A computational model for solid waste management with applications H. W. Gottinger Institute of Management Science. The University of Maastricht, PO Box 591, Maastricht, The Netherlands and Department of Systems Engineering, University of Virginia, Charlottesville, VA 22901, USA (Received June 1985)
A model for regional solid waste management is described as a network flow problem and a special purpose algorithm is developed. The model is applied to waste management and facility siting decisions in the Munich Metropolitan area, West Germany. Key words: solid waste management, facility siting, network flow problems, public sector planning Solid waste management today is made difficult and costly by the increasing volumes of waste produced, the need to control what are now recognized as potential serious environmental and health effects of disposal, the lack of land in urban areas for disposal purposes, partly due to public opposition to proposed sites. Waste management, once strictly a local and private sector matter, now involves regional, state and federal authorities. Various legislative initiatives and procedures have been activated within the last few years in the leading industrial countries.' The two primary reasons to have solid waste management on a regional instead of on the level of local towns and cities, which is the current practice, are economical and technical and political feasibility. One could define 'economies of scale' as declining average cost as scale increases and empirically establish the existence of economies of scale for transfer station, incineration and landfill operation. This has been established on the basis of various case studies. 2 Some of the 'core' cities of metropolitan areas often do not have adequate landfill capacity to dispose of their waste. This necessitates transporting the waste over long distances. If management is carried out on a regional basis, the 'core' cities could use the site capacities of peripheral counties of the metropolitan area. With increasing emphasis on resource recovery and the scarcity of land for disposal in recent years, many cities are turning towards incineration with power or steam generation, recovery of useful commodities from
330
Appl. Math. Modelling, 1986, Vol. 10, O c t o b e r
the waste (recycling), utilization of waste as a substitute for fuel in power plants, composting and other advanced technologies. Needless to say, these plants cannot be operated by small local governments. It should be noted here that, although regional management has distinct advantages, the communities asked to participate are usually reluctant to have waste from another part of the region disposed in their area and are unwilling to host waste processing facilities. There are many political problems associated with regional management, which, though important, are not of primary concern here. A broad class of models for regional waste management has been developed in the US. 3 There is a strong affinity between the present model and that of Walker et al., 4" the algorithm presented here being a modification of that of Marks. s Further references to related work include Clark and Gillian 6 and Dee and Liebman. 7 The type of models developed here and their application to a specific regional waste management system in the State of Bavaria (West Germany), should not be misjudged as an ultimate decision maker, but, rather, be used as an effective tool for efficient management. When the planner has made decisions regarding political, social and other practical aspects of the problems which are difficult to quantify, the models will help choose the most economical system.
0307904x/86/0533009/$03.00 © 1986 Butterworth & Co. (Publishers) Ltd
Solid waste management with applications: H. W. Gottinger
