LIBRARY
OF THE
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
^]AS3.
MULTIPLE CRITERIA PUBLIC INVEST>ENT DECISION MAKING
BY MIXED INTEGER PROGPJ^^IMING*
Jeremy
F.
Shapiro
WP 788-75
May 1975
*Multiple Criteria Decision Making
Proceedings of a Conference
Jouy-en-Josas, France
May 21-28, 1975
Jointly sponsored by
Centre d'Enseignement Superieur des Affaires (C.E.S.A.)
and European Institute for Advanced Studies
in Management (E.I.A.S.M.)
Edited by
H.
Thiriez and
New York: 1976
S.
Zionts
Springer-Verlag, Berlin-Heidelberg.
MULTIPLE CRITERU PL'BLTC INVESTMENT DECISION MAKING
BY MIXED INTEGER PROGRAMS, ING-<^
Jeremy
1.
F.
Shapiro
IKTRODVCT ION
Multicriterion optinization has long been an integral part of quantitative
The n>odels which have received the most attention
models for public planning.
consist of concave utility functions to be maximized (in a vector sense) subject
to
The convex structure of such models permits the development
convex constraints.
of con-.plete theories characterizing efficient solutions, and well-behaved algorithms
for finding these solutions.
In this paper, we consider a different
class of Dublic planning probleras for
which the assumption of convex constraints, or non-decreasing marginal costs of
production, is not valid.
There are at least three circumstances when this is the
First, there can be public investment alternatives such as dam or power
case.
Second, there can be important
plant construction involving large fixed costs.
returns to scale in construction and production costs.
Finally, public investment,
decision problems involving new technologies can have associated diffusion and
learning curves which are S-shaped; that is, increasing marginal costs are ex-
perienced at first, followed by a consolidation phase when marginal costs are
decreasing.
models.
All of these phenomena can be described by mixed integer programming
There is a grow^ing literature on decision models of this type; for example,
see Bhatia ';i975l, Dorfnan and Jacoby [1972], Goreux and Manne [1973], Kendrick
and Stoutesdijk [1975],
The
Kuhner and Harrington [197i], .Marks [1974], Westphal [1971].
of this paper is the development of procedures for multi-
main concern
criteria optiniization of inte;;er progracising problems.
consider is the vector maximization
Max
The specific model we will
proMem
Cx
s.t. Ax 5 b
I
where C is a P x N matrix, A
coefficients of A and
b are
i s
(1)
an M x N matrix, b is an M x
integer.
The columns a
vector, and the
1
of A correspond to projects
extending over time to be selected or rejected by P interested parties on the
basfs" of the P x
c
1
vector
c
measuring its value to these parties.
may equal the present net value of project
P
to the pth party.
j
computed with
a
The component
discount rate specific
The constraints in (1) can consist of capital budget constraints
for each of a number of time periods, other resource constraints,
«!UVI<IJb,llfi.W!.W^
logical constraints
imposing
unique starting period for a project, precedence constraints on subsets
a
For future reference, we define the set
of projects, and so on.
F =
(xiAx
b. X.
'.
say F
the set is finite,
«»
or 1);
"
{x
).
^i
(2)
•
Different and more general mult icriterion optimization problems than (1) in-
volving indivisible investment alternatives are clearly possible.
One can envision
mixed integer programming models involving continuous decision variables as well as
fixed charge variables.
In
party may be measured by
a
multicri terion
IP
addition, the value of a feasible solution x to the pth
utility function u
.
Such extensions to our basic
model and algorithms for finding efficient solutions to it are
possible and will be discussed in the conclusions section
of this paper.
As with convex vector maximization problems, there probably is not
a
vector
BaxiTnum solution to problem (1), and we are led to a study of efficient solutions.
Specifically,
a
solution x satisfying the constraints of problem (1) is said to be
efficient if there does not exist a feasible solution x satisfying the P inequalities.
Cx 2 Cx
The characterization of efficient solutions
vith strict inequality for some component.
of IP problem (1) is not as simple as it is for linear or convex multicriterion
'
problems, and moreover, the calculation of these solutions is radically different.
The plan of this paper is as follows.
