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Weighting in compromise programming: A theorem on shadow prices
Operations Research Letters 13 (1993) 325-329 North-Holland
June 1993
Weighting in compromise programming: A theorem on shadow prices Enrique Ballestero Department of Economics, Universidad Polit~cnica de Madrid, Madrid, Spain
Carlos Romero Department of Forestry Economics & Management, Universidad Polit~cnica de Madrid, Madrid, Spain
This paper attempts to justify a weighting system proposed as a normalizer in the literature by proving that weights inversely proportional to the ideal are good shadow prices in economic scenarios. Compromise programming; Objectives normalization; Shadow prices
1. Introduction
Compromise programming (CP) is a well established method in operations research (see Yu [5] Chap. 4). However, when a CP model is built, it is necessary to fix the weights attached to the attributes. In fact, weights within a CP context play the following double role: a) Normalizers of attributes. Normalizing is necessary in CP since attributes are usually measured in different units and the values achievable by the attributes are also frequently very different. Hence, without previous normalization, comparing a n d / o r aggregating the attributes is meaningless. Moreover, when the aggregate is optimized, biased solutions towards the attributes with higher achievable values may be obtained. b) Indicators of the decision-maker's preferences with respect to each attribute. Only normalization will be considered in what follows. There are two reasons for focusing on the normalizing problem: 1. Normalization is a technical question
Correspondence to: Professor Carlos Romero, E.T.S. Ingenieros de Montes, Unidad de Economla, Avenida Complutense s / n , 28040 Madrid, Spain.
whereas individual preferences are subjectively revealed or estimated. Hence, the analyst needs to determine normalization and preferential estimation separately. 2. In a traditional economic approach the decision-maker's preferences are actually given by the iso-utility curves defined in the n-dimensional attribute space and not through a weighting system. When CP is developed within this framework, a preferential weighting becomes not only superflous but contradictory as well [1]. Several normalizing systems have been proposed in the literature (e.g. [3] pp. 235-244; [4] pp. 35-43; and [6] Chaps. 6 and 7). The purpose of this paper is to formally justify the superiority of one of the systems commonly used in CP (weights inversely proportional to the ideal or anchor values) since it can be identified with shadow prices in economic scenarios.
2. Shadow prices as normalizers
We use the following notation: (xl . . . . . x i , . . . , xn) = vector of attributes or outputs (commodity-mix or basket),
0167-6377/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
325
Volume 13, Number 5
OPERATIONS RESEARCH LETI'ERS
k = aggregate input or resource level (index), [ T ( x l , . . . , X i , . . . , X n ) = k] = production-possibility frontier, (x* .....
x* .....
, • • ., Xi*,.
X 1 +
• ., Xn*)
= nadir values or anti-ideal point, R = aggregate cost, or internal value of aggregate input k to produce a commodity-mix (x~ . . . . . x i . . . . . x n) on the production-possibility frontier T = k, w~ = shadow price corresponding to output ith, = aggregate shadow value of outputs, or basket shadow value; that is, Y'.~'=1wixi . In economics, the usual approach towards aggregation, when we have a multidimensional basket with commodities such as cars, boats, sugar, etc., consists in taking a price system as the normalizer according to the following formula: Basket value = Y'~ Pixi i=1 where Pi is the price of the i-th commodity. The use of shadow prices in internal accounting when the market prices are not related to the production scenario is a well known procedure (e.g. [2]). In our scenario, a consistent " p e r f e c t " system of weight-shadow prices should satisfy the following conditions:
First condition. The basket shadow value R must be greater than or equal to the corresponding aggregate cost R, i.e.: n
R=
Ewixi>__R
(1)
i=1
f o r every mix on the frontier. Indeed, since all production processes usually entail an increment of wealth, every internal accounting estimation of the basket value (what shadow prices attempt) must satisfy a condition such as output value >_ input cost. In other words, any basket shadow value Y~'=lWiXi assigned to every o u t p u t (x 1. . . . . x i , . . . , x n ) which can be produced with the available resource level k must cover the cost R of k in the internal accounting of the process.
326
For instance, let us suppose a joint production process which includes cars (x 1) and trucks (x 2) and a frontier or trade-off curve given by:
x*)
= anchor values or ideal point, (XI*
June 1993
2X 2 =
30.