Modelling waste management systems The total system of waste management can almost be decoupled into two major subsystems. One is the solid waste collection system and the other is the regional management system. This is under the provision, in this particular case, that the present mix of waste management facilities is maintained for a given planning period and that there are no rapid changes in collection and treatment technologies. The assumption of decoupling is made for conceptual purposes, frequently there seem to be interactions among both systems and under these circumstances, a waste management solution may be vastly improved by modelling both systems as one. 8 A solid waste collection system is concerned with the collection of waste from sources, routing to trucks within the region, the frequency of collection, crew size, truck sizes, number of operating trucks, transportation of collected waste to a transfer station, an intermediate processing facility or a landfill and a host of other problems. The regional management concerns itself with the selection of the number and locations of transfer stations, intermediate processing facilities, landfill sites, their capacities, capacity expansion strategies and routing of the waste through the facilities to ultimate disposal on a macroscopic level. While the waste is generated all over the region, it is assumed that it is generated at point sources, each point source representing a neighbourhood. The routing of waste from these point sources to ultimate disposal is considered a part of regional management. It is true that solid waste collection is an important facet of overall management and is interrelated to the regional management, but the coupling is weak enough to analyse the regional management system separately. This paper deals with modelling such regional management systems.
Mathematical models The general problem of the static model of the waste management system can be described as follows: given the potential locations of intermediate processing facilities and landfills, the locations and capacities of existing facilities, the cost structurestransportation, processing and fixed costsand the quantities of waste generated at the sources, determine which facilities should be built and how the wastes should be routed, processed and disposed of, so that the overall cost of the system is minimized. A mathematical formulation of the problem is given below.
General mathematical model Problem 1 min
~
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~
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Z
~J<<'Cjyi
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where the symbols used denote:
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flow from i to j Oor l if j e Bi for i E F 1 if facility j is already in existence set of sources quantity of waste generated at source i set of intermediate pseudofacilities yet to be built and intermediate facilities already in existence L set of pseudofacilities at landfill sites that may be 'built' and landfill facilities that already exist F locations of potential facilities Bi approximations to facility cost curve at location i tq transportation cost from i t o j Pj processing cost at (pseudo)facilityj ai weight reduction factor at (pseudo)facilityj Fj fixed cost at (pseudo)facilityj Ck capacity of (pseudo)facility k The first two terms of the objective function (1) give the total transportation costs, the third term gives the costs of processing at facilities and the fourth gives the total fixed costs incurred. Equation (2) implies that all the waste generated at all sources should be shipped out. Equation (3) is a balance equation between the input and output at intermediate facilities. Equation (4) implies that at no facility can the amount of waste treated be greater than its capacity. At any potential facility location, only one pseudofacility can be built and this condition is reflected in inequalities (5) and (6). When represented in a graphical form, the flow network looks like the one shown in Figure 1. There are three sets of nodes: the sources S, the intermediate facilities I including the ones to be designed and the ones that already exist and the landfill sites L including the ones to be 'built' and the ones that already exist. There are arcs between S and I, S and L, and I and L but there are no arcs between nodes of the same set. The nodes of I and L have capacity restrictions, costs and 01 integer variables associated with them. In addition, flow is not necessarily conserved at intermediate nodes 1. In other words, if the input to node i e I is x, the output from the node is ape, where 0 <~ai <. 1. s
I
L
j~IUL
subject to:
Z
fq=Gi
(2)
ieS
uelUL
