An integer goal programming model for hazardous waste treatment and disposal

An integer goal programming model for hazardous waste treatment and disposal Abdulaziz

S. Alidi

King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia The improper handling and disposal of hazardous wastes cause threats to human health and the environment. One reason for the improper handling and disposal of these wastes is that not much consideration is usually given to the logistical aspects of hazardous waste systems. In this paper an integer goal programming model is developed that takes into consideration the multiple goals and needs of many groups involved in managing andplanning hazardous waste systems. The model can easily be implemented and can be used to address many of the issues related to facility location, recycling, treatment, and disposal of hazardous wastes.

Keywords: goal programming, optimization

hazardous waste management,

Introduction Substantial quantities of hazardous materials are produced daily as by-products of manufacturing processes. Much of the waste is generated by the chemical industry, which produces different types of chemicals that are very important to any industrialized society. Jain’ reported that there are more than 60,000 commercial chemicals on the market, and the chemical industry creates about 1000 new chemicals each year. In addition, approximately 35,000 pesticides, 8600 food additives, and many cosmetic and drug ingredients are currently available. Some of the chemicals, pesticides, food additives, and other items may in the long run be hazardous to human health and/or the environment. At present, hazardous wastes can be treated, disposed of, or stored. On the basis of a study prepared for the U.S. Office of Solid Wastes,’ an affiliate of the Environmental Protection Agency (EPA), the quantity of hazardous wastes treated, disposed of, or stored in the United States during 1980 was estimated to be 247.5 million metric tons. Many mathematical models have been developed to study the treatment of hazardous wastes by physical, chemical, thermal, and biological processes.3 Additionally, mathematical programming techniques such as linear programming, dynamic programming, and network models have been used to aid in managing the logistical aspects, such as finding the optimal location and size of facilities, of hazardous and nonhazardous Address reprint requests Petroleum and Minerals, Received 1992

15 February

to Dr. Alidi at King Fahd Dhahran, Saudi Arabia.

1991; revised

0 1992 Butterworth-Heinemann

5 March

University

1992; accepted

of

2 April

location-allocation

analysis, mathematical

wastes. For example, Pierce and Davidson4 pointed out that the linear programming technique in general and the EPA Waste Resource and Allocation Program (WRAP) are useful in management of hazardous and nonhazardous wastes. They developed a linear programming model to help in the identification of a costeffective configuration of transportation routes, transfer stations, processing facilities, and secure long-term storage impoundments. A dynamic programming model was developed by Baetz et al.’ to determine the optimal sizing and timing of facility expansions and the utilization levels for both landfill and waste-to-energy facilities. The objective of their model is to minimize all development costs plus operating costs less any energy revenues. A mathematical formulation to implement regional hazardous waste management planning, which can be optimized by using either cost or risk penalty functions for all network activates, was developed by Jennings and Sholar.6 The model allows one to specify a realistic inventory of chemical waste types and numerous onsite or off-site treatment and disposal processes. Alidi and Freedman’ developed a three-stage mathematical model for handling and transporting stochastically generated materials. Their model includes cost of environmental constraints and the stochastic nature of the supply. In managing and planning the logistical aspects of hazardous waste systems, multiple goals, such as community and environmental control goals, that have different priorities have to be taken into consideration. Reviewing the literature reveals that this consideration has not been addressed properly. The goal programming technique, defined by Zeleny ,’ attempts to minimize the set of deviations from prespecified multiple goals, which are considered simultaneously but are

Appl. Math. Modelling,

1992, Vol. 16, December

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An integer goal programming

model: A. S. Alidi

weighted according to their relative importance. This is an appropriate technique for managing and planning the logistical aspects of hazardous waste systems. In this paper an integer goal programming model for hazardous waste treatment and disposal is developed and tested. The model is based on previously developed models but includes many of the aspects that are missing from old models such as the need to consider the multiple goals of many groups involved in the management of hazardous wastes systems.