The cost of producing baskets such as: (30 cars, 0 trucks), (10 cars, 10 trucks), (20 cars, 5 trucks), and so on, is R = 500 dollars. Thus, we have: Basket shadow value = WlX 1 + W a X 2 = w1(30 - 2x 2) + WzX z > 500 dollars = input cost. This accounting condition can be extended to all 0 _
Second condition. The margin ( R - R) between the basket shadow value and cost must be estimated for internal accounting purposes in a prudential way, avoiding overestimations beyond the requirements of the first condition ( R > R).
The first and second condition can be put together as follows: Min s.t.
(R-R) (1)
(2)
where the restrained minimization is extended to any basket on the frontier. Thus, the shadow prices will minimize any discrepancy between a basket shadow value and the input cost, simultaneously upholding the rule that "you produce when your wealth grows (or at least, does not decreases)". Our aim is to demonstrate that weights inversely proportional to the anchor values (such as w l x ~' . . . . . wix* .... wnx* = R ) is the only system capable of satisfying (2). Therefore, they are consistent " p e r f e c t " shadow prices (see the basic theorem below). Indeed we will demonstrate that (2) has a single solution (a weighting system inversely proportional to the anchor values).
Theorem 1. Given a CP structure with a convex frontier intercepting every x i-axis o n point (0, • ' ' ' x *l ' ' ' " ,0), a null anti-ideal and a positive real number R, weights such as: WlX ? . . . . .
Volume 13, Number 5
OPERATIONS RESEARCH LETTERS
wix* . . . . w , x * = R are the only ones capable of minimizing the difference R-R=
~wixi-R. i=1
s.t. R>_R,
Proof. Let us consider the following family of hyperplanes: (3)
~wix i = R
hyperplane of the third group can be eliminated. b) Let us now choose any point (x ° . . . . . x/° , . . . , Xn°) on the frontier. The difference ( R - R) for this point if we are weighting according to the first group scheme, will be equal to R-R=~
for every vector of attributes on the frontier.
R-R
=
R wi = x----f Vi,
(5)
R wi = x * + ~.
(6)
being ~'i > 0 at least for one i = j. Systems of weighting (4), (5) and (6) will be associated to the first, second and third group of hyperplanes, respectively. a) Now, consider a hyperplane from the third group. In this kind of hyperplane, at least one of their coefficients must be R
with ~ > 0. By introducing the point (0 . . . . , x~ . . . . . 0) (which obviously belongs to the frontier T = k) into the difference R - R, we get --Rx*
-R
(8)
X?
As (7) > (8) the minimum of (R - R) (for every point on the frontier) corresponds to the second group scheme. c) The single hyperplane of the second group i=l X ?
is included in this group. Third group. The remaining hyperplanes from family (3), that is,
x?
(7)
R.
(4)
being ~ / > 0 (for every i). Second group. Only the hyperplane where
R-R=
-i:1
R wi =- x * - ~i
wj=
Rx° i=i x* =~i
If we are weighting according to the second group scheme, the analogous difference will be
i=1
Family (3) can be classified in the following three groups. First group. Hyperplanes where
June 1993
R<0.
Hence, condition R _ R for every point on the frontier does not hold. In consequence, every
R
(9)
0
divides the n-dimensional space into two regions A and B. If point (0 . . . . ,0 . . . . . 0) belongs to region A, then region B contains the convex frontier, except for their extreme points (0 . . . . , xi* ..... 0) (for every i) which belongs to (9). By introducing point ( 0 , . . . , 0 , . . . , 0) into the difference (R R), we obtain ( - R ) < 0. Therefore, if any point of the frontier is introduced into the difference (R - R), we must get ,~ - R > 0. Thus, condition R >_R holds and the theorem demonstrated. Corollary 1. For linear frontiers XI
x-? + "
Xi
+ x-? +
Xn
+
= l
the equality R = R with the weighting w l x ~ . . . . = wix* . . . . . wnx* = R for every point on the frontier can be straightforwardly obtained.