Z ieS
fiiai Z kEL
fJk=O
jel
(3)
Figure I Flow network in graphical form
Appl. Math. Modelling, 1986, Vol. 10, October
331
Solid waste management with applications: H. W. Gottinger
S
I
al ~
['
r"
(O,C,P) (0,=,,0)
L
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this additional arc will become apparent later on. The capacity and processing cost of a landfill are represented on the arc It introduced in the enlarged network for each I e L. Having described the network to be dealt with, the solution procedure for solving the mathematical program in equations (1)(7) can be outlined. The solution procedure is a branch and bound technique very similar to the one developed by Marks, 5 but a special purpose algorithm has been designed to solve the subproblems of each node of the branching tree.
F
O
/
/
/
V
Structure of facility costs
/ Flow Figure3
C = capacity
Costfunctionof pseudofacility
For example, if 100 ton of waste is sent to an incinerator, the output may be 20 ton of ash. To represent the capacities and costs at the nodes, and to represent the condition that all the waste generated in source tre S must be disposed of, the network is enlarged with a super source 0, a super sink t and additional arcs as shown in Figure 2. The arcs (O,o~, i e S, have a lower bound of zero and an upper bound equal to the amount of waste generated at source o'i. The costs on the arcs (0,oi), i ~ S, are set to zero. For each intermediate facility, three nodes and two arcs are introduced. The capacity and processing cost are introduced in the arc (il,i[) as shown in Figure 2. The reduction factor ai is introduced in another arc (i i ,i'{) as ki'i'[, the capacity of which is infinity and the cost is zero. This will be the only type of arc which has a reduction associated with it. The advantage of having
332 Appl. Math. Modelling, 1986, Vol. 10, October
Various types of cost function have been discussed in other studies, most notably by Walker et al. 4 but there are two different cases to be considered, whilst discussing the facility costs for the present model. One is the case when a facility already exists in the region and has a finite capacity. It is assumed here that such a facility has a fixed cost independent of flow through the facility and a linear processing cost. In the second case, the facility does not exist, but should be designed. The best strategy would be to build a facility with a capacity equal to the flow through the facility, when dealing with static models. This is based on the assumption that once the facility is built, the flow through it will be sufficiently large that any excess capacity would be uneconomical. The cost function is approximated by linear segments and the facility is assumed to be equivalent to n pseudofacilities, each with a fixed charge and a linear processing cost. A 01 integer variable is associated with each pseudofacility, with the condition that only one of them may be built. Actual plots are given in a later section. It is clear that if the integer variables y/s are known, then the mathematical program (equations (1)(7)) is a minimum cost network flow problem and appropriate network algorithms can be developed to solve the problem. This special structure of the problem is exploited in the solution procedure for the problem. The cost
Solid waste management with applications: H. Wo Gottinger curve of a pseudofacility is shown in Figure 3. At the beginning of the branch and bound procedure, the cost function of every pseudofacility is replaced by a modified cost MC given by MC= F/C+p as shown in Figure 3. The capacity of each pseudofacility is unbounded but is taken to equal to the total amount of the waste Table 1
Numbering of facilities Facility number
Facility type
Location
Landfill Landfill Landfill Landfill Landrill Landfill Incinerator Incinerator Incinerator Incinerator* (pyrolysis) Incinerator* Combination facility
Bockhorn/Erding Grosslappen Dachau Landsberg/L Ebersberg Gallenbach M0nchenNord M0nchenSLid Neufahrn/Freising Erding Weilheim/Schongau Geiselbullach
1 2 3 4 5 6 7 8 9 10 11 12
* In projection: planning horizon 1990
Table2
Source number 14/1:
Waste generation data (ton day 1 Quantity
Source number
1 2 3 4 5 6
688 1465 769 383 360 188
W3:
7 8 9 10 11 12
150 302 156 281 355 188
