the total cost for the optimal state at stage i. If P is very large, the etfcct is to rule out from further consideration all solutions which pass through this currently optimal state at stage i. The dynamic programming tables then need to be recalculated only for stages I through (i - I). The alternative solution is obtained by tracing through the new set of tables. Note that this alternative solution will have an objective function value within P of the previously optimal value. If no such alternative solution exists, the procedure would then return with the same solution. Finally, should another alternative solution be desired, the user would select a new penalty cost, P, add it to the total cost for the optimal states at stage i for all previously generated solutions, and the process would repeat.
COMPARISON
OF PROPOSED APPROACH TO EXISTING GENERATION APPROACH
ALTERNATIVE
The proposed approach contains improvements over a technique to generate alternative solutions utilizing dynamic programming originally proposed by Hopkins et al, [4]. The existing approach applies a penalty cost to all active links of the optimal solution, and then resolves the entire dynamic program with the transformed set of link costs. The proposed alternative approach contains improvements which relate to computational efficiency, variation of alternatives in the near term, and the explicit establishment of cost differences between alternative solutions. These features are discussed below, and are illustrated with respect to the hypothetical shortest route problem shown in Fig. 2. The features of the approach are described for this example problem without loss of generafity, since any dynamic programming problem may be viewed as a shortest route problem in an acyclic network. Link costs and minimum costs to proceed from a particular point to the end of the time horizon are shown, as is the optimal path with an objective function value of 105 units. Three iterations of the alternative generation approach are presented in Fig. 2. At each iteration, a P value of IO units was added to the total cost of the optimal states at penalty stage t = 2 for all previously generated solutions.
Generating alternative soiutions for dynamic programming-baa
It
Et(o8i
problems
29
tw the
cost fur 5ptfnmf St8ta fft
Spacu
Pensttg
lssef Stags f
(and aff stale
spsce
Penalty stage t and Penalty Cast P me User-Specffferi Fig. 1. Flow chart of solution generation approach.
The benefits of the proposed approach
Cimpar uliod
are as follows.
eficieff cy
instead of recalculating the optimal policy for the entire time horizon, as is done in the approach of [4]. the procedure shown in Fig. 1 recalculates the optimat policy only for the first stage up to the penalty stage. Depending upon the range of feasible state space points at each stage, this win generally result in ~ropo~iona~ computational savings For each iteration. As shown in Fig, 2, after the penalty cost is added to the optimal state level at the penalty stage in any iteration, only five nodes are considered to determine the near-optimal alternative solution, This five node requi~ment compares with thirteen nodes for solving the entire time horizon, yieiding an approximately 60% savings in computation. The corresponding financial savings could be appreciable for a real-life problem,
Intuitively, it may appear that the computational efficiency is achieved at the expense of reduced variation in the alternative generated solutions. Actually, the opposite is true, since the establishment of the penalty stage near the start of the time horizon for a backwards recursion will cause increased variation in the alternative solutions in the earlier time periods. This is often preferred by a decision-maker who is most interested in alternative strategies for near-term consideration and development, white evaIuating the tong-term consequences of the near-term options.