eg indexes a type of treated hazardous waste to be el

eo i

5 NZ) j k

Model development It is assumed that different types of hazardous wastes are generated as by-products of manufacturing processes at different industrial plants scattered over a geographical area. The rates of generation of these wastes at each industrial plant vary with time because of the stochastic demand for the manufactured products. The future generation of these wastes can be estimated by using forecasting techniques.” These wastes can be hauled to hazardous waste treatment stations, where recyclable materials can be recovered and sold to different markets and the nonrecyclable materials can be treated and made ready to be sent to landfill sites or to incineration plants, where they can be disposed of. The organization managing the hazardous waste system would like to satisfy as many groups as possible, such as the public, local government, hazardous waste generators, and others who have multiple goals and needs. These multiple goals can be related to environmental control aspects such as air and groundwater pollution; objections to siting of hazardous waste treatment stations, landfills, and incineration plants at certain locations; restrictions related to traffic flow of vehicles to minimize the risk of road spills, traffic congestion, and noise; and finally the needs of hazardous waste generators to haul the wastes as they are generated because of storage limitations and to minimize the risk of fire and explosions at their plants. The organization would also like to determine the optimal locations, in terms of minimizing the total cost of managing the hazardous waste system over the future, of treatment stations, landfills, incineration plants, and recyclable material markets. Additionally, the organization would like the periodic (e.g., monthly) distribution of hazardous wastes, treated hazardous wastes, and recyclable materials to be as efficient as possible. Mathematical

formulation

of the model

m

4 r s

xX+

F

G

L

0

P

XQ XJ XK

Indices

XR

Assume a fixed number of time periods NT such that 1,2,. . . ,NT.

ef

646

indexes a type of hazardous -W

ArmI. Math. Modellina.

waste (1 5 ef 5

1992. Vol. 16. December

indexes a site for a landfill for disposing of treated hazardous wastes (1 I j 5 NJ) indexes a site for an incineration plant for disposing of treated hazardous wastes (1 5 k 5 NK)

indexes a mode of transportation for hazardous wastes, treated hazardous wastes, and recyclable materials (1 5 m 5 NM) indexes a type of operational policy for a mode of transportation such as owning, renting, and/or contracting (1 5 q 5 NQ) indexes a potential market (1 5 r % NR) indexes a site for a hazardous waste treatment station (1 5 s I NS)

Variables

The following set of indices, variables, and constraints is used in the mathematical formulation of the model.

t=

disposed of at a landfill (1 5 eg % EG) indexes a type of treated hazardous waste to be disposed of by an incineration plant (1 i el 5 EL) indexes a type of recyclable material to be sold to a market (1 5 eo I EO) indexes a generator of hazardous wastes (1 5 i

xs Y

underachievement variable overachievement variable a O-l integer variable representing a proposed site of a hazardous waste treatment station (0: station will not be installed at the proposed site; 1: station will be installed at the proposed site) a O-l integer variable representing a proposed site of a landfill (0: landfill will not be installed at the proposed site; 1: landfill will be installed at the proposed site) a O-l integer variable representing a proposed site of an incineration plant (0: plant will not be installed at the proposed site; 1: plant will be installed at the proposed site) a O-l integer variable representing a potential market for recyclable materials (0: recyclable materials cannot be sold to market; 1: recyclable materials can be sold to market) quantity of recyclable material to be hauled to a market quantity of hazardous waste to be hauled from a hazardous waste generator to a treatment station quantity of treated hazardous waste disposed of at a landfill quantity of treated hazardous waste disposed of by an incineration plant quantity of recyclable material sold to a market quantity of hazardous waste treated at a treatment station quantity of treated hazardous waste to be hauled to an incineration plant

An integer goal programming Z WA

quantity of treated hazardous waste to be hauled to a landfill site quantity of water extracted from hazardous wastes