W4:
13
158
14 15
586 271
Ws:
16 17 18 19 20
368 16 109 78 227
14/6: 34 35 36 37 38
Table 3
Quantity
21 22 23 24 25
50 76 120 140 140
26 27 28 29 30 31 32
7 96 168 6 355 355 355
Source number 39 40 41 42 43
Quantity 358 356 419 527 126
to be sent through the network from the sources. Since this is the maximum possible flow through the pseudofacility, one can, for algorithmic purposes, treat the capacity of the pseudofacility as infinite. Setting the capacity of a pseudofacility is discussed further while dealing with different types of static models. At the moment it suffices to say that the capacity of a pseudofacility should be set at the maximum of the total amount of waste generated in the region. Setting a higher capacity would result in the decrease of modified cost and, consequently, a reduced lower bound for the optimal solution to the problem at each node of the branching tree. If the concave cost function of a facility is approximated by n linear segments, then the pseudofacility with the highest fixed cost and the least variable cost will have the lowest modified cost and, hence, if any flow is sent through a pseudofacility at a particular location, it would be sent through this pseudofacility. The other pseudofacilities, therefore, may be dropped from the network, reducing the number of nodes and arcs in the network. As will become clear, at any node of the branching tree, there will be only one pseudofacili.ty at each location in the network. With modified costs given to the pseudofacilities with the highest fixed cost and the least processing cost at each location and with the integer variables of these pseudofacilities set to 1 (the integer variables corresponding to other pseudofacilities are set to zero), problem 1 can be restated as:
Problem 2 rain
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keeL*
+ ,y___,
~.
fMik
(8)
j~l*
y_, P, ,y___,
J~ I ' U L °
jel'uL"
i~S'uI"
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fii = Gi
ieS
f~iai ~
f]a=0
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i~S
33
f~itij+ ~
Z j~I'UL *
f,.k <~Ck
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jcI*
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k 6 L*
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i~Sul*
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294 828 360 458 468
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(12)
ieS
f,j I>
Reduction factors at facilities
Waste type
Facility type
Reduction (%)
Municipal refuse Industrial waste Prunings Farm refuse Refuse from canning and preserving industry Demolition and construction debris
Incinerator Incinerator Incinerator Incinerator Incinerator Anaerobic digestion
90 80 95 95 95 50
(13)
where 1" is the set of pseudofacilities that have the highest fixed costs and the least processing costs at potential intermediate facility sites and the intermediate facilities already in existence. L* denotes the set of pseudofacilities that have the highest fixed costs and the least processing costs at potential landfill sites and those already in existence. Looking at the constraint sets in problems 1 and 2, it is clear that every feasible solution of problem 2 is also a feasible solution of problem 1. The constraints of problem 2 are more relaxed than those of problem 1. In addition, all the modified costs are lower than the costs of problem 1 and, hence, the cost of an optimal solution to problem 2 is a lower bound on the cost of an optimal solution to problem 1.
Appl. Math. Modelling, 1986, Vol. 10, October
333
Solid waste management with applications: H. W. Gottinger 600
25C
500
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( = 250 doys)
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8
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IOOO 2000
I
I
3000 4000 Flow(Ion doy "1)
I
5000
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!
6000
7000
Figure 4 Landfill operation: nonlinear part of cost function
=0 Vorioblecost =DM 2200 Ion"1doyI f~, oneyeor (= 250doys) Fixed cost
I
0
I
,50
I
I00
I
150 200 Flow (Ion dayI ) Figure 6 Anaeorbic digestion plant as part of combination facility: nonlinear part of cost function
