The MARS Model

The MARS model was developed by Litwin of the Institute of Global Climate and Environmental Monitoring in Moscow and Sinyak of IIASA. It is a modified version of the CLAIR model used

Suppose we divide the initial set into several subsets of which the dimensions are selected in a way t o reduce computation time t o a reasonable limit. Every measure Ai belongs t o only one of the disjoint subsets

Il ( i = 1, M ) , I2 ( i = MI

+

^{1, M2),}

^{. .} .,

I,, ( i = MN

-

1, M N )

.

I t is assumed that IlU12U..

.

IN = I ( i =

-

1, M). The measure has a discrete characteristic: it is either fully accomplished or never accomplished.

Let us call two measures alternative if they cannot be implemented simultaneously. Usually, the combined implementation of measures depends on a number of specific conditions and, in general, there is no formal means on how t o establish alternative pairs within a large set of measures A;, i = 1, M . Therefore, a table or tables which indicate the feasibility of different combinations are assumed to be constructed exogenously for each subset I,, in index pairs ( I , q), where

I,

^{q E} I and a,

+

^{a, 5 1.}

Suppose that there is a set. which includes no alternative pairs of measures. Let us call this set a permissible set which can be formed from the existing initial set I.

'see, for example, V. Litwin and S. Golovanov (1990), CLAIR: An Environmental Decision Support System for Atmospheric Air Pollution Simulation and Control, System Handbook, IIASA, Laxenburg, Austria.

With this in mind, a square symmetrical matrix (zCfl) of M x M can be formed where

Obviously, the idealistic apprwch is in finding and comparing all consecutive pairs (gK1, V1)

, . . .,

( g k d , r d ) ,

. . . ,

(ckm, vm) where gr,l

<

^gk2

< . . . <

gkm and vl

<

^~2

< . . . <

vm, k = 1 or 2 and (gkd, vkd) is a permissible set.

Therefore, we have the following model for the optimization of a C 0 2 abatement strategy:

Fl(a;) = ^{Cgkjai +} min; F2(a;) =

C

^{via; + min;}

i € I i€ I

(i) (j) (i) (j) ( j ) (j) (j) (j)

where (El

,

E2 ), (Ul

,

U2 ), (R,

,

R2 ), (BPI

,

Bp2 ) are bilateral constraints.

This model belongs t o the class of multicriteria, integer, combinatoric, distributive problems of a large dimension. When M is considerably large and if a number of alternative pairs is relatively small, the value of S can be too large t o be easily handled by today's computers.

Therefore, special algorithms are required. A practical way of solving these problems consists in splitting the entire set into several subsets (by using some combinatoric methods) and finding the best combinations (interrelations) between them (by using optimization methods, e.g., dynamic programming). The efficiency of the approach depends on the size of each subset.

Suppose the matrix of alternatives is diagonal, i.e., includes only zero elements ( X C J = 0) for ⁶

#

q. In this case any combination

cL,

ⁱ

-

⁼

^1,M

is possible for fulfillment of constraint (A6). In this case, the variable ai, i E I, j = 1, N in each block (group of variables) can be casually divided into subsets (taking into account, of course, some specific aspects of a problem which can later help in understanding and interpreting the results).

If any X'J = 1 for c

#

q, dividing the variables a;, i E I, j E J into groups can be done with an ordinary lexicographical procedure in arranging pairs of indexes ( 6 , q) E I, which later should be reshuffled according to increasing q. Therefore, the corresponding columns and rows in the symmetrical matrix X = (Xi") change their position and are consecutively renumbered.

As a result, the matrix X j becomes block-structured with the size of each block I:'), 1 =

m,,

where 1 equals the number of groups.

Thus, the following chain of problems appears:

Block problem j* with the relaxation of constraint (A6) for the subset of variables I;: p

<

15), an effective procedure for solving equations (A18) to (A24) can be suggested with the following steps:

1. Select consecutively a measure p = 1,2,.

. . ,

^p from an initial set of measures and examine if the vector components e,, u,, r,,

. . .

satisfy the corresponding constraints (A19) to (A22). If all requirements are fulfilled, the iteration number (step) h receives value h + l (at the beginning, h=O) and the vector is stored. Thus, the initial set of solutions HI is formed.

2. Produce consecutive combinations

C&

(iteration index S becomes S+1) from the measure indexes. As a result, we have a list of indexes of measures

sf),

where h is the iteration number (step). If all combinations are tried, we can proceed to step 7.