30
Generating alternative solutions for dynamic programming-based
problems
31
Variation of alternatives in the near-term may also be experienced with the existing dynamic programming-based approach [4], but not in all cases. Re-optimization with penalty costs added to all of the active links may lead to an alternative solution that varies from the previous optimal solution only in terms of the links in later time periods. For systems with fixed starting conditions, the penalty stage for the proposed approach must be set at least one stage away from the start of the time horizon. For systems where the starting condition is a decision variable, the penalty stage may be at the start of the time horizon or at any later stage. For the example problem, the penalty stage was set at the halfway mark of the time horizon, with the iterations yielding a range of alternative solutions that differ considerably in their initial characteristics. One potential disadvantage of the proposed approach is that it eliminates from consideration all solutions that are at the optimal state level of the penalty stage in previously generated solutions. Alternatives with considerable variation from previously generated solutions may actually pass through the same state at the penalty stage. The likelihood of this occurrence diminishes as the user-specified penalty stage is established nearer the start of the time horizon, since the paths of the alternative solutions will then have fewer opportunities to diverge from previously generated paths and then rejoin these paths at the penalty stage. The variation in alternative solutions in the first stages may be less critical for dynamic programming problems with stages which are not time-oriented. However, variation in the decision variables associated with the first stage up to the penalty stage will be obtained with the technique; these decision variables should be appropriately sequenced in order to generate differences in those choices where significant variation is desired. Establishment
of objective jiinction
value differences between solutions
The following discussion relates to the optimal solution and the first near-optimal alternative solution to be generated, but applies equally to any later iterations where an alternative solution has been generated and another alternative solution is about to be generated. Hopkins et al. [4] do not allow for the direct specification of the maximum objective value difference between the optimal solution and the alternative solution to be generated. This is due to the fact that penalties are attached to the optimal activity links, and it is thus not possible to determine which of these links will be selected for the alternative solution. If, as in our approach, a penalty P is attached to the optimal state space level at the penalty stage, the alternative solution will have an objective value within P of the optimal solution, or the alternative solution will be the same as the optimal solution (with the objective function value equal to the optimal value plus P). For example, in any of the iterations shown in Fig. 2, a penalty cost of less than five units would have resulted in the alternative solution being identical to the previously generated solution (since no alternative solution exists inside a five unit range of the previous solution for any iteration). The ability to set the maximum objective value difference can be used to terminate the alternative solution generation process if the alternative solution is equal to the previously generated solution. The penalty can be made dependent upon the number of alternative solutions previously generated. For example, one logical approach is to let P decrease as further alternative solutions are generated. Thus, after generating a number of alternative solutions, the objective value difference for remaining solutions would be made smaller, thereby decreasing the likelihood that a different alternative solution would be found within the adjusted penalty. The advantages of the proposed approach are thus its computational efficiency, the variation of alternatives in the early stages, and the capability of establishing an upper bound on the objective value difference between two successively generated alternatives. These advantages have been illustrated through the hypothetical shortest route problem, and may be similarly experienced in problems of greater size and increasing complexity. It is noteworthy that both the proposed and existing approaches are heuristics in that they do not guarantee the actual “second best” solution. Their purpose is, instead, to generate, in a computationally efficient manner, solutions that are close to the optimal in terms of objective function value but which are substantially different in terms of system characteristics.
BiuN W. EAETZet ai.
32
APPLICATION
OF THE GENERATION
OF ALTERNATIVES
APPROACH
We now apply the MGA approach shown in Fig. 1 to the facility capacity planning problem [l) within the area of municipal solid waste management, The purpose is to illustrate the approach’s potential for dealing with real-life problems. The stages here correspond to time intervals. while the state space represents the capacities of landfill and waste-to-energy facilities. The objective is to minimize all facility development costs plus operating costs less any energy revenues (all appropriately discounted). Decision variables at the start of each time period are the expansion capacities for both facility types. The optimal allocation of waste at each stage is determined through an embedded trans~rtation problem within the dynamic program. Parameters for typical solid waste management conditions were input into the optimization program, and an optimal solution and alternative near-optimal solutions were generated. The penalty cost was set to a very large value, and the penalty stage was set at i = 2. Thus, the optimal policies were recalculated for the first two stages (6 years) of a 30 year planning horizon. The time requirements for the optimization runs were as shown in Table 1. Each alternative solution was generated in approximately 15% of the original optimum solution time for this particular application. The resulting alternative expansion patterns for the waste-to-energy facility and landfill facility are shown in Fig. 3. These optima1 and near-optimal alternatives are characterized in Table 2 in terms of total discounted cost, front-end financing requirements, excess waste-to-energy capacity, and landfill capacity requirements. For a more general problem, regulatory r~uirements, demand center proximity, and technological reliability may also be important elements to be considered in examining tradeoffs between alternatives. Excess waste-to-energy capacity is a positive characteristic since excess capacity is a marketable commodity to outside municipalities, while landfill capacity requirement is a negative characteristic due to the potential impacts of landfill development upon the natural and social environments. In terms of total discounted cost, the optimal solution has the lowest front-end ~nancing levels, but has low excess waste-to-energy capacity and the highest landfill capacity requirements. The first near-optimal alternative has higher cost and increased financing levels, with lower excess waste-to-energy capacity and similar landfill development. The third and fourth alternatives also have higher cost and financing levels, but offer greater excess waste-to-energy capacity while both have decreased landfill development levels. Assuming that financial constraints are not limiting, the third near-optimal alternative is attractive from the standpoint of excess waste-to-energy capacity and landfill requirements, with a total cost within 12.5% of the optimal solution. The information summarized in Table 2 would allow decision-makers to evaluate and utilize tradeoffs between alternative options to meet future waste disposal demand on the basis of overall cost, financing requirements, waste-to-energy capacity levels, and landfitl siting requirements.