At any general node of the branching tree, the mathematical program to be solved may be written as follows: define T as the set of integer variables set to 0 or 1 and T as the set of integer variables not yet decided upon. T LJT is the complete set of integer variables and
1000(3 800(3 A
T fl T is a null set.
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4000
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I
I
I
3000 4000 5000 Flow (ton doyI)
I
6000
I
+ ~.FiYJ+ jeTuT
7000
~.PiZfo jeluL
i
subject to:
Incinerators: nonlinear part of cost function
Z If upon solving problem 2, all the facility flows are either zero or equal to the capacity, then an 'exact solution' has been attained and the optimal solution of problem 2 is an optimal solution of problem 1. If the flow in a pseudofacility, however, is less than the capacity but greater than zero, the cost of this solution is less than the cost of the optimal solution of problem 1. One can now branchon this facility. One at a time, each pseudofacility is assumed to be ready built and the problem is solved with its true linear processing and fixed costs. In addition, another problem in which it is assumed that there exists no facility at this location, is also solved. All the solutions are placed on a master list and the least cost solution is taken. If the solution is such that, for all pseudofacilities with Yi values yet to be chosen, the flow through the pseudofacility is either zero or equal to the capacity (in other words, if the solution is exact), an optimal solution to problem 1 has been obtained. Otherwise, one can branch again on a pseudofacility which has a flow greater than zero but less than the capacity, as described above. The procedure terminates because all the different combinations of 01 integer variables will be formed by the branching routine.
334 Appl.Math.Modelling,1986,Vol.10,October
f,J= G,
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(15)
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(17)
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j e T fl l
(20)
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Instead of solving problem 3, a lower bound to the cost of the optimal solution to the problem can be
Solid waste management with applications: H. W. Gottinger obtained by solving a problem wherein the pseudofacilities for which Yi values are unknown are replaced by pseudofacilities with the least modified costs. The yj values of other pseudofacilities are set to zero. This approach is similar to solving problem 2 instead of problem 1.
General algorithm Step 1 Input data m n Bi
number of locations of intermediate facilities and landfills number of sources number of straight lines used to approximateconcave cost function of a facility at site i, number of pseudofacilities at site i
m* total number of pseudofacilities, ~
Bi
i=l
F~ fixed cost of pseudofacilityj (DM) P~ processing cost of pseudofacilityj (DM ton I) Fr highest fixed cost of all pseudofacilities located at site i= max/= 1.... n, {Pj} Pr lowest processing cost of all pseudofacilities = mini =1 .... B, {Pi} (DM ton I) i* pseudofacility having fixed cost F~. and processing cost Pitit transportation cost from i to j (i • S, j • / ) (i • S, j • L) and (i • l, j • L) ( D M t o n I) S set of sources T set of intermediate facilities L set of landfills Step 2 Define: T 1
set of integer variables y~. set to zero or one set of all integer variables which are not decided upon set of all integer variables
TUT TnT T' 7~J
set of integer variables set to 1 set of integer variables set to 0, T' 13 7~ = T
Step 4 Use one of the special purpose algorithms for this type of problem to solve the minimum cost flow network problem. If there is no feasible solution and this is the first time through, then stop. No feasible solution exists for problem 1. If it is not the first time through, no branching will be carried out from this node and hence the information about this need not be stored. Go to step 6 if this is the last of the problems set up in step 7, otherwise continue with the next problem defined in step 7 and go to step 3. If an optimal solution is obtained (the solution cannot be unbounded because the problem is to send a finite amount of flow through the network and even if the costs are .negative, the minimum cost of the solution cannot be infinitely negative). Compute: Cost = minimum cost of network problem
