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Socio-Economic Planning Sciences 34 (2000) 85±99
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Project prioritization: a resource-constrained data
envelopment analysis approach
Wade D. Cook a, Rodney H. Green b,*
a

Schulich School of Business, York University, Toronto, Canada
b
School of Management, University of Bath, UK

Abstract
The problem considered in this paper is that of selecting, from a larger set of proposals, a subset of
projects to be undertaken. Each project is expected to make use of input resources under a number of
headings to produce outputs under a number of headings. It is desired to establish a subset of projects
that can be justi®ed as making the best use of available resources. There is no a priori agreement
amongst all concerned about how inputs and outputs should be weighted and combined to facilitate the
evaluation and selection of the projects. In essence, our approach treats each subset of the projects that
could feasibly be selected within the resource constraints as a single, composite project. These composite
projects are then evaluated, by data envelopment analysis, against a `production technology' de®ned by

the available projects. In fact, evaluation and selection are combined in a single model by placing the
data envelopment analysis model within a mixed-binary linear programming framework. This model is
illustrated using Oral, Kettani and Lang's [1, Management Science 1991; 37(7): 871±883] data on 37
R&D projects. Extensions to the basic model are discussed in the context of prioritizing highway safety
retro®t projects. # 2000 Elsevier Science Ltd. All rights reserved.
Keywords: Multicriteria decision making; Data envelopment analysis; DEA; Mixed-binary linear program

1. Introduction
Data envelopment analysis (DEA) was originally conceived by Charnes, Cooper and Rhodes
* Corresponding author. Tel.: +44-1225-826742; Fax: +44-1225-826473.
E-mail address: mnsrhg@bath.ac.uk (R.H. Green).
0038-0121/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 3 8 - 0 1 2 1 ( 9 9 ) 0 0 0 2 0 - 8

86

W.D. Cook, R.H. Green / Socio-Economic Planning Sciences 34 (2000) 85±99

(CCR) [2] as a tool for examining the relative eciency of production units on the basis of ex
post data on outputs produced and inputs consumed. It has proved remarkably successful; the

publication of [2] in 1978 had, by 1994 [3], led to nearly 1000 papers dealing with applications,
methodological extensions and computational issues. Whilst many studies have been concerned
with the eciency of (past) production, it is clear that DEA is now playing a wider role. In
particular, DEA-inspired approaches have assumed status within the toolkits of investigators
concerned with multicriteria decision making [e.g. 4±8]. In these circumstances, it is desired to
compare some set of alternatives characterized over multiple dimensions, with a view to:
. identifying the best alternative,
. ranking the alternatives,
or, perhaps more likely,
. establishing a shortlist of the better alternatives for detailed scrutiny.
The application discussed here falls ®rmly into the multicriteria decision making arena. It
involves the selection, from a larger set of proposals, of a subset of projects to be undertaken.
Individual projects are expected to produce bene®ts under a number of headings and, in so
doing, will make use of resources under a number of headings. It is desired to establish a
subset of projects that can be justi®ed to all concerned as making the best use of the available
resources.
This prioritization problem, in various forms, has received substantial attention over the past
several decades [9]. Our approach to the problem has its seeds in Bunch et al. [10] but is
speci®cally related to that of Oral, Kettani and Lang (OKL) [1]. OKL's point of departure is
identical to ours: the CCR DEA model.

In the interests of fairness to each of the proposed projects, OKL erect a rather complex
multi-stage collective evaluation and selection model on this foundation. Our approach, which
combines evaluation and selection in a single stage, remains substantially faithful to CCR DEA
and is somewhat less complex.
Our approach is presented in the third section, after ®rst discussing various preliminary
matters. An example application follows in the fourth section where, for comparative purposes,
we use OKL's data. The next two sections re¯ect on our choice of underlying DEA model and
consider various elaborations to the prioritization model, in the context of highway safety
retro®t projects. Concluding remarks follow in the ®nal section.

2. Preliminaries
A set P (={1. . .k . . . vPv}) of project proposals is to somehow be evaluated over a set O
(={1. . .j . . . vOv}) of outputs and a set I (={1. . .i . . . vIv}) of inputs. Project k is characterized by
the magnitudes of its outputs ykj (e0) to be produced and its inputs xki (e0) to be consumed.
There is a limit Li on the quantity of input i available to the set of projects as a whole and we
assume that at least one project satis®es these limits. It is desired to select a subset of projects,
s WP, which can be justi®ed as making the best use of the available resources.
It is assumed that all the projects are, in principle, supportable; all would be undertaken in

