1996 huang et al a grey hop, skip, and jump approach: generating alternatives for expansion planning...

A grey hop, skip, and jump approach: generating alternatives for expansion planning of waste management facilities

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 ii 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 be 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).

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 be incorporated within an optimization model) to bear o n the problem (Gidley and Bari 1986).

Methods for 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

Can. I. Civ. Eng. 23: 1207 - 1219 (1996). 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-to- optimal 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, socio- economic, 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;] 1 Vi} xi E ('$3') I X ~ [2.3] xi = {x: = [x-, x$] I 'v'i, j } XF E { ( S Z ~ } I J ' X I ~ rJ 8 JJ

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) I b:, 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 t and V j

where xf can be discrete or continuous variables. The detailed solution algorithms for the GMP models have been provided by Huang (1994).

A simple example for the GMP model can be presented as follows:

Minimize

subject to

where f* is a grey objective function, x: are 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

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

------- constraint [2 8b]

constrarnt [2 8c]

A = feaslble for [2 8b] and [2 8c] B = feaslble for [2 8b] + softly-feaslble C = feasrble for [2 8c] + softly-feasible

for 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~ble for 12 8b] + Infeasible for [2.8c] H = softly-feasible for [2 8c] + Infeasible for [2 8b] I = infeas~ble for [2 8b] and [2 8c]

0 I U 1 2 3 4 0 = solution set

x2

subject to

(ii) Minimize

subject to

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

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 be 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 xyopt an$ 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:

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

I uncertain ~arameters I

interactive solution algorithm e and objective function value implicit knowledge

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

potential decision scheme I - interest groups:

I justifcation of the generated alternat~ves I

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

. .

Municipality 3

Facility 2

/ , * * , U W E Facility 1 1

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

Minimize

subject to

represent uncertain inputs and outputs. This is particularly meaningful for practical applications because (i) it is typi- cally 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).

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. Application to solid waste decision making

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-

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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 k = 3 Symbol k = l k = 2 k = 3

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

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.

3.2. Model formulation 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

Waste generation (tld) WG:, (Municipality 1) [200, 2501 [225, 2751 [250, 3001 WGk (Municipality 2) [375, 4251 [425, 4751 [475, 5251 WG:, (Municipality 3) [300, 3501 [325, 3751 [375, 4251

Cost of waste transportation to the landfill ($It) TR:,, (Municipality 1) [12.1, 16.11 [13.3, 17.71 [14.6, 19.51 TRf,, (Municipality 2) [10.5, 14.01 t11.6, 15.41 [12.8, 16.91 TR:,,(Municipality 3) [12.7, 17.01 [14.0, 18.71 [15.4,20.6]

Cost of waste transportation to WTE facility 1 ($It) TR:,, (Municipality 1) [9.6, 12.81 t10.6, 14.11 [11.7, 15.51 TR&, (Municipality 2) [10.1, 13.41 [11.1, 14.71 [12.2, 16.21 TR:,, (Municipality 3) [8.8, 11.71 [9.7, 12.81 [10.6, 14.01

Cost of waste transportation to WTE facility 2 ($It) TR:,, (Municipality 1) [12.1, 16.11 [13.3, 17.71 [14.6, 19.51 TR?,, (Municipality 2) [12.8, 17.11 [14.1, 18.81 [15.5, 20.71 TR;,, (Municipality 3) [4.2, 5.61 [4.6, 6.21 [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 ($It) OP:, (Landfill) [30, 451 [40, 601 [50, 801 OP: (WTE facility 1) [55, 751 [60, 851 [65, 951 OP$ (WTE facility 2) [50, 701 [60, 801 [65, 851

subject to 3 k' 3

[31b] C C L,[X',~+ C x&FE] j = l k = ~ i=2

k' < C ALC*y: + LC*; k t = 1, 2, 3 -

k= l

(landfill capacity constraints)

i = 2 , 3 and k' = 1, 2, 3 (WTE facility capacity constraints)

(waste disposal demand constraints)

[3.le] C z:,,~ 5 1; i = 2, 3 and V k Ill= I

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(only one WTE facility expansion may occur in any given time period)