+ZF, jeT'
Compute also the upper bound to the problem defined on the node as: Upper bound = Cost + Z
F/
j e TG
where G is the set of facilities which have flows greater than zero.
Step 5 Test the 'exactness' of the solution. A solution is 'exact', if in all facilities belonging to the set T, the flow is either zero or equal to the capacity. If the solution is not exact, then compute the ratio of the total flow into the facility to the capacity of the facility, for all facilities belonging to set T and having flow between zero and the capacity. Choose the facility having this ratio closest to 0.5. Store all the information about the problem. If this is the last of the problems defined in step 7, go to step 6, otherwise continue with the next problem defined in step 7 and go to step 3.
Step 6 Select the problem which has the least storage cost and check if it is exact. If it is, terminate as the optimal solution has been found. If not, go to step 7.
Initialize:
T
set of all integer variables, corresponding to pseudofacility i* for each location i, i = 1 . . . . . m complement of T, T = 7~ U T' = ~J
T
Step 3 For all the pseudofacilities which belong to T, take the real processing and fixed costs and set the modified cost as:
MCj Pi
•
jeT'
and capacity as: Cj=0
V
j • T°
Step 7 The least cost problem chosen from the master list in step 6 has a facility which was chosen to be branched upon in step 5. Set up problems a I . . . aBi and a 0, where Bi is the number of straight lines used to approximate the concave cost function of the facility. Problems a l . . . aBi would have the facility location i, the corresponding pseudofacility with a fixed charge and linear processing cost introduced into the set T; the corresponding Yi is set to one. For problem a0, all the pseudofacilities at location i will have their Yi values set to zero. Repeat steps 35 and then go to step 6.
A case study for the Munich Metropolitan area Several areas adjacent to the city of Munich (Landeshauptstadt Mtinchen) form the Munich Metropolitan
Appl. Math. Modelling, 1986, Vol. 10, October
335
Solid waste m a n a g e m e n t with applications: H. W. Gottinger Table 4
Transportation costs (DM (250 ton) 1
Destination/
Origin
F1
F2
F3
F4
F5
F6
F7
Fa
/:9
/:10
Fli
/:12
S1 S2 S3 S4 S5 S6 S7 Sa S9 Sl0 Sll Sl~ S13 S14 S15 S16 S17 Sis S19
218 157 131 83 115 184 277 226 278 280 369 549 218 157 115 151 220 226 283 549 157 184 277 384 384 157 184 220 283 384 384 386 131 218 157 131 115 184 220 226 280 369 384 104 150 276 223 550
375 369 280 223 221 131 223 383 549 374 383 562 375 369 277 156 183 383 377 562 369 131 223 559 378 369 131 183 377 559 378 380 280 375 369 280 221 131 183 383 374 383 559 278 182 220 380 563
383 375 368 278 276 278 130 552 550 283 372 377 383 375 281 280 156 552 286 377 375 278 130 381 280 375 278 156 286 381 280 220 368 383 375 368 276 278 156 552 283 372 384 281 283 150 549 378
154 183 223 280 284 381 554 104 130 280 224 369 154 183 277 371 384 104 283 369 183 381 554 286 549 183 381 384 283 286 549 556 223 154 183 223 284 384 384 104 280 224 286 224 572 554 130 371
280 281 289 372 377 554 378 226 221 180 131 217 280 284 368 383 369 226 154 217 281 554 378 182 283 281 554 369 154 182 283 371 284 250 284 284 377 554 369 226 180 131 182 286 384 378 277 184
281 283 372 378 383 554 369 370 254 183 180 102 281 283 374 550 374 369 180 102 283 554 369 131 156 283 554 374 180 131 156 218 372 284 283 372 383 554 374 369 183 180 131 374 552 369 372 115
2135 2158 2205 2176 2141 2070 2010 2073 2108 2067 2011 1911 1894 1918 1948 1894 1826 1832 1774 1670 1422 1335 1274 1178 1003  913  824  820  769
2014 2068 2109 2160 2176 2141 2014 2007 1921 1920 1911 1731 1773 1827 1897 1980 1830 1766 1676 1491 1322 1405 1278  999 1171  822  895  825  671
1913 1920 2008 2075 2068 2107 2208 1905 1907 2013 1921 1919 1672 1679 1773 1774 1935 1664 1769 1678 1184 1371 1472 1178 1279  674  861  930  764
2176 2140 2109 2068 2015 1919 1907 2208 2176 2068 2011 1913 1935 1899 1830 1769 1675 1967 1824 1672 1404 1183 1171 1180  998  894  673  670  829
2005 2023 1914 1908 1742 1731 1920 1919 1923 2071 2138 2208 1764 1682 1672 1496 1678 1678 1886 1967 1187  995 1181 1463 1375  677  481  667  860
0 0 0 0 0 