Note that although a linear frontier is a simple model it is not the most accurate model for describing joint production processes. 2. I f the nadir values are other than zero then the weighting system Wl(Xt - - X l * ) . . . . . Wi(X ? -- Xi*) . . . . . Wn(X* n -- Xn*) holds (2). Corollary
Proof. By taking the anti-ideal point as the new origin of coordinates and denoting the new coor327
Volume 13, N u m b e r 5
dinates by
X i, we
O P E R A T I O N S R E S E A R C H LEqT~ERS
get:
X i =X i --Xi*.
(10)
In the CP scenario with the new coordinates, the anti-ideal is zero. Therefore, we can apply T h e o r e m 1 to the system
June 1993
feasible baskets. To prove this remark, it is enough to r e m e m b e r that n
E WiXi i-1
=
RX~//X~
=
R
on basket ( O , . . . , x * . . . . . 0), since weights are: wix* . . . . . wnx* = R (see Theorem 1).
WxX~ . . . . . [~wiXi-r ]
Min
i=1 n
(11)
E wiXi ~ r
s.t.
i-1
for every mix on the frontier, being r a positive real number. In consequence, solution (11) holds r
-
w,
r
-
x7
x* - x i ,
Vi.
(12)
This internal evaluation is full of economic meaning since the least shadow value (i.e. R dollars) is assigned to the most disbalanced mix. In fact, an extremely disbalanced allocation such as (0 ..... x * . . . . . 0) implies an unsuitable shift of all resources towards the i-th output (a risky policy, specially within aggregate joint production scenarios).
Let us select the following value for r 3. S o m e i l l u s t r a t i v e e x a m p l e s
r=R/ 1+ i= x*-x i. " By introducing in (11) its solution (12), and taking into account (10) and (13), we obtain n
r
i=l
Xff --Xi,
E --'xi>-R
(14)
[~-~wixi-R ] (15)
n
~ wi x i > R, i=1
is the set of weights r = x,* - x i ,
(0,3,0),
R
~
(0,0,2),
(0.7, 1.4, 1)
and (3.36, 0.7, 0.7).
i=1
s.t.
(6,0,0),
(2.8, 0.7, 0.93),
for every mix on the frontier. Hence, the solution of the system Min
To check the T h e o r e m and Corollary 2, the following numerical examples are suggested. 1) Let us suppose a convex frontier with the following extreme efficient points:
Xi*
(x*-xi.) l+i=lx,_xi.
] "
1 (16)
Obviously, the ideal is (6, 3, 2) and the antiideal (0, 0, 0). To achieve any output on the frontier we can assign a cost R = 100 dollars to the aggregate input. By developing (2) we get the following six linear programs: 1) Min(6w I - 100); 2) Min(3w 2 - 100); 3) Min(2w 3 - 100); 4) Min(2.8w I + 0.7w 2 + 0.93w 3 - 100); 5) Min(0.7w 1 + 1.4w 2 + w 3 - 100); and 6) Min(3.36w 1 + 0.7w 2 + 0.7w 3 - 100) all subject to the restraints: 6w 1 > 100,
Thus, Corollary 2 is demonstrated. Remark. Weights according to T h e o r e m 1 also hold the following interesting property: 'shadow values of extreme baskets (0, • ' ' ' x *l ' ' " ., 0) = R'. T h a t is, the baskets (0, " ' ' ' x *I ' ' ' " ,0) are internally evaluated at the lowest level among every 328
3w 2 > 100, 2w 3 > 100,
2.8w I + 0.7w 2 + 0.93w 3 > 100, 0.7w 1 + 1.4w 2 + w 3 > 100, 3.36w x + 0.7w 2 + 0.7w 3 >__100.
Volume 13, Number 5
OPERATIONS RESEARCH LETTERS
A (0,10)
all subject to the restraints 10w 2 > 1000,
I 0~,10)
_*_A°_._s)_ ~ ................... _
June 1993
6w 1 + 8w 2 > 1000, 10w I + 5 w 2 > 1000. By solving the above linear programs, weights Wl = 50 and w 2 = 100 inversely proportional to the ranges A I - C = 10 and A I - A = 5 (as formula (16) predicts) are obtained in the three cases. The basket shadow values /~ for every mix on the frontier are: 1000, 1100 and 1000, respectively.