(landfill expansion may only be considered once) [3. lg] xSk r 0; V i, j , k

(nonnegativity constraints)

= integer; V k

= integer; i = 2, 3 and V m , k (nonnegativity and binary constraints)

where i is the type of waste management facility (i = 1, 2, 3, with i = 1 for the landfill, and 2 and 3 for WTE facilities 1, and 2, respectively); j is the municipality, j = 1, 2, 3 (Fig. 2); k is the time period, k = 1, 2, 3; m is the expansion option for the WTE facilities, m = 1, 2, 3, 4; x ik is the waste flow from municipality j to facility i during period k (tld); y: is the binary decision variable for landfill expansion at the start of period k ; z:f,k is the binary variable for WTE facility i with expansion option m at the start of period k , i = 2, 3; C t k is the total cost of waste management for waste flow from municipality j to facility i in period k ($It):

Ci: = TRL + OP$; for i = 1 and V j , k ilk ilk C$ = TR$ + OP$ + FE(FT$ + OP&) - RE:;

for i = 2, 3 and V j , k FE is the residue flow rate from the WTE facility to the landfill (% of incoming mass to the WTE facility); FLC: is the capital cost of landfill expansion in period k ($); FT: is the transportation cost for "WTE facility i - landfill" waste flow during period k , i = 2 , 3 ($It); FTC,,,,k is the capital cost of expanding WTE facility i by option m in period k , i = 2, 3 ($); Lk is the length of time period k (d); LC* is the existing landfill capacity (t); OP$ is the operating cost of facility i during period k ($It); RE: is the revenue from the WTE facilities during period k ($It); TC! is the existing capacity of WTE facility i, i = 2, 3 (tld); TR& is the transportation cost for waste flow from municipality j to facility i during period k ($It); WG$ is the waste generation rate in municipality j during period k (tld); ALC is the amount of capacity expansion for the landfill (t); and ATC,,,,,, is the amount of capacity expansion (option m) for WTE facility i at the start of period k , i = 2, 3 (tld).

Equation [3. la] implies that the objective is to minimize system cost which is related to the benefits and costs of different waste management activities and capital costs for related facility expansions. Constraints [3. lb] and [3. lc] stipu-

late that the upper limit for waste treatment and disposal in any time stage is determined by both the existing and expanded capacities for the landfill and WTE facilities. This dynamic nature is related to economic development, population increase, and environmental management activities. Constraint [3. Id] states waste disposal demand for the three municipalities. Constraint [3. le] requires that only one WTE facility expansion may occur for any given time period, and constraint [3 . l f ] stipulates that the landfill can only be expanded once for the entire planning time horizon. Constraints [3.lg] to [3. l i] define technical relationships for the decision variables.

The detailed solution algorithm for the above GIP model is provided by Huang et al. (1995). Generally, interactive relationships between objective and constraints, between decision variables and parameters, and between different decision variables are analyzed and quantitatively presented. Submodels corresponding to the upper and lower bounds of the objective function value are formulated based on the interactive relationships. Grey solutions are then generated through interpretation of solutions from the two submodels.

The solutions for the grey binary variables have four possible representations ([0, 01, [ I , 11, [O, 11, and [ l , 01). These variables represent the related grey decisions that reflect potential system condition variations caused by the input uncertainties. For example, if an incinerator has two options for capacity expansion in a given time period: 500 or 600 t/d capacity (assuming that only one expansion is allowed in the period, and the option of 600 tld has a higher capital cost than that of 500 t/d), let the 500 t/d option correspond to a binary variable x , , and the 600 t/d option correspond to another binary variable x2. When both x , and x2 are deterministic variables, we will have one of the following three possible solutions: (i) no expansion (xl = 0 and 12 = 0); (ii) expanding by 500 t/d (xl = 1 andx2 = 0); o r (iii) expanding by 600 tld (x, = 0 and x2 = 1). Thus, the solutions are deterministic and cannot effectively reflect the effects of uncertainties.