489

668

670

958


660

769

448

865


659

819

480

825


992 . . . . . . . . . . . . . . .
1093 . . . . . . . . . . . . . . .

898
0
S20
S21 S2z
S23 S24
S2s S26 S27
S28 S29
S3o
S31 S32 S33 S34 S35
S36 S37 S~
S39 S~ S41 S42
S~
F7
Fa F9 ~o ~1
668
 495  493 1192 . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
1093 . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
(11 F1F12 refer to facilities (Table 1). (2) Municipal waste is generated at sources 112; industrial waste at sources 1320; prunings at sources 2125; farm refuse at sources 2632; refuse from canning and preserving industry at source 33 and demolition and construction debris at sources 3443. (3) No entry in (i,I] means transportation from o r i g i n / t o destination j is not allowed. (4) Transportation costs include actual transportation costs plus linear part of processing costs at appropriate facilities. Costs are expressed in DM (ton day 1) for one year's operation, i.e. DM (250 ton) 1.
Region, which consists of six counties: Dachau, Freising, Ebersberg, Mfmchen Stadt (Munich City), Starnberg and Fiirstenfeldbruck. They have been chosen as the study region for model 1. In general, the chosen regions are roughly equivalent to the identification of 'agglomeration centres' (Verdichtungsr~ume) according to regional planning guidelines laid down by the Ministry for the Environment of the State of Bavaria. More than thirty towns or cities are situated in the Munich Metropolitan Region, with a combined population of 1.5 million in the year 1980. The data on waste generation, processing and fixed costs of various facilities are based on published material about solid waste management, mainly by the Bavarian Environmental
336
Appl. Math. Modelling,
1986,Vol.
10, O c t o b e r
Protection Office (Bayerisches Landesamt fi.ir Umweltschutz), 197378. In some cases, the data had to be recomputed and adjusted to suit the modelling needs for this paper. Transportation costs are computed on the basis of actual charges to the user which, according to the charge profile (Geb~hrenspiegel), vary considerably depending on container size, transportation distance and, last but not least, on whether public or private transportation is available. Six types of waste have been identified: (1) municipal refuse, WL (2) industrial waste, W2 (3) prunings, W3
Solid waste management with applications: H. W. Gottinger Table5 Optimal solution for problem: facility loading Facility type
Location
Waste type
Quantity (ton day 1)
Incinerator
M0nchenNord/MI] nchenSud MfinchenNord/MLinchenSud Neufahrn/Erding Weilheim Geiselbullach
Municipal refuse Industrial waste Prunings Refuse from canning and preserving industry Total
5285 1813 517
Landfill
Bockhorn Dachau/Bockhorn Dachau/Landsberg
Farm refuse Demolition debris Residue from incinerator Total
271 3122 911 4304
Landfill
Gallenbach G rosslappen/Ebersberg
Farm refuse Demolition Tot a I
1071 1072 2143
Combination facility
89 7704
Table 6 FIowassignments Waste type
Sources
Intermediate facility
Location of intermediate facility used MOnchenNord/MfinchenSud MLinchenNord/M LinchenSud Neufahrn/Erding
Landfill used
Municipal refuse
112
Incinerator
Industrial wastes
1320
Incinerator
Prunings Farm refuse
2125 26, 27 and 28 2932
Incinerator Combined facilities 