NM

NQ NR Constants A B

CJ CK CR cs D

E

FNJ FNK FNR FNS H HE HF

HG M NB

NG

NH

NJ NK NL

fixed cost of installing a landtill at a site unit transportation cost of hazardous wastes from a waste generator to a treatment station capacity of a landfill capacity of an incineration plant capacity of a market capacity of a treatment station unit transportation cost of treated hazardous wastes from a treatment station to a landfill unit transportation cost of treated hazardous wastes from a treatment station to an incineration plant fixed number of landfills to be selected fixed number of incineration plants to be selected fixed number of markets to be selected fixed number of treatment stations to be selected percentage of water in hazardous wastes percentage of water in treated hazardous wastes to be disposed of at a landfill percentage of water in treated hazardous wastes to be disposed of at an incineration plant percentage of water in recyclable materials to be sold to markets unit net income obtained from selling a recv* clable material to a market number larger than the capacity of an incineration plant by one unit (to be used in the set of constraints for incineration plant selection) number larger than the capacity of a market by one unit (to be used in the set of constraints for market selection) number larger than the capacity of a landfill by one unit (to be used in the set of constraints for landfill site selection) total potential sites of landfills total potential sites of incineration plants a number larger than the capacity of a treat-

NS

QK $Y

QS u V W

model: A. S. Alidi

ment station by one unit (to be used in the set of constraints for treatment station selection) total modes of transportation total types of operational policies total potential markets total potential sites of treatment stations limitation on treated hazardous waste haulage to an incineration plant limitation on hazardous waste haulage limitation on recyclable material haulage to a market limitation on treated hazardous waste haulage to a landfill site fixed charge for providing a market with recyclable materials fixed cost of installing an incineration plant at a site fixed cost of installing a treatment station at a site

The objective function of the integer goal programming model consists of minimizing the variables from a prespecitied target. All the variables to be minimized are included in the objective function. For example, the term Pr(xl + , xl -) in the objective function gives the decision maker the choice of minimizing one of the two variables xl’ or xl but not both. As expressed in the objective function, there are ten goals to be considered. A list of these goals is presented in Table I. Each goal can be assigned a certain priority depending on the belief of the decision maker about that goal. For example, if the decision maker would like the main goal of managing the hazardous waste system to be that it not exceed the allocated budget, the first priority will be to minimize the variable xl + first. Based on the above definitions of indices, variables, and constants the mathematical representation of the objective function can be expressed as the minimization of all or some of the following, depending on the decision maker’s choice: [Pr(x 1+ , x 1 ) , Pr(x2,t,, x2,), Pr(x5J

, Pr(xSJ ,

Pr(xl&,), Wx14rJ, Pr(xlf&,,xW& Wx173:j,,, X’7lj.t), WxK:,,,, ~18,t,,,), Pr(xl9.~,,,,~19~,,,)1 subject to Hazardous

(1)

waste system cost limitation:

Appl. Math. Modelling,

1992, Vol. 16, December

647

An integer goal programming

Hazardous

model: A. S. Alidi

waste supplies:

x 2 c c XQi,s,ef.m,y,, + x2;

- x2; = Qi,,

i = 1,2,. . . ,NZ;

t = 1,2,. . . ,NT

(3)

4 m es s

Hazardous

waste minimization:

C C [C q

m

C

ef

Hi,,,,f,,,q,txQi,s,ef,m.q.t-

i

-

c

C

a

j

HEs.j.eg.m.q.fZs,j.eg,m.q.f-

r

1,2,...

A

Treatment

XQi,s.eAm,q,r

-

,NS;

=

0

?= 1,2 ,...,

NT

(5)

NS;

(6)

station selection:

NZ,Ff - C C C x C f

q

m

ef

XQi,s,eAm,q,tZ

0

q

m

cl

1,2,...,NS

Ys,k,el,m,q,t

C C C C ‘s,j,eg,m,q,t - xJ,,< = O 4 m eg s NJ; t = 1,2,. . . , NT j= 1,2,...,

(8)

NJ;

t = 1,2, . . . , NT

(9)

Landfill site selection:

k= 1,2,...,

XK,,, + X11,, = k= 1,2,...