In comparison, when x, and x2 are grey binary variables, we have x l = x: and x2 = x$. Assume that the objective is to minimize f* , and that x; and x; correspond to f - . Thus we will have one of the following six possible solutions (through formulating the related constraints): (i) no expansion (x; = 0 , x; = 0; and x: = 0, x: = 0); (ii) expanding by 500 t/d (x; = 1, x; = 0 ; and xj = 1 ,

x: = 0); (iii) expanding by 600 t/d (x; = 0, x; = 1; and x: = 0,

x; = 1); (iv) expanding by [O, 5001 t/d (x; = 0, x; = 0; and x: =

1, x; = 0); (v) expanding by [O, 6001 t/d (x; = 0, x; = 0; and x: =

0 , x; = 1); (vi) expanding by [500, 6001 tld (x; = 1, x; = 0; and

x: = 0, x; = 1). The solutions in (i) to (iii) are the same as those when x , and x2 are deterministic. However, the grey solutions in (iv) to (vi) reflect potential system condition variations caused by the existence of input uncertainties. The lower bound expansion values (and thus lower capital costs) correspond to advantageous system conditions (e.g., conditions when recycling, reduction, and reuse (3R) initiatives are effective in controll-

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Table 3. Optimal solutions obtained through the grey integer programming (GIP) model.

Table 3 (concluded). --

Symbol Facility Expansion Period Solution Symbol Facility Expansion Period Solution

x * ,, , WTE facility 2 1 x:,? WTE facility 2 1 x WTE facility 2 1

Binary decision variable Landfill 1 1 1 Landfill 1 2 0 Landfill 1 3 0

WTE facility 1 1 1 0 WTE facility 1 1 2 0 WTE facility 1 1 3 0





WTE facility 2 2 1 11, 01 WTE facility 2 2 2 1 WTE facility 2 2 3 0

WTE facility 2 3 1 10, 11 WTE facility 2 3 2 0 WTE facility 2 3 3 0


yf opl y: opt Y: opt

.r f 321 WTE facility 2 2 ~ $ 2 WTE facility 2 2 x$, WTE facility 2 2 x:~, WTE facility 2 3 x:~? WTE facility 2 3 x;~, WTE facility 2 3

z:I I opt z:12 op1

':I3 opt

z:21 op1

~ $ 2 2 opt 6 2 3 opt

System cost, f* ($lo6):

':3I opl

z:z opt z:3 opt

Fig. 4. Expansion scheme for waste-to-energy (WTE) facilities (optimal solution).

(a) 800 -1 I '$41 opt

~ $ 4 2 opt

~ f l l opt

G I 2 opt

'$13 opt

z:2, apt

z;22 opt

'f23 opt

'$31 opt z:3? opt

~ $ 3 3 opt

z:, opt z:2 opt

z:43 opt

0 5 10 15 Time (year)

(b) 800 -

S 700- B Continuous decision variable (tld)

Landfill 1 1 0 Landfill 1 2 0 Landfill 1 3 0

Landfill 2 1 [263, 2711 Landfill 2 2 [51, 721 Landfill 2 3 [125, 1371 Landfill 3 1 0 Landfill 3 2 0 Landfill 3 3 0

I I I 5 10 15

Time (year) WTE facility 1 1 1 [200, 2381 WTE facility 1 1 2 122 WTE facility 1 1 3 150 ing waste generation), and the upper bound expansion values (and thus higher capital costs) correspond to m o r e demand-

ing system conditions. For example, [500, 6001 tld in ( v i ) means that the expansion level is quite flexible and can be either 500 o r 600 t/d, which correspond to different system conditions.

Thus, through the GHSJ approach as applied to model [3.1], decision alternatives (close-to-optimal solutions) based o n the GIP model can be obtained by minimizing the sum of

WTE facility 1 2 1 87 WTE facility 1 2 2 [374, 4031 WTE facility 1 2 3 [350, 3631 WTE facility 1 3 1 0 WTE facility 1 3 2 0 WTE facility 1 3 3 0

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121 4 Can. J. Civ. Eng. Vol. 23, 1996

Table 4. Close-to-optimal solutions under different target constraints for the cost objective (tld). - --