~ c (10,s)
Acknowledgement xl fIRST ATTRIBUTE
Fig. 1. A polygonal frontier and a non null anti-ideal
By solving the above linear programs the weights: W1 16.66, W2 33.33 and w 3 = 50 inversely proportional to the anchor values (as the theorem predicts) are obtained in the six cases. The basket shadow values R for every mix on the frontier are: 100, 100, 100, 116.48, 108.32, and 114.31, respectively. 2) Now suppose a convex frontier with a nonnull anti-ideal defined by the following extreme efficient points: =
A(0, 10),
B(6, 8),
=
C(10, 5)
Obviously, the ideal is the point I(10, 10) and the anti-ideal is point AI(0, 5) (see Figure 1). Considering the input cost R = 1000 dollars, we get the following three linear programs: 1) Min(10w 2 - 1000); 2) Min(6w I + 8w 2 - 1000); and 3) Min(10w 1 + 5w 2 - 1000)
Comments raised by the reviewer are appreciated. Thanks are given to Ms. Christine Mendez for her English editing. The work of Carlos Romero was supported by Spanish 'Comisi6n Interministerial de Ciencia y Tecnolog~a (CICYT)' under project PB91-0035 and Junta de Andalucia (Research Group 2081).
References [1] E. Ballestero and C. Romero, "A theorem connecting utility function optimization and compromise programming", Oper. Res. Lett. 10, 421-427 (1991). [2] W.J. Baumol and R.E. Quandt, "Dual prices and competition", in: G.C. Archibald (ed.), The Theory of the Firm, Penguin Books, New York, 1971, 422-447. [3] A. Goicoechea, D.R. Hansen and L. Duckstein, Multiob-
jective Decision Analysis with Engineering and Business Applications, John Wiley and Sons, New York, 1982. [4] C. Romero, Handbook of Critical Issues in Goal Programming, Pergamon Press, Oxford, 1991. [5] P.L. Yu, Multiple-Criteria Decision Making. Concepts, Techniques and Extensions, Plenum Press, New York, 1985. [6] M. Zeleny, Multiple Criteria Decision Making. McGraw Hill, New York, 1982.
329
June 1993
Weighting in compromise programming: A theorem on shadow prices Enrique Ballestero Department of Economics, Universidad Polit~cnica de Madrid, Madrid, Spain
Carlos Romero Department of Forestry Economics & Management, Universidad Polit~cnica de Madrid, Madrid, Spain
This paper attempts to justify a weighting system proposed as a normalizer in the literature by proving that weights inversely proportional to the ideal are good shadow prices in economic scenarios. Compromise programming; Objectives normalization; Shadow prices
1. Introduction
Compromise programming (CP) is a well established method in operations research (see Yu [5] Chap. 4). However, when a CP model is built, it is necessary to fix the weights attached to the attributes. In fact, weights within a CP context play the following double role: a) Normalizers of attributes. Normalizing is necessary in CP since attributes are usually measured in different units and the values achievable by the attributes are also frequently very different. Hence, without previous normalization, comparing a n d / o r aggregating the attributes is meaningless. Moreover, when the aggregate is optimized, biased solutions towards the attributes with higher achievable values may be obtained. b) Indicators of the decision-maker's preferences with respect to each attribute. Only normalization will be considered in what follows. There are two reasons for focusing on the normalizing problem: 1. Normalization is a technical question
Correspondence to: Professor Carlos Romero, E.T.S. Ingenieros de Montes, Unidad de Economla, Avenida Complutense s / n , 28040 Madrid, Spain.
whereas individual preferences are subjectively revealed or estimated. Hence, the analyst needs to determine normalization and preferential estimation separately. 2. In a traditional economic approach the decision-maker's preferences are actually given by the iso-utility curves defined in the n-dimensional attribute space and not through a weighting system. When CP is developed within this framework, a preferential weighting becomes not only superflous but contradictory as well [1]. Several normalizing systems have been proposed in the literature (e.g. [3] pp. 235-244; [4] pp. 35-43; and [6] Chaps. 6 and 7). The purpose of this paper is to formally justify the superiority of one of the systems commonly used in CP (weights inversely proportional to the ideal or anchor values) since it can be identified with shadow prices in economic scenarios.