IP
Section
2
discusses mathematical pro-
The following section contains a brief review of
perties of efficient solutions.
duality theory which provides the constructive means for implementing some of the
results of section
Section
2.
4
contains
used to compute efficient solutions.
description of how IF duality theory is
a
Section
5
contains sensitivity analysis for
compiiting the convex region in which a given efficient solution is optimal in a
related
IP
problem.
Some concluding remarks and areas for future research are
discussed briefly in section
2.
6.
MATHEMATICAL PROPERTIES OF EFFICIENT SOLUTIONS
The usual practice in generating efficient solutions to a multicriterion mathe-
matical programming problem is to give positive weights to each of the P objective
functions and combine them (for example, see Rjhn and Tucker Cl951J, or Karlin
To this end, let
1:1959].
S-lXcR
P
r
T
,
p-l
•
=1.\
\
i
^!gifgaP«gaE»^ | W^^
-:;ijagS^S^^f^j!
\
P
P
i
i
'•-
W
^M M!iti!rt^g- ip
|
"*«' '" ^«ft* Jg
'
-
J
g^^
int
=
S
(aU
and
S
c
>
a
for all p).
P
For
A
E
IP problem
int S, define the
v(X) = iCx
(3)
s.t. X
Theorem
For
I:
c
X
F
s
any maximal solution to (3) is efficient.
int S,
Let x denote a maximal
Proof:
a
coaponent.
int S,
Since
>
e
solution to (3) and suppose it is not efficient;
y e F such that Cy t Cx «ith strict inequaT,ity in some
that is, there exists
this clearly icolies
(Cy)
X
)
P
p-1
>
P
T
p=l
(Cx)
X
P
P
contradicting the optimality of x in (3).||
As is well known,
the difficulty with the characterization of Theorem
efficient solutions is that it
The following example, due to Gabriel Bitran, illustrates
solution to be efficient.
The
this point.
IP
of
1
a sufficient but not necessary condition for a
is
problem is
1,10,9
1
10,1,9
(4)
1-3
"2 '
1
X.
-
'^3
or 1,
j
- 1,2,3
J
There are seven feasible solutions to this problem, and three efficient
=
solutions X
(1,1,0), x~ = (1,0,1), x
(0,1,1). Let
X
be the weight on the
top objective function in (4), and 1-X be the weight the bottom objective function.
It is
X
easy to verify that x
e[0,l/2], and x
3
is optimal in the IP
is optioal for X
t
problem (3) derived
[1/2,1], but x
1
is not optimal
froni
for
(4)
for any
X
e
S.
rnis anomaly of efficient IP solutions needs further study and will not be
Instead, we focus on parametric
reconciled here.
Given any x
e
v(X) = \Cx
F,
for
XCx
comoutatlon of v(X) for
x
g int S.
is a oiecewisc linear convex function on S.
It can easily be shown that v())
X
satisfying
1,
P
(5)
p-1
-ll- l
J-W Jgi^^W,» l'l/ Wijy^U JKJW^' 'M
i
i
-
'>^- t'' -'-'»'^t«W!,^^
!
m
.
s(x
)
dennte the region of
Theorem
1
if Six'')
Let
Cnc S
r.
'
S
defined in (5); clearly x
e
int S for mul ticri tericn LP is possible by
is efficient by
.
!<
Paranetric variation of
>
sensitivity analysis which relies in turn on LP duality theory.
The IP problem (3)
can be expressed as the LP problem
V (\)
=»
max
>Cx
s.t.
X
(6>
where "[
1"
[F]
e
This last statement is true because all extreme
denotes convex hull.
points of [F] are solutions in
F.
fact, the zero-one structure of problem (1)
In
implies all solutions in F are extreme points of CF];
problem (3) to
a
that is, x is an extreme
The difficulty with this apparent reduction of
point of [F]if and only if x eF.
more manageable optimization problem is that [F] is generally
impossible to characterize in any real practical sense.
The
IP
duality theory
reviewed in the next section, and used in subsequent sections, can be interpreted
as approximating [F] in a neighborhood of an optimal
IP
solution.
provides insights on how to perform sensitify analysis as
problem
3.