Symbol Facility Expansion Period Alternative 1.1 Alternative 1.2 Alternative 1.3

':I1 opt

'$12 opt '$13 opt

'$21 opt ~ $ 2 2 opt z:23 op1

';4I opt 'i42 opt

'$3 op1

':I l opt

' f l 2 opt

':I3 opt

'$21 opt ':22 opt

'i23 opl

z:31 opt

':32 opt

z:33 .pi

~ f 4 I opt

'$42 apt ':43 opt

x T I l op1

xT12 opt

'?I3 opt

xT21 opt

xT22 op1

~ T 2 3 o p ~

'$31 opl ~ T 3 2 opt

x T 3 3 op1

~ $ 1 l opt x:12 op1

~ 1 1 3 opt

~ $ 2 1 opt x:22 apt

x;23 opt

~ : 3 l opt

x:3? opt

~ $ 3 3 opt

~ f l l opt

x:12 opt

':I3 opt

Landfill Landfill Landfill

WTE facility 1 WTE facility 1 WTE facility 1








Landfill Landfill Landfill Landfill Landfill Landfill Landfill Landfill Landfill

WTE facility 1 WTE facility 1 WTE facility 1 WTE facility 1 WTE facility 1 WTE facility 1 WTE facility 1 WTE facility 1 WTE facility 1


Binary decision variable 1 1 1 1 1 2 0 0 1 3 0 0

1 1 0 0 1 2 0 0 1 3 0 0

2 1 0 0 2 2 0 0 2 3 0 0

3 1 [1,01 [ I , 01 3 2 1 1 3 3 1 1

4 1 10, 11 10, 11 4 2 0 0 4 3 0 0

1 1 0 0 1 2 0 0 1 3 0 0

2 1 0 0 2 2 0 0 2 3 0 0

3 1 1 1 3 2 1 1 3 3 1 0

4 1 0 0 4 2 0 0 4 3 0 [O, 11 Continuous decision variable (tld) 1 1 0 0 1 2 [O, 421 LO, 421 1 3 0 0 2 1 262 262 2 2 59 5 9 2 3 116 116 3 1 0 0 3 2 0 0 3 3 0 0

1 1 [200, 2501 [200, 2381 1 2 122 [122, 1301 1 3 125 150 2 1 [63, 751 88 2 2 366 366 2 3 359 359 3 1 10, 501 [O, 501 3 2 10, 501 [O, 501 3 3 10, 501 [O, 501 1 1 0 [O, 131 1 2 [103, 1111 103 1 3 [125, 1751 [loo, 1501

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Table 4 (concluded). Symbol Facility Expansion Period Alternative 1.1

x:,, WTE facility 2 2 x : , ~ WTE facility 2 2 x:,, opt WTE facility 2 2 x : ~ , opt WTE facility 2 3 x : ~ , opt WTE facility 2 3 x:,, opt WTE facility 2 3 System cost, f* ($lo6)

Alternative 1

[25, 751 10, 501 [O, 501 300 325 375

Alternative 1.3

different combinations of nonzero variables that appeared in the previous solutions subject to different target constraints on the cost objective. 3.3. Modelling results Table 3 and Fig. 4 contain and illustrate the results for the initial optimal solution obtained through the GIP model. Table 4 presents close-to-optimal solutions obtained through the GHSJ approach, where alternatives 1.1, 1.2, and 1.3 were generated under the conditions where the sum of all nonzero continuous variables in the initial solution was minimized subject to different target constraints on the cost objective ( a = 0.02, 0.05, and 0.08). The solutions for alternative 1.3 ( a = 0.08) are illustrated in Fig. 5. Table 5 contains the close-to-optimal solutions for the conditions when the sum of different combinations of continuous variables (waste flows to landfill, WTE facility 1, and WTE facility 2, respectively) in the initial solution was minimized for an a value of 0.05. It is indicated that solutions for the objective function value and many decision variables are grey numbers.

