A grey hop, skip, and jump approach: generating ...

A grey hop, skip, and jump approach: generating alternatives for expansion planning of waste management facilities Can. J. Civ. Eng. Downloaded from www.nrcresearchpress.com by Renmin University of China on 06/04/13 For personal use only.

I

G.H. Huang, B.W. Baetz, and G.G. Patry

Abstract: A grey hop, skip, and jump (GHSJ) approach is developed and applied to the area of municipal solid waste management planning. The method improves upon existing modelling to generate alternative approaches by allowing uncertain information to be effectively communicated into the optimization process and resulting solutions. Feasible decision alternatives can be generated through interpretation of the GHSJ solutions, which are capable of reflecting potential system condition variations caused by the existence of input uncertainties. Results from a hypothetical case study indicate that useful solutions for the expansion planning of waste management facilities can be generated. The decision alternatives obtained from the GHSJ solutions may be interpreted and analyzed to internalize environmental-economic tradeoffs, which may be of interest to solid waste decision makers faced with difficult and controversial choices. Key words: grey programming, modelling to generate alternatives, hop-skip-jump approach, waste management planning, uncertainty, public sector decision making.

Resume : Une approche dite de triple saut de zones incertaines (GHSJ) est dCveloppte et appliquCe au domaine de planification de gestion de dCchets municipaux. La mtthode amtliore les rnodkles existant pour gtntrer des approches alternatives en permettant i des informations incertaines d'Ctre communiquCes efficacement dans le processus d'optimisation et les solutions rtsultantes. Les alternatives de dtcisions faisables peuvent Ctre gCnCrCes B travers l'interprttation des solutions de la GHSJ, qui sont capables de refltter les conditions de variations potentielles du systkme causCes par l'existence des incertitudes introduites. Des risultats d'un cas hypothttique d'Ctude indiquent que des solutions utiles pour la planification d'expansion des amtnagements de gestion de dtchets peuvent Ctre gCnCrCes. Les alternatives de dCcision obtenues des solutions de la GHSJ peuvent &tre interprCtCes et analysies pour assimiler les compromis environnementaux et Cconorniques, qui peuvent Ctre inttressants pour les dtcideurs de gestion de dCchets solides confrontCs avec des choix difficiles et controversCs. Mots elks : programmation d'incertitudes, modtlisation pour gCnCrer des alternatives, approche de triple saut,

planification de gestion de dCchets, incertitudes, prise de dtcision en secteur publique. [Traduit par la rCdaction]

1. Introduction The planning of municipal solid waste management systems to satisfy increasing waste disposal and treatment demands is often subject to a variety of impact factors. Therefore, optimization may b e useful for reflecting the effects of these factors and generating optimal solutions. However, due to the presence of uncertainty and many nonquantifiable factors relating to environmental and economic objectives, and the possibility that public opposition may eliminate the optimal Received October 12, 1995. Revised manuscript accepted June 6, 1996. G.H. Huang. Faculty of Engineering, University of Regina, Regina, SK S4S OA2, Canada. B.W. Baetz. Department of Civil Engineering, McMaster University, Hamilton, ON L8S 4L7, Canada. G.G. Patry. Faculty of Engineering, University of Ottawa, ON KIN 6N5, Canada.

Written discussion of this paper is welcomed and will be received by the Editor until April 30, 1997 (address inside front cover). Can. I. Civ. Eng. 23:

1207 - 1219 (1996).

solution from further consideration, solid waste decision makers faced with difficult and controversial choices may prefer a set of alternatives s o that they can bring implicit knowledge (i.e., knowledge that cannot b e incorporated within an optimization model) to bear o n the problem (Gidley and Bari 1986). Methods f o r modelling to generate alternatives (MGA) have been proposed in response to the above situation. T h e M G A approaches provide an optimal solution and several near-optimal alternatives for a planning problem. Preferably, the alternatives are close to the optimal solution with respect to the objective function value, but vary considerably from the optimal solution in terms of system variables. A decision maker can then review the generated alternatives and internalize the tradeoffs between the differences i n the objective function value and the differing system characteristics. Previously, a number of M G A approaches have been proposed and applied. Brill (1979) developed a technique named hop, skip, and jump (HSJ) for linear and mixed integer programming problems to generate alternatives that are good with respect to the model objective and different from one another with respect to the specified decisions. Church and

Printed in Canada / Imprimt au Canada

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Can. J. Civ. Eng. Vol. 23, 1996