2. Shadow prices as normalizers
We use the following notation: (xl . . . . . x i , . . . , xn) = vector of attributes or outputs (commodity-mix or basket),
0167-6377/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
325
Volume 13, Number 5
OPERATIONS RESEARCH LETI'ERS
k = aggregate input or resource level (index), [ T ( x l , . . . , X i , . . . , X n ) = k] = production-possibility frontier, (x* .....
x* .....
, • • ., Xi*,.
X 1 +
• ., Xn*)
= nadir values or anti-ideal point, R = aggregate cost, or internal value of aggregate input k to produce a commodity-mix (x~ . . . . . x i . . . . . x n) on the production-possibility frontier T = k, w~ = shadow price corresponding to output ith, = aggregate shadow value of outputs, or basket shadow value; that is, Y'.~'=1wixi . In economics, the usual approach towards aggregation, when we have a multidimensional basket with commodities such as cars, boats, sugar, etc., consists in taking a price system as the normalizer according to the following formula: Basket value = Y'~ Pixi i=1 where Pi is the price of the i-th commodity. The use of shadow prices in internal accounting when the market prices are not related to the production scenario is a well known procedure (e.g. [2]). In our scenario, a consistent " p e r f e c t " system of weight-shadow prices should satisfy the following conditions:
First condition. The basket shadow value R must be greater than or equal to the corresponding aggregate cost R, i.e.: n
R=
Ewixi>__R
(1)
i=1
f o r every mix on the frontier. Indeed, since all production processes usually entail an increment of wealth, every internal accounting estimation of the basket value (what shadow prices attempt) must satisfy a condition such as output value >_ input cost. In other words, any basket shadow value Y~'=lWiXi assigned to every o u t p u t (x 1. . . . . x i , . . . , x n ) which can be produced with the available resource level k must cover the cost R of k in the internal accounting of the process.
326
For instance, let us suppose a joint production process which includes cars (x 1) and trucks (x 2) and a frontier or trade-off curve given by:
x*)
= anchor values or ideal point, (XI*
June 1993
2X 2 =
30.
The cost of producing baskets such as: (30 cars, 0 trucks), (10 cars, 10 trucks), (20 cars, 5 trucks), and so on, is R = 500 dollars. Thus, we have: Basket shadow value = WlX 1 + W a X 2 = w1(30 - 2x 2) + WzX z > 500 dollars = input cost. This accounting condition can be extended to all 0 _
Second condition. The margin ( R - R) between the basket shadow value and cost must be estimated for internal accounting purposes in a prudential way, avoiding overestimations beyond the requirements of the first condition ( R > R).
The first and second condition can be put together as follows: Min s.t.
(R-R) (1)
(2)
where the restrained minimization is extended to any basket on the frontier. Thus, the shadow prices will minimize any discrepancy between a basket shadow value and the input cost, simultaneously upholding the rule that "you produce when your wealth grows (or at least, does not decreases)". Our aim is to demonstrate that weights inversely proportional to the anchor values (such as w l x ~' . . . . . wix* .... wnx* = R ) is the only system capable of satisfying (2). Therefore, they are consistent " p e r f e c t " shadow prices (see the basic theorem below). Indeed we will demonstrate that (2) has a single solution (a weighting system inversely proportional to the anchor values).
Theorem 1. Given a CP structure with a convex frontier intercepting every x i-axis o n point (0, • ' ' ' x *l ' ' ' " ,0), a null anti-ideal and a positive real number R, weights such as: WlX ? . . . . .
Volume 13, Number 5
OPERATIONS RESEARCH LETTERS
wix* . . . . w , x * = R are the only ones capable of minimizing the difference R-R=
~wixi-R. i=1
s.t. R>_R,
Proof. Let us consider the following family of hyperplanes: (3)
~wix i = R
hyperplane of the third group can be eliminated. b) Let us now choose any point (x ° . . . . . x/° , . . . , Xn°) on the frontier. The difference ( R - R) for this point if we are weighting according to the first group scheme, will be equal to R-R=~
for every vector of attributes on the frontier.
R-R
=
R wi = x----f Vi,
(5)
R wi = x * + ~.