The results indicate that the landfill should be expanded at the start of period 1 ( y+ = [ l , I]), but not expanded further in periods 2 and 3 ( y$ and y$ are both equal to [O, 01). The amount of expansion is the [1.55, 1.701 x lo6 t level input into the model (Table 1). Figure 4 shows the optimal expansion schemes for WTE facilities 1 and 2 , respectively. It is indicated that WTE facility 1 should be expanded by 200 tld in both periods 1 and 2 (z:~, = [I , 11 and z:~, = [ l , 11) and WTE facility 2 should be expanded by 1150, 2001 tld in period 1 and 150 tld in period 2 (z:,, = [ I , 01, z:,, = 11, 11, and zz3, = [O, 11). The expansion of 1150, 2001 tld in period 1 means that there are two alternatives for the expansion, where 150 tld corresponds to f - , and 200 tld corresponds to f +. Thus, when the decision scheme tends toward f - under advantageous conditions, it may be applicable to expand WTE facility 2 by 150 tld in both periods 1 and 2; and when the scheme tends toward f + under more demanding conditions, it may be suitable to expand WTE facility 2 by 200 tld in period 1 and 150 tld in period 2. No expansion should be carried out in period 3 for either of the facilities, since sufficient capacity has been developed in the previous periods (see Fig. 4 for details).

For the grey continuous variable solutions, it is indicated in the optimal solution that the landfill accepts wastes only from municipality 2 because of its close proximity to the municipality and landfill capacity limitations, in addition to residues from the WTE facilities. All waste flows from municipality 3 and delivered to WTE facility 2 due to its

Fig. 5. Expansion scheme for waste-to-energy (WTE) facilities (alternative 1.3).

(a) 800 -1 I

Time (year)

% 300

g 200 I00

0 5 10 Time (year)

close proximity to the facility. WTE facility 2 also accepts a portion of the flows from municipalities 1 and 2. The remaining waste flows from municipalities 1 and 2 are determined to be hauled to WTE facility 1. As shown in Table 6 , [30.1, 26.4]%, 132.8, 31.7]%, and 137.1, 41.91% of the total waste are determined to be routed to landfill, WTE facility 1, and WTE facility 2, respectively, in period 1. Similar distribution patterns can also be found in the solutions for periods 2 and 3, although direct flows to the landfill are reduced owing to capacity limitations.

Alternative 1.1 was generated by minimizing the sum of all nonzero continuous variables in the initial solution subject to a target constraint off* 1 1.02 f&,. It is indicated that

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Table 5. Close-to-optimal solutions when different variable combinations are optimized.


--

Symbol Facility Expansion Period Alternative 2 Alternative 3 Alternative 4

Binary decision variable Landfill 1 1 1 Landfill 1 2 0 Landfill 1 3 0

Y: opt Y: opt Y'

3 opt

':I1 opt

6 1 2 opt

':I3 opt

':*I opt

z:22 opt

':23 opr

':I opr

';32 opr

z:33 opt

'&I opr

':42 opt

G 4 3 opt

'$1 l opt 2:12 opt

':I3 opt

'$21 opt 2:22 opt

'$23 opt

':3l opt

'$32 opr '$33 opr

'$41 opr '$42 opt '$43 opt



WTE facility 1 3 1 1 WTE facility 1 3 2 [I, 01 WTE facility 1 3 3 0

WTE facility 1 4 1 0 WTE facility 1 4 2 [o, 11 WTE facility 1 4 3 0

WTE facility 2 1 1 0 WTE facility 2 1 2 [I , 01 WTE facility 2 1 3 0

WTE facility 2 2 1 0 WTE facility 2 2 2 [o, 11 WTE facility 2 2 3 0

WTE facility 2 3 1 [1,01 WTE facility 2 3 2 0 WTE facility 2 3 3 1

WTE facility 2 4 1 [o, 11 WTE facility 2 4 2 0 WTE facility 2 4 3 0

Continuous decision variable (tld) 1 1 0 0 1 2 0 0 1 3 0 0 2 1 237 262 2 2 34 [59, 621 2 3 112 116 3 1 0 0 3 2 0 0 3 3 o [O, 391

x:l I opt

~ f l 2 opt

x i 1 3 opt

x i 2 1 opt

x:22 opt

'?23 opt

'731 opt

~ ? 3 2 opt

x i 3 3 opr

Landfill Landfill Landfill Landfill Landfill Landfill Landfill Landfill Landfill

x:l l opr

x i 1 2 opt

~ ; 1 3 opt

~ $ 2 1 opr ~ $ 2 2 opr x:23 opt

x:l opt

x:2 opt

x;33 opt

WTE facility 1 WTE facility 1 WTE facility 1 WTE facility 1 WTE facility 1 WTE facility 1 WTE facility 1 WTE facility 1 WTE facility 1