Huber (1979) used a reverse heuristic to find close-tooptimal solutions for maximal covering location problems. Falkenhausen (1979) employed a heuristic evolution strategy to generate alternative solutions for a regional wastewater treatment system planning problem. Chang et al. (1980) developed a random technique for generating decision alternatives by maximizing the sum of several randomly selected decision variables. Brill et al. (1981) applied the HSJ method to a multiobjective linear programming problem to generate alternatives for a hypothetical land use planning problem. Chang et al. (1982) discussed an approach named branch and boundlscreen (BBS) for obtaining good and different alternatives by first generating many solutions efficiently and then applying a screening process to determine the alternatives. More recently, Baetz et al. (1990) developed an MGA approach for dynamic programming-based planning problems and applied it to solid waste management planning. The major deficiency with the existing MGA approaches is that they are based on deterministic mathematical programming models, which may not be able to effectively communicate uncertain information into the optimization framework. Therefore, a grey hop, skip, and jump (GHSJ) approach is developed in this paper to mitigate this problem. The GHSJ approach can directly communicate uncertainty into the optimization process and the resulting solutions, such that optimal and close-to-optimal solutions for the decision variables and the objective function value can be obtained (Huang et al. 1995). Thus, decision alternatives can be generated by adjusting different combinations of the decision variable values within their solution intervals according to projected applicable conditions, which will reflect potential system condition variations caused by the existence of input uncertainties. The purpose of this paper is to develop the GHSJ approach and apply it to a hypothetical case study of municipal solid waste management planning. The results will be interpreted and analyzed to show the potential applicability of the developed methodology to waste management planning and other types of public sector decision making problems.

2. The grey hop, skip, and jump approach In solid waste management systems, uncertainties may exist in many system components related to environmental, socioeconomic, and resources concerns, and the associated information may not be known with certainty but instead as follows: "the capital cost for expanding the composting facility will be in the range of $1 000000 to $1 200000," "the waste generation rate is approximately 90 to 100 tonnes per week," "the incinerator has a capacity of 2000 to 2500 tonnes per week", and so on (Inuiguchi et al. 1990). Difficulties may arise when modelling such systems with deterministic mathematical programming methods. Therefore, a GHSJ method is now developed, where concepts of grey systems and grey decisions are introduced into an HSJ modelling framework to reflect the effects of uncertainties, and interactive solution algorithms are used for solving the related grey programming problems (Huang 1994). First, we introduce some definitions. Let x denote a closed and bounded set of real numbers. A grey number x* is defined as an interval with known upper and lower bounds but unknown distribution information for x:

where x- and x + are the lower and upper bounds of x i , respectively. When x- = x + , x + becomes a deterministic number, i.e., x i = x- = x + . Let '$3' denote a set of grey numbers. A grey vector (or matrix) is defined as [2.2]

X+ = {x:

=

[x;, x;]

[2.3]

xi = {x:

=

[x-, x$] rJ JJ

1

I

Vi} 'v'i, j }

xi E

('$3')

I

X

XF E {(SZ~}IJ'XI~

8

A grey system is a system containing information presented as grey numbers, and a grey decision is a decision made within a grey system. Thus a grey mathematical programming (GMP) model can be defined as follows (Huang 1994): Minimize

subject to

where X i is a grey vector of decision variables, f*(X*) is a grey objective function, and g: (Xi) Ib:, Vi, are grey constraints. When model [2.4] is linear, integer linear, or quadratic, it has interval solutions as follows:

[2.6]

x;~pt

=

[ ~ j , xTOpt], ~~'

~

xTOpt> x j O p tand V j

where xf can be discrete or continuous variables. The detailed solution algorithms for the G M P models have been provided by Huang (1994). A simple example for the G M P model can be presented as follows: Minimize

subject to

where f * is a grey objective function, x: a r e grey decision variables, and the coefficient [2, 31 represents a grey number with its lower and upper bounds being 2 and 3, respectively (and so on for the other coefficients). According to the GMP solution algorithm proposed by Huang (1994), the above model can be converted to two submodels as follows: (i) Minimize

Huang et al. Fig. 1. Graphical depiction of a grey mathematical programming problem and its solution.

-------

constraint [2 8b]

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constrarnt [2 8c]

0 I

1

2

3

A = feaslble for [2 8b] and [2 8c] B = feaslble for [2 8b] + softly-feaslble for C = feasrble for [2 8c] + softly-feasible for D = feasible for [2.8b]+ infeasible for [2.8;] E = softly-feasible for [2.8b]and [2.8c] F = feasible for [2.8c]+ infeasible for [2.8b] G = softly-feas~blefor 12 8b] + Infeasible for [2.8c] H = softly-feasible for [2 8c] + Infeasible for [2 8b] I = infeas~blefor [2 8b] and [2 8c] U4 0 = solution set