=

o

NK;

1,2,...,NT

t=

(11)

CKk,,

,NK;

t= 1,2 ,...,

NT

(12)

plant selection: f

q

m

el

Ys,k,el,m.q.t

Lo

s

k= 1,2,...,NK

(13)

Quantities of recyclable material arriving at markets: c 2 x 2 Z-‘s,r,eo,m,q,t - XR,, = 0 4 m eo s Y= 1,2,... ,NR; t= 1,2 ,...,

XR,,, + x14, = CR,,, r= 1,2,..., NR;

NGO,

NT

(14)

Goal No.

I

(15)

q m eo 3

Allocated budget Hazardous waste supplies Treatment station capacities Landfill site capacities Incineration plant capacities Market capacities Hazardous waste haulage Treated hazardous waste haulage to landfill sites Treated hazardous waste haulage to incineration plants Recyclable material haulage

1992, Vol. 16, December

1,2,...,NR

r=

C C C Related to

Appl. Math. Modelling,

t= 1,2,...,NT

- c 2 2 c c Ps,r,eo,m,q,r2 0

(10)

Table 1. A list of goals considered

648

xKk,,

plant capacities:

Environmental

10

waste arriving at

Market selection:

NH,C, - 2 x x x C Zs,j,eg.m,q,t20 f 4 m eg s j= l,Z,...,NJ

9

(4)

Market capacities:

Landfill site capacities:

1 2 3 4 5 6 7 8

-

. . ,NT

s

(7)

waste arriving at

t = I,&.

hazardous

N&J% - 2 2 c 2 x

i

hazardous

XJj,, + ~8jrt = CJj,, j= 1,2,...,

Quantities of treated incineration plants:

Incineration

s=

Quantities of treated landfill sites:

s = 132, . . . ,NS;

Incineration t= 1,2,...,NT

~~s,~,e~.m,q,r~s,~,e~,m,q.t

k

c x c c

station capacities:

x&s,,, -I-x5,, = cs,,, s= 1,2,..., Treatment

Xss,r

C

el

Quantities of hazardous waste treated at each treatment station: F z z C

C

HGs,,,eo,m,q,fPs,r,eo,m,s.r 1= 0

x

eo

C

4

m

(16)

control:

XQi,s,e~.m,q.t

+

x16,,,

-

x16G.t

=

QNi,s.r

ef

foreach i, s, t 7 x C Zs,j.tx,m.q.r + x17,j.t - xl7,tj,, = m

(17)

QSs,j.r

ek-

for each s,j, t T c t;:

Ys.k,e,.m,q.t

+

x18,k,,

-

xls:k.t

=

(18)

QK(s.k.r

m

for each s, k, t T C 2 m

Ps,j.eo,m,q,t

+

X193~r.t

-

X19Zr.t

=

(19)

QR3,r.t

e*

foreachs,

r, t

(20)

An integer goal programming

Siting limitations: x F+NS s

(21)

x

Gj5N~

(22)

x

L,sNK

(23)

within live different priority structures for the example problem, as follows:

are considered

Priority Structure 1

20,s~~ Selection

model: A. S. Alidi

(24)

Prl:

Pr2:

of fixed number of sites:

2 F+-FNS

(25)

x

G,zFNJ

(26)

c L,rFNK k

(27)

2 O,ZFNR

(28)

Pr3:

Pr4:

Pr5:

Minimize the overachievement of the flow of hazardous wastes xl 6it,,,, treated hazardous wastes ~17,:~,,, xlS,F& and recyclable materials x19,t,,, as defined by equations (17)-(20) inclusive Minimize the underachievement of collecting the hazardous waste quantities x2,7, as defined by equation (3) Minimize the underachievement of the treatment station capacities x5,, as given by equation (6) Minimize the underachievement of the capacities of landfillsx8jT,, incineration plants xl I&, and marketsx14,1t as defined by equations (9), (12), and (15), respectively Minimize the overachievement of hazardous waste system cost limitation xl + as given by equation (2)

Priority Structure 2

Overall material balance:

Prl:

- c c x ~~~~~~~~~~~~~ - WA = 0 > s r co

(291

0- 1 integer variables:

Pr2:

Pr3:

F, = Oor 1

s=

1,2,...,NS

Gj=Oorl

j=

1,2

(30)

. . >NJ

L, = Oorl

k= I,;;...