'$13 opr


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Table 5 (concluded). Symbol Facility Expansion Period Alternative 2 Alternative 3 Alternative 4

xi2, opt WTE facility 2 2 1 [50,100] [50, 1001 25 xi2? opt WTE facility 2 2 2 0 to, 471 0 xi2, WTE facility 2 2 3 [o, 501 [o, 501 0 xi,, op, WTE facility 2 3 1 [300, 3501 [300, 3501 [300, 3271 xt3, WTE facility 2 3 2 [325, 3751 [325, 3751 325 4 3 3 O ~ I WTE facility 2 3 3 [375, 4251 [375, 3861 375 System cost, f+ ($lo6) [405.1, 725.41 [405.1, 725.41 [405.1, 725.41 '

both WTE facilities 1 and 2 should have more expansion capacity in this alternative than in the optimal solution, which leads to an increase in system cost. The detailed allocation for "municipality - facility" waste flows in this alternative is also different from the optimal solution. However, the general percentage-distribution pattern for waste flows to the landfill, WTE facility 1, and WTE facility 2 is similar to that of the optimal solution (see Table 6). This is an option that provides alternative waste flow allocation patterns and increased waste management capacity relative to the optimal solution based on a 2% increase in the system cost. Alterna- tive 1.2 was subject to a target constraint off * 2 1.05 f:pt. Generally, the facility-expansion and waste-flow-allocation solutions in alternative 1.2 are similar to those in alternative 1.1. Alternative 1.3 was subject to a target constraint of f * r 1.08 f Zpt. It is indicated that, compared to the optimal solution and alternatives 1.1 and 1.2 solutions, more waste flows to WTE facilities (i.e., sum of waste flows to WTE facilities 1 and 2) and more expansions of the two WTE facilities are determined, which corresponds to the target 8% increase in the system cost. Since more flow to WTE facilities and less flow to the landfill are provided, this alternative is an option that emphasizes land resource conservation under an assumption that increased utilization of the WTE facilities (and thus increased system cost) is allowed.

When the a value is increased, a higher system cost will be incurred for the waste management activities. The increased cost can be devoted to increased expansion of waste management facilities to provide more capacity for waste disposal and treatment in the future, and (or) allocation of increased waste flows to the WTE facilities (and thus less flows to the landfill) for the purpose of land resource conservation.

Alternatives 2, 3, and 4 in Table 5 were generated by minimizing the sum of different combinations of decision variables in the initial solution (min Cj Ck x:,~, min Cj Ck x % ~ , and min C, Ck x $ ~ for alternatives 2, 3, and 4, respectively), for an increase in system cost of 5% for all three cases. When the sum of flows to a facility is minimized, the relevant solutions for these flows will be lower than those in other alternatives. For example, in alternative 2, waste flows to the landfill are generally lower than those in other alternatives, since Cj Ck x S . ~ is minimized (see Table 6 for detailed compari- sons). ~ h u s , based on increased system costs, alternative 2 is an option that emphasizes land resource conservation under an assumption that increased utilization of the WTE facilities is allowed; alternative 3 relates to the situation when environmental regulations or community perception may influence

the utilization of WTE facility 1 (where more waste flows are determined to be routed to WTE facility 2); and alternative 4 relates to the situation when the utilization of W T E facility 2 is affected, where more waste flows are routed to WTE facility 1.

Generally, the results indicate that, through the proposed modelling approach, uncertain information can be effectively communicated into the GHSJ optimization processes and resulting solutions. Thus, decision alternatives can be generated from the GHSJ solutions according to projected planning situations. For each alternative individually, lower decision variable values (i.e., lower flows to waste management facilities and lower expansion for the WTE facilities) within their solution intervals should be used under advantageous system conditions (e.g., conditions when recycling, reuse, and reduction initiatives are effective in reducing waste disposal and treatment demands), and higher decision variable values within their solution intervals should be used under more demanding system conditions. This GHSJ solution feature may be favored by decision makers because of the increased flexibility and applicability for determining final decision schemes.