SUMMARY An approach has been developed to generate alternative near-optimal solutions to planning problems in which dynamic programming is used as the solution method. The proposed technique improves on an existing approach to generate alternatives utilizing dynamic programming with respect to computational efficiency, variation of alternatives in the near-term, and the control of the maximum objective value differential between alternative solutions. The technique can be used to efficiently generate near-optima1 alternatives that differ considerably from the optimal solution Table
i.
Computationa requirements for mukiplc soiutions
Computation time (CPU seconds: IBM 3081) :I: 64 71
Number of rolutions generated Optimal Optimal Optimal Optimal
only + I alternative + 2 akmativrs + 3 altcmative
Generating
alternative
solutions for dynamic
programming-based
problems
* /,////,/ : / /
///
// ///
llu
i-
Demand Optlmsl Pettern
0
3
6
i-o
l-l
l-2
L
Pattern
#I
Near-Optlmal
Pattern
#2
Near-Optlmal
Pattern
03
i-4
l-3
penalty
Near-Optimal
i-5
JO
TIME (years)
i-6
STlEES
stage (a)
LEGEND
0
3
6
-
Optlmal Pattern
WH n n
Near-Optimal
Pattern
*I
0 0 0 0
Near-Optlmal
Pattern
82
NH R/Y
Near-Optlmal
Pattern
#3
II
16
23
30
TIME (years)
l-4
l-5
l-6
STACES
W Fig. 3. (a) Alternative solutions for waste management capacity capacity. (b) Alternative solutions for waste management capacity depletion patterns.
planning planning
example: waste-to-energy example: landfill capacity
33
BRIANW. BAETZer al.
34
Table 2. Characteristics of optimal and near-optimal alternatives for waste management applicatmn Pattern Consideration Total discounted cost (MS) Front-end financing requirements (MS) Capability to provide waste-to-energy capacity to other regions (excess capacity in 1000 ton units) Landfill capacity requirements (acres)
Opttmal 70.2 49 Low (350) 220
Near-optimal 72.7 61 Minimal (70) 210
I
Near-optimal 2
Near optimal 3
74. I 69 Low to moderate (470) 135
79.0 80 High (1350) 130
in their near-term characteristics. Such alternative solutions may be of great interest to decisionmakers faced with difficult and controversial choices. Acknowledgemenrs-This vork has been supported by the Natural Sciences and Engineering Research Council of Canada. the Department of Civil and Environmental Engineering. Duke University, and the Business Foundation of North Carolina.
REFERENCES I. B. W. Bactz (1988) Capacity planning for production facilities and alternative facilities with consumable capacity: application to waste management systems. Unpublished Ph.D. dissertation, Department of Civil and Environmental Engineering. Duke University, Durham. N.C. 2. J. L. Cohort. Mulriobjective Programming and Planning. Academic Press, New York (1978). 3. J. S. Gidley and M. F. Bari. Modelling to generate alternatives. Am. Sot. Cit. Engrs Warer Forum ‘86, 1366-1373 (1986). 4. L. D. Hopkins, E. D. Brill and B. Wong. Generating alternative solutions for dynamic programming models of water resources problems. War. Resow. Res. 18(4), 782-790 (1982).