(6)
being ~'i > 0 at least for one i = j. Systems of weighting (4), (5) and (6) will be associated to the first, second and third group of hyperplanes, respectively. a) Now, consider a hyperplane from the third group. In this kind of hyperplane, at least one of their coefficients must be R
with ~ > 0. By introducing the point (0 . . . . , x~ . . . . . 0) (which obviously belongs to the frontier T = k) into the difference R - R, we get --Rx*
-R
(8)
X?
As (7) > (8) the minimum of (R - R) (for every point on the frontier) corresponds to the second group scheme. c) The single hyperplane of the second group i=l X ?
is included in this group. Third group. The remaining hyperplanes from family (3), that is,
x?
(7)
R.
(4)
being ~ / > 0 (for every i). Second group. Only the hyperplane where
R-R=
-i:1
R wi =- x * - ~i
wj=
Rx° i=i x* =~i
If we are weighting according to the second group scheme, the analogous difference will be
i=1
Family (3) can be classified in the following three groups. First group. Hyperplanes where
June 1993
R<0.
Hence, condition R _ R for every point on the frontier does not hold. In consequence, every
R
(9)
0
divides the n-dimensional space into two regions A and B. If point (0 . . . . ,0 . . . . . 0) belongs to region A, then region B contains the convex frontier, except for their extreme points (0 . . . . , xi* ..... 0) (for every i) which belongs to (9). By introducing point ( 0 , . . . , 0 , . . . , 0) into the difference (R R), we obtain ( - R ) < 0. Therefore, if any point of the frontier is introduced into the difference (R - R), we must get ,~ - R > 0. Thus, condition R >_R holds and the theorem demonstrated. Corollary 1. For linear frontiers XI
x-? + "
Xi
+ x-? +
Xn
+
= l
the equality R = R with the weighting w l x ~ . . . . = wix* . . . . . wnx* = R for every point on the frontier can be straightforwardly obtained.
Note that although a linear frontier is a simple model it is not the most accurate model for describing joint production processes. 2. I f the nadir values are other than zero then the weighting system Wl(Xt - - X l * ) . . . . . Wi(X ? -- Xi*) . . . . . Wn(X* n -- Xn*) holds (2). Corollary
Proof. By taking the anti-ideal point as the new origin of coordinates and denoting the new coor327
Volume 13, N u m b e r 5
dinates by
X i, we
O P E R A T I O N S R E S E A R C H LEqT~ERS
get:
X i =X i --Xi*.
(10)
In the CP scenario with the new coordinates, the anti-ideal is zero. Therefore, we can apply T h e o r e m 1 to the system
June 1993
feasible baskets. To prove this remark, it is enough to r e m e m b e r that n
E WiXi i-1
=
RX~//X~
=
R
on basket ( O , . . . , x * . . . . . 0), since weights are: wix* . . . . . wnx* = R (see Theorem 1).
WxX~ . . . . . [~wiXi-r ]
Min
i=1 n
(11)
E wiXi ~ r
s.t.
i-1
for every mix on the frontier, being r a positive real number. In consequence, solution (11) holds r
-
w,
r
-
x7
x* - x i ,
Vi.
(12)
This internal evaluation is full of economic meaning since the least shadow value (i.e. R dollars) is assigned to the most disbalanced mix. In fact, an extremely disbalanced allocation such as (0 ..... x * . . . . . 0) implies an unsuitable shift of all resources towards the i-th output (a risky policy, specially within aggregate joint production scenarios).
Let us select the following value for r 3. S o m e i l l u s t r a t i v e e x a m p l e s
r=R/ 1+ i= x*-x i. " By introducing in (11) its solution (12), and taking into account (10) and (13), we obtain n
r
i=l
Xff --Xi,
E --'xi>-R
(14)
[~-~wixi-R ] (15)
n
~ wi x i > R, i=1
is the set of weights r = x,* - x i ,
(0,3,0),
R
~
(0,0,2),
(0.7, 1.4, 1)
and (3.36, 0.7, 0.7).
i=1
s.t.