As a result it
varies in int
in
S
(3)
REVIEW OF
In
X
IP
DUALITY TtiEORY
this section we reciew briefly the
duality theory developed in Shapiro
TP
[1971], Fisher and Shapiro [1974], Bell !:i973a3[1973bl
We consider the zero-one
,
Bell and Shapiro [1975].
problem
ff
max ex
s.t
Ax +
.
=
b
X.
=
or
1
s.
»=
0.1,2,
Is
(7)
..., U.
which is identical to problem (1) except
bound en the value of the slack s..
c
is
1
The integer U. is an upper
x N.
The slacks in (7) are integer variables
because A and b are integer.
A dual problem to (7) is constructed by reformulating it as follows.
Let G
be any finite abelian group with the representation
«Z
G = Z
q^
e
...
eZ
q,
"^r
where the positive integers q.
satisfy
q.?
the cyclic group of order q..
Let
denote the order of G; clearly
|g|
-
r
n
i"l
^
q..
I
Let
e
1-
^n
^"^
]gJ
2, q.
q.
,,
i"l,
.... r-1, and Z
is
^"Y elements of this group and for any M-vector
iww 'WJiwi'PBBiBiPtBMp'Wtgww^ ^^
'ii
^
174
f,
define the element $(t) " o
£
G by
H
(f)
-
y
f,.e..
i=l
It
can easily be shown that (7) is equivalent to (has the same feasible region as)
max ex
(8a)
(8b)
(8c)
It
can easily be shown that surh a pair is optimal in the rtspective primal
For a piven IP dual problen, there is no (Euarantee that
and dual problems-
the optijiality conditions can be established, but attention can be restricted to
optimal dual solutions tor which we try to find a complementary optimal primal
solution.
V
<
IP problen cannot -be used
the dual
If
V and we say there is
Solution of the
IP
a
IF dual problems analogous to
The group G
= 2^,
= {(x,z)|x.
Y
the primal problem,
problem (7) by dual methods is constructively achieved
by generating a finite sequence of groups {G
and
to solve
duality gap.
dual problem can be shown to be
)
,
„,
sets fY
}
.
(9)
with minimal objective function value w
=
or
tlie
groups here have the property that G
k
I
,
s
=
.
.
0,1,2,...U.) and the corresponding
linear prograiraning relaxation of (7). The
k+1
implying directly that
,
is a subgroup of G
^k+1
supergroup of G
.
The critical step in this approach to solving the
IP
problem (7) is that if an
optimal solution to the kth dual does not yield an optimal integer solution, then
k+1
k+1
k
we are able to construct the supergroup G
c Y
so that Y
Moreover, the
.
construction eliminates the infeasible
combination by the IP
IP solutions
£
which are used in
Y
dual problen to produce a fractional solution to the
optimality conditions.
Since the set of feasible solutions to (7) is finite,
the process must converge in a finite number of
an
(x,s)
IP dual problem constructions to
ipdual problem yielding an optimal solution to (7) by the optimality conditions.
The
IP dual problem (9) is actually a large scale linear programming problem,
Let Y = {x
t
T
t
s
,
w
be an eniraieration of Y.
]
=
min
The LP
formulation of (9) is
\>
V > ub +
(c-uA)x
t
- us
= 1.
(10)
..., T
The linear programing dual to (10) is
V = max
y
(ex
)!i.
(Ax
+
t=l
s.t. I
t=l
b
" b
)u3
(11)
^
T
as* ai)iP!W J«w iiti <»
«!T!
i
!
(
ft
i
fi4*
ii
i
mmmt
The nunb«r of rows T in (10), or columns in (11), is enonaous.
The solution
methods given in Fisher and Shapiro ri974], generate colunms as needed
(11) as a primal dual pair.
algorithras for solving (10) and
by descent
Thu columns are
generated by solving the La^raneean problem which can be solved in a matter of a
few seconds for values of
|g|
see Glover [1969] or Gorry, Northup
up to 3000;
and Shapiro [1973].
The formulation (11) has a convex analysis interpretation.
Specifically, the
feasible region in (11) corresponds to
= b,
{(x,s)|Ax+Is
s
X.