Grosslappen/Bockhorn Dachau/Bockhorn Bockhorn/Gallenbach 
33 3440 4143
Incinerator 
Weilheim 
Landsberg Dachau/Landsberg Grosslappen/Ebersberg
Refuse from canning and pruning industry Demolition debris
Dachau/Grosslappen
Total profits in one year's operation = DM 3 675 198.40 Profit per ton of waste processed = DM 0.77.
(4) farm refuse, W4 (5) waste from canning and preserving industries, Ws (6) demolition and construction debris, W6 The waste is assumed to generate at point sources over the region. Each point source generates only a single waste type and many point sources may occupy the same geographical location. The potential locations of intermediate facilities and landfills are chosen. There are six landfill sites, five incinerators and one combination facility (anaerobic digestion and incineration for residual waste). Table 1 gives the locations of these facilities and the numbers assigned to them. Only refuse from the canning and preserving industry, Ws, can be treated by an anaerobic digestion plant and demolition and construction debris, W6, cannot be incinerated. The other waste types may be incinerated or sent directly to landfill. For each waste type, a network is formed with facility locations and sources as nodes with arcs connecting the nodes forming transportation arcs. Processing costs are associated with the nodes corresponding to the facilities. Table 2 gives waste generation data and Table 3 weight reduction factors at intermediate facilities for each waste type. For example, a reduction, factor of 90% for municipal refuse sent through an incinerator plant means that if 100 ton of municipal refuse are incinerated, 10 ton of ashes will be the output.
Processing costs (a) Landfills Costs of disposing wastes at landfills are dependent on the type of waste disposed. From the cost estimations it was found that the total costs at a landfill can be conveniently divided into two parts: a linear cost depending on the type of waste and a nonlinear concave cost independent of the type of waste. The linear costs are lumped together with appropriate transportation costs on the arcs of the network for each waste type. The nonlinear cost function is plotted in Figure 4 and is approximated by a straight line as shown. For the purposes of this example, there is a fixed cost of DM 100 000 and a variable processing cost of D M 8 0 for 1 ton day 1 operation for a year, for all landfills. It should be noted that although landfills at different locations have different costs, the costs are reduced to a common value and any deviation from this value is lumped together with appropriate transportation costs.
(b) Incinerators As with landfills, incinerator cost is a function of the type of waste processed. Here again, from engineering cost estimations, the total costs are classified into two parts: a linear cost depending on the type of waste and
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Solid waste management with applications: H. W. Gottinger a nonlinear concave cost function. Steam is generated at these incinerators and the amount of steam generated per ton of input is a function of the waste type, as are also reclamation profits. The linear cost, profits from steam, and reclamation are all lumped together with appropriate transportation costs on the arcs of the network for each waste type. The nonlinear cost function is shown in Figure5 which also gives the approximation used in this model. (c) Anaerobic digestion plant The cost function and its approximation are shown in Figure 6. Transportation costs, modified as previously explained, are listed in Table 4. All the steam produced by incineration activity is assumed to be saleable. On the basis of engineering evaluation of solid waste technology the price of steam is taken to be DM 1.50ton 1 and the waste type considered produce in the range of 2.53.5 ton of steam for 1 ton of input. In the optimal solution, which is tabulated in Tables 5 and Table 6, waste allocation is performed through all intermediate facilities and landfills. Municipal refuse, industrial waste, prunings and refuse from the canning and preserving industry from all sources go through the incinerators (as available and projected) to the landfills associated with the incinerators. Farm refuse from some sources is partially treated by the combination facility or sent directly to the landfills. Demolition debris from some sources is sent untreated directly to either the Grosslappen or Ebersberg landfills. With the assumption that all the steam produced could be sold, each ton of waste processed brings in a profit of DM 0.77. The problem required 99 iterations and was solved on an IBM 370/165 in 7.85 s using the general algorithm. The time includes the time for compiling the FOR
338 Appl. Math. Modelling, 1986,Vol. 10, October
T R A N program, loading, execution, and printing the output.
Conclusions A general mathematical model of the fixed change integer programming type has been applied to a specific case of solid waste management. The minimal cost combination option is shown in a regional context, for a fixed time horizon. The model can be interpreted as a technology choice model in which various waste treatment technologies, in combination with transfer stations and landfilling, can be substituted for each other to provide an overall optimal solution.
References 1 Clark, R. M. and Gillean, J. I. 'Analysis of solid waste management in Cleveland. Ohio: a case study', Interfaces, 1975, 6, 3242 2 Dee, N. and Liehman, J. C. 'Optimal location of public facilities', Nay. Res. Logist. Q. 1972, 19,753759 3 Kemper, P. and Quigley, J. M. 'The economics of refuse collection', Ballinger, Cambridge, Massachusetts, 1976 4 Liebman, J. "Models of solid waste management', Chapter 5 in 'Mathematical models in government planning', (S. Goss, Ed.), Princeton. New Jersey, 1975, 139164 5 Marks, D. H. and Liebman, J. C. 'Mathematical analysis of solid waste collections', Public Health Service PubL No. 2065, MS Dept. of Health, Education and Welfare, 1970 6 OECD 'Economic instruments of solid waste management', Paris 1981 7 Walker, W., Aquiline, M. and Schur, D. "Development and use of a fixed charge programming model for regional solid waste planning'. Rand Corp. Paper, p. 5307, Santa Monica, California, October 1974 8 Ward, J. E. and Wendell, R. E. 'A million dollar annual savings from a transfer station analysis in Pittsburgh', Proc. 9th Ann. Pittsburgh Conf. Modelling Simulation, April 1978, 9, 13051308