(6,0,0),
(2.8, 0.7, 0.93),
for every mix on the frontier. Hence, the solution of the system Min
To check the T h e o r e m and Corollary 2, the following numerical examples are suggested. 1) Let us suppose a convex frontier with the following extreme efficient points:
Xi*
(x*-xi.) l+i=lx,_xi.
] "
1 (16)
Obviously, the ideal is (6, 3, 2) and the antiideal (0, 0, 0). To achieve any output on the frontier we can assign a cost R = 100 dollars to the aggregate input. By developing (2) we get the following six linear programs: 1) Min(6w I - 100); 2) Min(3w 2 - 100); 3) Min(2w 3 - 100); 4) Min(2.8w I + 0.7w 2 + 0.93w 3 - 100); 5) Min(0.7w 1 + 1.4w 2 + w 3 - 100); and 6) Min(3.36w 1 + 0.7w 2 + 0.7w 3 - 100) all subject to the restraints: 6w 1 > 100,
Thus, Corollary 2 is demonstrated. Remark. Weights according to T h e o r e m 1 also hold the following interesting property: 'shadow values of extreme baskets (0, • ' ' ' x *l ' ' " ., 0) = R'. T h a t is, the baskets (0, " ' ' ' x *I ' ' ' " ,0) are internally evaluated at the lowest level among every 328
3w 2 > 100, 2w 3 > 100,
2.8w I + 0.7w 2 + 0.93w 3 > 100, 0.7w 1 + 1.4w 2 + w 3 > 100, 3.36w x + 0.7w 2 + 0.7w 3 >__100.
Volume 13, Number 5
OPERATIONS RESEARCH LETTERS
A (0,10)
all subject to the restraints 10w 2 > 1000,
I 0~,10)
_*_A°_._s)_ ~ ................... _
June 1993
6w 1 + 8w 2 > 1000, 10w I + 5 w 2 > 1000. By solving the above linear programs, weights Wl = 50 and w 2 = 100 inversely proportional to the ranges A I - C = 10 and A I - A = 5 (as formula (16) predicts) are obtained in the three cases. The basket shadow values /~ for every mix on the frontier are: 1000, 1100 and 1000, respectively.
~ c (10,s)
Acknowledgement xl fIRST ATTRIBUTE
Fig. 1. A polygonal frontier and a non null anti-ideal
By solving the above linear programs the weights: W1 16.66, W2 33.33 and w 3 = 50 inversely proportional to the anchor values (as the theorem predicts) are obtained in the six cases. The basket shadow values R for every mix on the frontier are: 100, 100, 100, 116.48, 108.32, and 114.31, respectively. 2) Now suppose a convex frontier with a nonnull anti-ideal defined by the following extreme efficient points: =
A(0, 10),
B(6, 8),
=
C(10, 5)
Obviously, the ideal is the point I(10, 10) and the anti-ideal is point AI(0, 5) (see Figure 1). Considering the input cost R = 1000 dollars, we get the following three linear programs: 1) Min(10w 2 - 1000); 2) Min(6w I + 8w 2 - 1000); and 3) Min(10w 1 + 5w 2 - 1000)
Comments raised by the reviewer are appreciated. Thanks are given to Ms. Christine Mendez for her English editing. The work of Carlos Romero was supported by Spanish 'Comisi6n Interministerial de Ciencia y Tecnolog~a (CICYT)' under project PB91-0035 and Junta de Andalucia (Research Group 2081).
References [1] E. Ballestero and C. Romero, "A theorem connecting utility function optimization and compromise programming", Oper. Res. Lett. 10, 421-427 (1991). [2] W.J. Baumol and R.E. Quandt, "Dual prices and competition", in: G.C. Archibald (ed.), The Theory of the Firm, Penguin Books, New York, 1971, 422-447. [3] A. Goicoechea, D.R. Hansen and L. Duckstein, Multiob-
jective Decision Analysis with Engineering and Business Applications, John Wiley and Sons, New York, 1982. [4] C. Romero, Handbook of Critical Issues in Goal Programming, Pergamon Press, Oxford, 1991. [5] P.L. Yu, Multiple-Criteria Decision Making. Concepts, Techniques and Extensions, Plenum Press, New York, 1985. [6] M. Zeleny, Multiple Criteria Decision Making. McGraw Hill, New York, 1982.
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