£
1,
?
s.
1
U.}
[Y]
n
where the left hand set is the feasible region of the LP relaxation of (7).
Thus,
in effect, the dual approach approximates the convex hull of the set of feasible
integer points [F] by the intersection of the LP feasible region with the polyhedron
When the
[Y].
IP
dual problem (9) solves the
IP
Problem (7), then [Y]has cut away
enough of the LP feasible region to approximate the convex hull of feasible integer
solutions in a neighborhood of an optimal
solution.
IP
CALoruaios of efficent solutions
4.
f
Solution methods for the mul ticri terion
of
A
c
of efficient solutions as
IP
IP
problem
\
for
(3)
F
int S are basically methods for solving arbitrary
a
fixed value
problems.
The generat
varies in int S, however, facilitates the use of the
dual methods outlined
solution with this value.
s. t
M
.
Ax +
X.
-
Is
= b
or 1;
(12)
s.
•=
0,1,
... ,U.
q=l,....Q
v(X
)
is an initial
lower bound on the value of an optimal solution to (12).
As discussed in the previous section, a dual problem to
(12)
is
constructed from
abelian group G inducing a set Y which includes all feasible solutions to (12),
M
problem
but which eliminates many infeasible solutions.
For u t R , the Lagrangean
an
—
"
..„
.
...u.
.
UN..,.
"
I.
..
"i" '»i
i
Jll
i
«^" *^^><Wm***'MW**
177
(13)
(x.sUY
Y = {(x,s)|
and the
TP
a.x.
T
e.x. -
"
e.s.
y
t-
or 1,
s.
-
0,
.... U.}.
1,
dual problem is
w(X
= mill
)
L(ulx
s.c. u
)
R
e
(lA)
.
For future reference, the maximization in (13) is written as ub + G(eiu,X ); the
algorit'ons given by Shapiro [1968], Glover [1969], Gorry, Northup and Shapiro [1973],
compute G(o|u,\
for all a
)
e
G as a natural by-product of computing G(g|u,X ).
For any \c S, ve define the set
J(X) = (j|x. =
The
IP
ontiaal
diial
methods at
=
\
in
,0
1
(12) without loss of optimality}
solution is found bv the ontimalitv conditions.
j
)
until an
Alternativelv. the
IP
anv ooint and branch and bound nethodi can be
dual methods can be abandoned at
oerforued usine onlv orojects
(15)
can be viewed as building uo the set J(X
c
for
c
j
J
fX
).
Without loss of ootiraalitv. the oroiect k can be eliirinated from
Theoren
M
consideration in (12) (i.e. k c J(X )) if. for any u c R ,
2:
X^c*" * u(b-a"'')
Proof:
Let v(X
t
x,
+
G(3
-
a^|u,X°)
^
denote the value of (12)
= 1)
N
v(\°ly^
=
= x"c'' + max
1)
^
j-1
N
*
Is
« b - a
j=l
or 1,
0,1
j
jl
k
U
'
''«Pi»8W:i*^-«^^!y«^.i,lii»^
•miism*''^^-
(16)
0(X°)
v.-itb x^ »
1
.
We have
]
178
Dualizing, we obtain the following upper bound on v(X
X°c^
*
uCb-a"")
max
+
-
|x
11
(X°cJ - uaJ)x.
I
J
j=l
N
J
j=l
J
i-1
'
(17)
^
s.t.
^
-Oorl,jjlk
X
J
s.
= 0.1,2,. .,U.
.
1
1
But the objective function in
(.17)
Thus, if the upper bound on v(X
|
simply the left hand side of inequality (16).
is
x^ = I) is less than v(a ),
the project k can be ignored without loss of optimality
5.
the best known solution,
.
]
SENSTTTJITY ANALYSE OF EFFICIENT SOLUTIONS
For convex multicriterion optimization problems involving more than
it is generally not possible to describe completely the subsets S(x
is an efficient solution (see (5)).
efficient solution at
^^
...
2
criteria,
.