x2

subject to

(ii) Minimize

subject to

the upper bounds of their solutions, we get a lower f* value but a higher possibility of violating the constraints. Conversely, when xf and x $ approach the lower bounds, a higher f * value but lower possibility of violation can b e obtained. In an application to solid waste management, we can assume that x$ and x $ represent waste flows, and f * is system cost. Thus, there exists a tradeoff between system cost and solution feasibility. A conservative strategy will correspond to estimation of higher waste flows and higher operation and transportation costs. In comparison, an optimistic strategy will relate to lower waste flows and lower operation and transportation costs. Consequently, decision alternatives can be generated through interpretation of the grey solutions according to projected applicable system conditions. The GMP solution algorithms are significantly different from ordinary best or worst case analysis. In the GMP, the solution corresponding to f - (lower bound of the objective function value) can be first solved (when the objective is to be minimized), and the relevant solution corresponding to f (upper bound of the objective function value) was proven to be feasible as one of the two bounds of the desired grey solution (Huang 1994). Thus, the results corresponding to f and f - lead to a set of optimal and stable grey solutions (the grey solutions are stable if the objective function value varies between f; and f ip, as the decision variables change between xyoptan$ xTOpJ.In a bestlworst case analysis, in comparison, the major concern is the solution of the objective function value, while decision variable solutions for the best and worst cases may not necessarily construct a set of feasible and stable grey solutions (i.e., when the best case (corresponding to f-) is first solved, the relevant worst case solutions for decision variables may be infeasible as one of the two bounds of the grey solution; conversely, when the worst case (corresponding to f +) is first calculated, poor grey solutions may be generated). To obtain a second grey solution which is different from the optimal solution, we introduce a GHSJ approach, where the sum of nonzero variables in the initial solution is minimized subject to a target constraint on the cost objective as follows: +

+

where x j and xf represent the lower and upper bounds of x:, respectively. x ; , and x ; , are the solutions obtained from solving submodel [2.9]. Thus, the solution of model [2.8] is

Figure 1 shows a graphical depiction of this problem, where "softly feasible" means that the feasibility is dependent upon the location of the decision variables {xf,x $ ) and the conditions of the grey constraints. The solution set for x: and x 3 is presented as a rectangle. When x: and x $ are close to

,

Can. J. Civ. Eng. Vol. 23, 1996

-

Fig. 2. Modelling process for the grey hop, skip, and jump (GHSJ) approach.

I uncertain ~arametersI

Fig. 3. Hypothetical study municipalities and waste management facilities. Munici~alitv . .2 Municipality 3

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interactive solution algorithm

and objective function value

Facility 2

implicit knowledge

/

examination of different solutions and different combinations of decision variable values within their solution intervals

I

,* * ,U W E Facility 1

1

potential decision scheme

interest groups:

I justifcation of the generated alternat~vesI

Minimize

subject to

- - - -,

Municipality 1 municipal solid waste residue from waste-to-energy (VVTE) facility

represent uncertain inputs and outputs. This is particularly meaningful for practical applications because (i) it is typically more difficult to specify distributions than to define fluctuation intervals; and (ii) the existing optimization methods that deal with distribution uncertainties have difficulties in solution algorithms, computational requirements, and results interpretation (Inuiguchi et al. 1990; Marti 1990).

3. Application to solid waste decision making

where xf E X * , and a is the increment for the target constraint. Normally, even though unmodelled issues are considered, good alternative solutions are unlikely to be more than 10% worse than the initial optimal solution (Chang and Brill 1982). Additional alternative solutions can be obtained by minimizing the sum of different combinations of nonzero variables that appear in one or more of the previous solutions. Figure 2 shows the general modelling process for the GHSJ approach. The GHSJ provides alternative solutions represented as grey numbers for the decision variables and objective function value, which can be further used to generate several deterministic alternatives by adjusting different combinations of decision variable values within their solution intervals. For each GHSJ solution, when the decision variable values vary within their solution intervals, the objective function value will change within its solution interval correspondingly. The GHSJ has an advantage of low computational requirements, since it does not lead to more complicated intermediate models due to the characteristics of the GMP solution algorithm (Huang 1994). Moreover, the method does not require distribution information, since grey numbers are used to

3.1. Overview of the hypothetical problem A hypothetical problem has been developed to illustrate the GHSJ modelling approach based upon representative cost and technical data from the solid waste management literature. The study region is assumed to include three municipalities, as shown in Fig. 3. Three time periods are considered with each having an interval of 5 years. At the beginning of the time horizon, an existing landfill and two waste-to-energy (WTE) facilities are available to serve the region's solid waste disposal needs. The landfill has an existing capacity of [0.625, 0.7751 x lo6 t, and WTE facilities 1 and 2 have capacities of [loo. 1251 and [200, 2501 t/d, respectively. The WTE facilities generate residues of approximately 30% (on a mass basis) of the incoming waste streams, and their revenues from energy sales are approximately [15, 251 $/t combusted. Over the 15-year planning horizon, the landfill capacity can be expanded once by an increment of [ 1 .55, 1.701 X lo6 t, and the WTE facilities can be expanded by any of four options in each of the three time periods (see Table 1 for detailed information), with a maximum expansion limit of 250 t/d. Table 1 also shows the capital costs for capacity expansions for the three facilities, which are expressed in terms of present value dollars, with the costs being escalated to reflect anticipated conditions and then discounted to gener-

Huang et al

Table 1. Capacity expansion options and their costs for the landfill and WTE facilities.