3. Examine combination h. If the combination contains a t least one pair X'@'*w2 = l ( p l

#

p2), then h receives value h - 1 and the procedure goes to step 2; otherwise we proceed t o step 4.

4. Calculate an evaluating vector for the combination h:

5. Check if components eh, uh of the evaluating vector satisfy constraints (A19) to (A22). If any of the constraints are not fulfilled, then h receives the value h - 1 and the procedure goes t o step 2, otherwise t o step 6.

6. Estimate the effectiveness of combination h in compliance with those already existing (set of solutions HI containing not more than h - 1 numbers from the preceding steps). This procedure, based on a double-criteria analysis, also consists of several steps:

(a) If gh

<

gp and vh

>

v,, then combination h is added to the current set of solutions as no analogs are available; the procedure goes t o step 2;

(b) If gh 5 gp and vh

<

v, or gh

<

g, and vh 5 v,, combination h replaces existing corresponding components of vector Al selected according t o the indexes of the initial measures.

Thus, an interconnecting problem for groups in block j* (connection problem I ) has the following form:

LJ* Lj*

F ~ ( x ) =

C gfl

^{) A!}

-

min i&(X) =

c

^{vf)XB) + min}

Because all effective sets inside every group are fully independent, it is possible to solve any model as consequent steps of dynamic programming problems.

(1) (1) (1) (1) (1)

Let k be fixed and the criteria evaluations (gl

,

^g2

, . . .

^,gH ) and (vl

,

v2

, . . . ,

vz)) corre- spond t o the optimal solution of group problem I . Then the following multistage procedure can be organized: satisfy constraints (A27) t o (A31). If any of the constraints are not fulfilled, return to step 3.

5. Compare the current pair of evaluations (gg), vi')) with already existing ones, indexes are stored in Q!:). In selecting pairs (g,, v,) for p E Q:), we have three cases:

(a) Ifg:)

<

gp and

VP) ^>

v p , then pair combination q, having no analogs, completes the current list of indexes and the procedure goes t o step 3.

(b) If

gg) <

g, and viT)

<

v, or

gg) ^<

g, and v,') (

<

v,, then pair combination q, being more effective, replaces the existing combination p in the current list of indexes Qj:) (

and we go to step 3.

(c) If the previous two cases are not fulfilled for all p E Q$), then combination q is not effective and we go t o step 3.

When

SL

(1) i d a r g e and components (gh, vh) have small differences, the size of the index vector can increase considerably. In such cases approximate solutions should be found by introducing the sensitivity thresholds €1 and €2 for g and v.

6. Sort components g(T)and v('), belonging t o the list of indexes Q$), according t o increased costs. Then we obtain a systematic list of solution indexes Q!:) a t the step r and an

where elements =

Xi

gki, k = 1,2 ; up) =

C~

^{v v ) ; etc.}

,(j) = ~ ( ' 1

,

3

where L*

<

L is a subset of group indexes from j, which entered the optimal set q, and are the indexes of measures entering the optimal combination h selected from group 1 into set q.

Thus, the linking problem (interrelation problem 11) receives the following form:

n Q j n Q j

(j)x(j) min; F~()L) =

C c

^{v~)x!) 3 min}

n ( x ) =

C

^{C 9 k P 9}

To solve model (A33) t o (A35), the multistage procedure described above is used. As a result of solving problems (A33) t o (A35), we have an effective solution for the initial problems

-

( A l ) t o (A9) which consists of m sets (d = 1, m) and corresponds to the list of measures Fl

, . . . ,

Fz,

. . . ,

Fm from I =

Uj

I,. A characteristic vector of every effective set is:

where elements gkd =

Co

^{gki, k = 1,2;}

vd = C i ; ed =

Ci

li, etc.;

Fd = UjcJmWPj (1) = UjEJ*Shl, 1

c

L:

.

Thus, we have optimal pairs (gkl, vl), (gk2, v2),

. . . ,

(gkm, urn) as the solution to the initial model (Al) t o (A9).

The block scheme of the procedure is shown in Figure A l .

Input o f i n i t i a l data:

Figure A 1: Block scheme: Algorithms o f decomposition and information flow.

I-emission

Appendix 2

Initial Assumptions, Input Data,

and Final Energy-Demand

Im Dokument Global Energy Strategies to Control Future Carbon Dioxide Emissions (Seite 109-116)