Thus the GHSJ solutions provide various alternatives that can be evaluated from the stand point of environmental and economic tradeoffs. Decision makers can compare a range of systems with respect to cost and relative utilization of various facilities, and can obtain optimal system operation characteristics for a range of policy alternatives (e.g., minimize landfill usage) that will in turn assist in the development of a preferred operational strategy under certain conditions.

4. Concluding remarks A GHSJ approach has been applied to a hypothetical case study of municipal solid waste management planning. The method improves upon the existing HSJ approach by allowing uncertain information, presented as interval numbers, to be effectively communicated into the optimization process and resulting solutions. Feasible decision alternatives can be generated through interpretation of the optimal and close-to-optimal grey solutions (presented as stable intervals) according to projected applicable system conditions. Results from the hypothetical case study indicate that potentially useful solutions for the expansion planning of solid waste management facilities have been generated. The decision alternatives obtained may be interpreted and analyzed to internalize environmental and economic tradeoffs, which may be of interest to public sector decision makers faced with difficult and controversial choices.

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Table 6 . Distributions of waste flows to different waste management facilities in time period k.

k = l k = 2 k = 3

Lower Upper Lower Upper Lower Upper Alternative bound bound bound bound bound bound

Optimal solution To landfill (tld) 263 27 1 5 1 To WTE facility 1 (tld) 287 325 496 To WTE facility 2 (tld) 325 429 428 To landfill (%) 30.1 26.4 5.2 To WTE facility 1 (%) 32.8 31.7 50.9 To WTE facility 2 (%) 37.1 41.9 43.9

Alternative 1.1 (a = 0.02) To landfill (tld) 262 262 59 To WTE facility 1 (tld) 263 375 488 To WTE facility 2 (tld) 350 388 428 To landfill (%) 29.9 25.5 6.1 To WTE facility 1 (%) 30.1 36.6 50.0 To WTE facility 2 (%) 40.0 37.9 43.9

Alternative 1.2 ( a = 0.05) To landfill (tld) 262 262 59 To WTE facility 1 (tld) 288 376 488 To WTE facility 2 (tld) 325 388 428 To landfill (%) 29.9 25.5 6.1 To WTE facility 1 (%) 33.0 36.6 50.0 To WTE facility 2 (%) 37.1 37.9 43.9

Alternative 1.3 ( a = 0.08) To landfill (tld) 247 247 34 To WTE facility 1 (tld) 278 328 488 To WTE facility 2 (tld) 350 450 453 To landfill (%) 28.2 24.1 3.5 To WTE facility 1 (%) 31.8 32.0 50.0 To WTE facility 2 (%) 40.0 43.9 46.5

Alternative 2 (minimize flows to landfill) To landfill (tld) 237 237 34 To WTE facility 1 (tld) 288 325 500 To WTE facility 2 (tld) 350 463 441 To landfill (%) 27.1 23.1 3.5 ToWTEfaci l i ty l (%) 32.9 31.7 51.3 To WTE facility 2 (%) 40.0 45.2 45.2

Alternative 3 (minimize flows to WTE facility 1) To landfill (tld) 262 262 59 To WTE facility 1 (tld) 263 263 463 To WTE facility 2 (tld) 350 500 453 To landfill (%) 29.9 25.5 6.1 To WTE facility 1 (%) 30.1 25.7 47.5 To WTE facility 2 (%) 40.0 48.8 46.5

Alternative 4 (minimize flows to WTE facility 2) To landfill (tld) 262 298 59 To WTE facility 1 (tld) 288 375 488 To WTE facility 2 (tld) 325 352 428 To landfill (%) 29.9 29.1 6.1 To WTE facility 1 (%) 32.9 36.6 50.0 To WTE facility 2 (%) 37.0 34.3 43.9

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Acknowledgments This research has been supported by the Natural Sciences and Engineering Research Council of Canada. We are also grateful to the anonymous reviewers for their helpful comments.

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1996 huang et al a grey hop, skip, and jump approach: generating alternatives for expansion planning...

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