.
of S in which x
)
k
Instead, the usual procedure is to find an
and see how it changes in the direction
V
k
Steuer [1973], Ceoffrion et al [1972]). For the
(Evans and
8X
multicriterion optimization
IP
problem (3), we are able to perform this sensitivity analysis through the dual
problem.
Specifically, suppose we have constructed an
and let
(x
,
s
)
:
Y denote the optimal
)
=
dual problerj which solves (3),
IP
solution to (3) found by the
IP
dual.
We
have
v(X
in
^
w(0
1
=
)
Cx'
and moreover.
T
w(x'^)
" max
(X°Cx')j
y
t=l
T
^•'^-
(Ax' +
I
IsSoi^. = b
(18)
t=l
T
u
y
t=l
that is,
= 1,
u<
all other u
=
'
' 1;
is optimal
M
5 0;
in this linear programming problem.
Let w(\) denote the maximal objective function value in (18) as
functions v{\) and v(\) are convex functions satisfying w(\)
£
X
varies.
The
v(X) by duality.
Ii
principle, we have only to perform linear programming sensitivity analysis on (18)
^WM
*P' t-^ -i^'*
'
'
W^-'L^WW
.l^L.i!SMWWBtWWtUJj..i.-W>SWN"'JWU
ji
^
..^i
J,
r-" ^" ''''^**^^?'
!a»r»)g3
W i5U«»ij
ii^t)eii.i»»i i5,<(»jit
f«
,
to compute the maximal value of
with
X»
problem (3).
6uch that
2
and therefore that
Ob)
V
(x
,s
•"
u.
1
remains optimal in (18)
remains optimal in the
)
IP
mul ticriterion
The difficulty is that the number of columns in (18) vill be enormous
and column generation procedures are required.
We oait further details except to
state that generalized progranmiing, otherwise known as Dantzig-Wolfe decomposition
(see Lasdon [1973],
Magnanci, Shapiro and Wagner [1973], can be used for this
purpose.
6.
CONCLUS TONS
In
this paper, we have discussed multicriterion IP problems and how recent
results in IP duality theory can be used to generate efficient solutions to these
problems.
The results presented here are far from complete and much research and
practical experimentation remains to be done.
As ve saw in Section 2, the IP dual methods are theoretically complete in
the sense that an IP dual problem can always be constructed by a finite procedure
which solves
a
given IP problem.
Practical experience (Fisher, Northup and Shapiro
[l97i], however, has shown that the
IP dual methods sometimes,
is to combine the dual methods with branch and bound
The procedure of section
In
5
but not always,
The natural resolution of this difficulty
require excessive numerical calculations.
(see Fisher and Shapiro [197h]1
needs to be reconciled with this practical consideration.
addition, the branch and bound procedures should be designed so that only one
pass is made through the tree of enumerated solutions in computing the efficient
V
More generally, theoretical research is
optimal in (3) as X varies in int S.
x'
needed into the structure of the family of IP duals (14) generated as
X
varies in
S.
There are properties of efficient solutions to problem (1) which it is important to study further and try to characterize.
able to identify the set of all projects
solutions.
j
For example, we would like to be
which are included in all efficient
Conversely, we would liVe to identify the set of all projects excluded
from all efficient solutions.
Another phenomenon to be studied which we have ignored here are the dynamics of
public investment decision making.
We eliminated an explicit study of dynamics by
formulating the static model (1) of
a T
period problem.
As mentioned in the introduction, some public Investment decision models in-
volve distribution as well as location sub-problems implying the appropriateness of
mixed
P multicriterion
The
models.
decomposition method for mixed
IP
IP
duality theory can be combined with Benders'
(Shapiro
[1974] ).
Thus,
it
appears that the
approach suggested in this paper can be extended to these problems.
nay wish to use
max (u (x)
^^^f^VfS^-
a
concave utility function u
(x)) instead of max Cx.
^
n£#^^iilV Ji^M
J>itJ^jjejjl^!j
Finally, we
in the objective function of
(l);that is
The study of such problems is another
.
180
area of future research.
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(1973b), "isproved Bounds for Integer Programs: A Supergroup Approach,
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,
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ferlin, S.
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<«
i
u »iujjiB^^j».i>
!i g[,
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