Table 2. Waste generation, transportation costs, and facility operating costs. Time period

Time period Symbol

k = l

k = 2

Capacity expansion option for WTE facility i, i = 2, 3 (tld) ATC, (option 1) 100 100 100 ATC,,, (option 2) 150 150 150 ATCi,, (option 3) 200 200 200 ATCi4, (option 4) 250 250 250

,,

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Symbol

k = 3

Capacity expansion option for the landfill (lo6 t) ALC [1.55, 1.701 [1.55, 1.701 [1.55, 1.701

*

Capital cost of WTE facility expansion, i = 2, 3 ($lo6 present value) FTC,,, (option 1) 10.5 8.3 6.5 FTC,,, (option 2) 15.2 11.9 9.3 FTC,,, (option 3) 19.8 15.5 12.2 FTC,, (option 4) 24.4 19.1 15.0 Capital cost of landfill expansion ($lo6 present value) FLC: t13, 151 [13, 151 [13, 151

ate present value cost terms for the objective function. Table 2 contains waste generation values for the three municipalities, operating costs of the three facilities, and transportation costs for the waste flows between municipalities and facilities in the three time periods. It is indicated that the municipal solid waste generation rates and the costs for waste transportation and treatment vary temporally and spatially. Therefore, the problem under consideration is how to obtain preferred facility expansion alternatives during different periods and how to effectively allocate the relevant waste flows, in order to minimize total system cost. Since the majority of data for the system have uncertain features and are known only as intervals without distribution information, the GHSJ approach is considered to be appropriate for this problem.

k = l

k = 2

k

=

3

Waste generation (tld) WG:, (Municipality 1) [200, 2501 [225, 2751 WGk (Municipality 2) [375, 4251 [425, 4751 WG:, (Municipality 3) [300, 3501 [325, 3751

[250, 3001 [475, 5251 [375, 4251

Cost of waste transportation TR:,, (Municipality 1) [12.1, 16.11 TRf,, (Municipality 2) [10.5, 14.01 TR:,,(Municipality 3) [12.7, 17.01

to the landfill [13.3, 17.71 t11.6, 15.41 [14.0, 18.71

($It) [14.6, 19.51 [12.8, 16.91 [15.4,20.6]

Cost of waste transportation to TR:,, (Municipality 1) [9.6, 12.81 TR&, (Municipality 2) [10.1, 13.41 TR:,, (Municipality 3) [8.8, 11.71

WTE facility t10.6, 14.11 [11.1, 14.71 [9.7, 12.81

1 ($It) [11.7, 15.51 [12.2, 16.21 [10.6, 14.01

Cost of waste transportation to TR:,, (Municipality 1) [12.1, 16.11 TR?,, (Municipality 2) [12.8, 17.11 TR;,, (Municipality 3) [4.2, 5.61

WTE facility [13.3, 17.71 [14.1, 18.81 [4.6, 6.21

2 ($It) [14.6, 19.51 [15.5, 20.71 [5.1, 6.81

Cost of residue transportation from the WTE Facilities to the landfill ($It) FT& (WTE facility 1) [4.7, 6.31 [5.2, 6.91 [5.7, 7.61 FT& (WTE facility 2) [13.4, 17.91 L14.7, 19.71 i16.2, 21.71 Operational cost ($It) OP:, (Landfill) [30, 451 [40, 601 OP: (WTE facility 1) [55, 751 [60, 851 OP$ (WTE facility 2) [50, 701 [60, 801

[50, 801 [65, 951 [65, 851

subject to 3

[31b]

3.2. Model formulation

k'

3

C C L,[X',~+ j=l k = ~

C x&FE] i=2

k'

In the municipal solid waste management system under consideration, grey decision variables include two categories: continuous and binary. The continuous variables represent "municipality facility" waste flows over the time horizon, and the binary variables represent facility expansion decisions. The objective is to achieve optimal planning for facility expansion and relevant municipal solid waste flow allocation with minimum system cost. The constraints include all relationships between the decision variables and the waste generation and management conditions. A grey integer programming (GIP) model can be formulated as follows: Minimize