O,=Oorl

r= 1,2,...,NR

,NK

(31)

Pr4:

(32) (33) Pr5:

Model testing A hypothetical but representative example problem is used to illustrate use of the model. It is assumed that there are only four generators of hazardous wastes (NZ = 4), three possible sites for treatment stations (NS = 3), two possible sites for landfills (NJ = 2), two possible sites for incineration plants (NK = 2), and two possible markets for recyclable materials (NR = 2). Additionally, it is assumed that there is only one type of hazardous waste (EF = l), only one type of treated hazardous waste to be disposed of at a landfill (EG = l), only one type of treated hazardous waste to be disposed of by an incineration plant (EL = I), and only one type of recyclable material (EO = 1). It is also assumed that there is only one type of hazardous waste vehicle available (NM = 1) and only one operational policy for vehicles (NQ = 1). The model is simulated for only one future cycle (NT = 1). Only five goals

Minimize the overachievement of the flow of hazardous wastes x16&,, treated hazardous wastes x17,&,, x18&,, and recyclable materials x19,f,,, as defined by equations (17)-(20) inclusive Minimize the underachievement of collecting the hazardous waste quantities x2,7* as defined by equation (3) Minimize the overachievement of hazardous waste system cost limitation xl + as given by equation (2) Minimize the underachievement of the capacities of landfills xSJyt,incineration plants xl lk,,, and markets x14, as defined by equations (9), (12),and(E),respectively Minimize the underachievement of the treatment station capacities x5,, as given by equation (6)

Priority Structure 3 Prl :

Pr2: Pr3:

Pr4:

Minimize the overachievement of the flow of hazardous wastes x16&,, treated hazardous wastes x17,tj,,, x18,:,,,, and recyclable materials XI?&,, as defined by equations (17)-(20) inclusive Minimize the overachievement of hazardous waste system cost limitation xl + as given by equation (2) Minimize the underachievement of the capacities of landfills x8,~~,incineration plants xl l;,,, and markets x14, as defined by equations (9), (12), and(l5), respectively Minimize the underachievement of collecting the hazardous waste quantities x2, as defined by equation (3)

Appl. Math. Modelling,

1992, Vol. 16, December

649

An integer goal programming

Pr5:

model: A. S. Alidi

Minimize the underachievement of the treatment station capacities x5,, as given by equation (6)

Priority Structure 4 Prl :

Pr2:

Pr3:

Pr4:

Pr5:

Minimize the overachievement of the flow of hazardous wastesxl6&,, treated hazardous wastes X17,tj,t, x18,t,,,, and recyclable materialsx19,t,,, as defined by equations (17)-(20) inclusive Minimize the underachievement of the treatment station capacities x5,, as given by equation (6) Minimize the underachievement of collecting the hazardous waste quantities x2; as defined by equation (3) Minimize the overachievement of hazardous waste system cost limitation xl + as given by equation (2) Minimize the underachievement of the capacities of landfills x8]rC,incineration plants xl l/i,,, and markets x14, as defined by equations (9), (12), and (IS), respectively

Priority Structure 5 Prl :

Pr2:

Pr3:

Pr4:

Pr5:

Minimize the underachievement of collecting the hazardous waste quantitiesx2, as defined by equation (3) Minimize the overachievement of hazardous waste system cost limitation xl + as given by equation (2) Minimize the underachievement of the treatment station capacities x5,, as given by equation (6) Minimize the overachievement of the flow of hazardous wastes x16&,, treated hazardous wastes x17,tj,,, ~18~:~.,,and recyclable materials x19,7,,, as defined by equations (17)-(20) inclusive Minimize the underachievement of the capacities of landfills ~8~;(,incineration plants xl I,& and markets x14, as defined by equations (9), (12),and(l5),respectively

The developed integer goal programming model can be solved by using the modified simplex procedure, and any available linear programming code can be used to solve it. The solution procedure entails the partitioning of the objective function according to priority levels

Table2.

and the sequential solution of the resultant linear programming models. The solution obtained at each priority level is used as a constraint at the lower level. This sequential goal programming algorithm is outlined by Ignizio” and Kornbluth.” The model has been solved for each of the five priority structures by using the Linear Interactive and Discrete Optimizer (LINDO) computer package. l2 The priority goals that are fully achieved and the results of the siting of treatment stations, landfills, and incineration plants and the locations of markets are presented in Table 2. The results shown in Table 2 indicate that the model is highly insensitive to the order in which the various goals are considered. After determination of the best sites for treatment stations, landfills, and incineration plants and locations of markets, the periodic flow of hazardous wastes, treated hazardous wastes, and recyclable materials throughout the hazardous waste system network can be determined by using a standard linear program similar in structure to the above mixed-integer goal program. The objective function of this linear program is to minimize the transportation cost of hazardous wastes, treated hazardous wastes, and recyclable materials. A summary of typical results of the periodic flow considering the three sites of treatment stations, site number 1 for the landfill, site number 1 for the incineration plant, and the two markets is shown in Table 3. Based on the hypothetical data used, the total transportation cost is found to be $556.56. Conclusions The manufacture of many chemicals and other types of products results in most cases in the generation of substantial quantities of hazardous/toxic materials. Much of the waste is generated by the chemical industry, which produces different types of chemicals needed by any industrialized society. At present there is much concern about hazardous waste generation and disposal. There is also a need for establishing effective environmental control measures. In managing and planning the logistical aspects of hazardous waste systems, multiple goals with different priorities must be taken into consideration. In this paper an integer goal programming model is developed. The model can be used to address many of the problems involved in the management and control of hazardous waste systems such as determining the optimal

Results of the siting of facilities Site number selected for

Priority structure number

Priority goals fully achieved

Treatment stations

Landfills

Incineration plants

Markets

1 2 3 4 5

I,5 I,3 I,2 1.4 I,3

1,283 lf2.3 lr2.3 1,2,3 1,2,3

1 1 1 1 1,2

I,2 1 1 1 1

1,2 I,2 I,2 182 1,2

650

Appl.

Math. Modelling,

1992, Vol. 16, December

An integer goal programming Table3.

Typical results of the

model:

A. S. Alidi

periodic flow of hazardous wastes, treated hazardous wastes, and recyclable materials

= 90.0 units of hazardouswaste type 1 to be hauled from hazardouswaste generator 3 to treatment station 1 by vehicle type 1 XQ3.1.1.1.1.1 under operational policy 1 during period 1 XQ 1.2.1.1.1.1 = 44.0 units of hazardous waste type 1 to be hauled from hazardous waste generator 1 to treatment station 2 by vehicle type 1 underoperational policy 1 during period 1 = 75.0 units of hazardous waste type 1 to be hauled from hazardous waste generator 2 to treatment station 2 by vehicle type 1 XQ2.2.1.1.1.l under operational policy 1 during period 1 XQ~.~,~.u., = 20.0 units of hazardous waste type 1 to be hauled from hazardous waste generator 3 to treatment station 2 by vehicle type 1 under operational policy 1 during period 1 X0 ,,3,1,,,,,,= 56.0 units of hazardous waste type 1 to be hauled from hazardous waste generator 1 to treatment station 3 by vehicle type 1 under operational policy 1 during period 1 XQ~.,.,.,.,., = 80.0 units of hazardous waste type 1 to be hauled from hazardous waste generator 4 to treatment station 1 by vehicle type 1 under operational policy 1 during period 1 Z 1.1.1.1.1.1 = 55.6 units oftreated hazardous waste type 1 to be hauled from treatment station 1 to landfill site 1 by vehicle type 1 under operational policy 1 during period 1 Z 2,,,,,,,,,, = 34.4 units of treated hazardous waste type 1 to be hauled from treatment station 2 to landfill site 1 by vehicle type 1 under operational policy 1 during period 1 Y2,,,,,,,,,, = 48.6 units of treated hazardous waste type 1 to be hauled from treatment station 2 to incineration plant 1 by vehicle type 1 under operational policy 1 during period 1 P 1.1.1.1.1.1 - 50.0 units of recyclable material type 1 to be hauled from treatment station 1 to market 1 by vehicle type 1 under operational policy 1 during period 1 P9.2,1.1,1,1 = 40.0 units of recyclable material type 1 to be hauled from treatment station 3 to market 2 by vehicle type 1 under operational policy 1 during period 1

locations of treatment transfer stations, landfill sites, incineration plant sites, and the sites of potential markets for recyclable materials. The multiple and conflicting goals of many groups involved in the management of hazardous waste systems are also considered in the model. The model has been tested by using a hypothetical but representative example problem, and the results obtained show that the model is a viable tool for managing and planning hazardous waste systems. The mathematical structure of the model is not complicated and can easily be used by managers and planners involved with the management of hazardous waste systems.

2

3

4

ardous waste management. J. Environmenfa[ ASCE 1982,lOS(EES), 1014-1026 5

6

Acknowledgments

Jain, R. K. Overview of hazardous/toxic ofHazardous

Div.

Baetz, B. W., Pas, E. I. and Neebe, A. W. Trash management: Sizing and timing decisions for incineration and landfill facilities. Interfaces 1989,19(6), 52-61 Jennings, A. A. and Sholar, R. L. Hazardous waste disposal network analysis. J. Environmental Engrg. 1984, 110(2), Alidi, A. S. and Freedman, R. W. A three-stage mathematical model for handling and transporting stochastically generated materials. Appl. Math. Modelling 1988,12(5), 495-502 Zeleny, M. Multiple Criteria Decision Making. McGraw-Hill, New York, 1982, pp. 281-306 Kmenta, J. Elements of Econometrics, 1st ed. Macmillan, New York, 1971 Ignizio, J. P. A review of goal programming: A tool for multiobjective analysis. J. Oper. Res. Sot. 1978.32,1109- 1119 Kornbluth, J. A survey of goal programming. OMEGA 1973,1, 193-205 Schrage, L. LINDO: Linear Interactive and Discrete Optimizer. The Scientific Press, Palo Alto, Calif., 1981

References Land Disposal

Engrg.

325-342

The author would like to thank the Department of Civil Engineering at the University of Newcastle upon Tyne, U.K., and in particular Professor David J. Elliot, for hospitality and support during the author’s residence as a guest faculty member during the summer of 1990, when this paper was prepared.

1

Schofield, and R. K. Jain. Ellis Horwood, Chichester, England, 1988,~~. 11-18 National survey of hazardous waste generators and treatment, storage and disposal facilities regulated under RCRA in 1981. Prepared for the U.S. EPA, Office of Solid Waste, by Westat, Inc., Rockville, Md., USA, 1984 van den Oosterkamp, P. F., Blomen, L. J., Ten Doesschate, H. J., Laghate, A. S. and Schaaf, R. Dechlorination of PCBs, dioxins and difurans in organic liquids. Hazardous Waste Management, eds. S. P. Maltezou, A. K. Biswas and H. S. Sutter. Tycooly, London, 1989, pp. 90- 103 Pierce, J. J. and Davidson, G. M. Linear programming in haz-

waste management.

Waste, eds. J. R. Gronow, A. N.

Appl.

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651