W.D. Cook, R.H. Green / Socio-Economic Planning Sciences 34 (2000) 85±99

87

the absence of the resource constraints. It is also assumed that the projects are neither
synergistic nor interfering, in the sense that, if both projects a and b were selected, the outputs
thus produced would be the sum of their respective outputs and similarly for the inputs used.
If some function:
Yk ˆ Y…x kl , . . . , x ki , . . . , ykl , . . . , ykj , . . .†
were available, such that it were possible to arrive at an `objective' evaluation Yk of each
project, a net bene®t say, this could be used to rank the projects. Further, s could be obtained
in a relatively straightforward manner. A natural representation for this situation might be as a
binary knapsack problem [e.g.11] along the following lines:
Maximize Sk2P ck Yk
subject to
Sk2P ck x ki ELi
ck 2 f0, 1g

i2I
k2P

…1†

Diculties arise because of the non-availability of Y or, at the least, as a result of
disagreement amongst the various interested parties concerning its form and detail.
A DEA-based approach circumvents these diculties by allowing each project proposal to
evaluate itself, relative to all the projects under consideration. Essentially, each project k is
allowed to rate itself as highly as possible via a kind of bene®t/cost ratio:
hk ˆ …Sj2O ukj ykj †=…Si2I vki x ki †,
by choosing the weights ukj and vki to be applied to its outputs and inputs, respectively. The
only restriction imposed is that no project p is allowed to receive a rating greater than 1 with
those weights. This self-evaluation is achieved by solving the following linear program on
behalf of each project k [2]. (Also see [12] for a formal treatment of situations where one or
more of xki, ykj might be zero.)
Maximize hk ˆ Sj2O ukj ykj
subject to
Si2I vki x ki ˆ 1
Sj2O ukj ypj ÿ Si2I vki x pi E0
ukj , vki e0 j 2 O, i 2 I

k2P

…2†
p2P

Whilst self-evaluation in this way is entirely notional, there is an implicit fairness in the
process. The ratings achieved depend only on the data for each project relative to the data for
the other projects.
The values of hk might now be used to rank the projects, but the problem of how to select a
subset to support within the resource constraints persists.
It is tempting to simply replace Yk by hk in the binary knapsack problem of Eq. (1).
However, this would, in general, be misguided, as the following example indicates. Imagine the

88

W.D. Cook, R.H. Green / Socio-Economic Planning Sciences 34 (2000) 85±99

situation where there is a number of projects each with a single output and a single input.
Three of the projects, A, B and C, have the following values for x.1, y.1 and h., respectively:
A
B
C

400,
300,
100,

400,
225,
26,

1.00
0.75
0.26.

It can easily be seen that from the viewpoint of the knapsack model, project A is inferior to
a combination of projects B and C. However, the latter combination is obviously inferior to A,
in terms of the quantity of output produced per unit of input, and should never be chosen in
preference to A.
Clearly, whilst hk may provide a meaningful ranking of the projects, to the limits of the
discrimination available given an upper limit of 1, it is not appropriate to treat these values
additively as in Eq. (1). In order to retain the apparent ¯exibility and fairness o€ered by a

DEA-based approach, we combine evaluation and selection into a single prioritization model,
as described in the following section.
Before proceeding, however, it is worth pointing out that our proposed model attempts to
draw a compromise between what might be regarded as two opposing views of optimal
selection in this context. One view is the traditional bene®t/cost ratio approach to evaluating a
set of choices (e.g. projects). This approach concentrates on the output per unit of input;
project a is thus preferred to project b if the bene®t/cost ratio of the former exceeds the latter.
No direct consideration of budget limitations on the inputs is given at the evaluation stage;
these must, somehow, be considered at the selection stage.
An alternative view is that typi®ed by the usual mathematical programming approach where
bene®t/cost ratios are not a direct consideration; rather, satisfying budget constraints on the
input resources whilst maximizing some measure of total output (bene®t) is the goal. Said
another way, if two groups of projects both meet the resource constraints (perhaps more than
one such constraint) and yield equal aggregate bene®ts, we would be indi€erent as far as the
desirability of these two groups was concerned. From the bene®t/cost viewpoint, however, the
group with the smaller cost would be preferred. So, on the one hand (the bene®t/cost
approach), resources have a value and the less used, the better. On the other hand (the
mathematical programming approach), we essentially assume resources have no value, except
when we try to exceed their budgetary limits. What we propose herein is an approach to
evaluation and selection that tries to capture both of these aspects.

3. A prioritization model
Given that the individual projects are independent, neither synergistic nor interfering, any
subset s of the set P of projects can be thought of as a single, composite project. The outputs
of this composite project Ysj are the combined-by-addition outputs of its constituent projects;
similarly for its inputs Xsi. The focus of interest now, of course, is an evaluation of each
composite project relative to the set of all such composite projects. This latter set we will call

W.D. Cook, R.H. Green / Socio-Economic Planning Sciences 34 (2000) 85±99

89

P(P ) and is essentially the so-called power set of P [13] (excluding the empty set b). The
individual projects constituting the highest rated composite(s), satisfying the resource
constraints, are then candidates for selection. Thus, noting that Ysj=Sk$sykj and Xsi=Sk$sxki,
Eq. (2) becomes:
Maximize hs ˆ Sj2O usj ysj
subject to
Si2I vsi x si ˆ 1
Sj2O usj ypj ÿ Si2I vsi x pi E0

usj , vsi e0

s 2 P…P†
…3†
p 2 P…P†
j 2 O, i 2 I

As the number of elements (composite projects) in P(P ) is 2jPj ÿ 1, which is large even for
relatively modest vPv, Eq. (3) does not represent a practical proposition. However, as a ®rst
step toward practicality, the number of ``E'' constraints in Eq. (3) can be reduced from 2jPj ÿ
1 to vPv.
Imagine dividing the `` E '' constraints in Eq. (3) into two groups: the ®rst group is
associated with the singleton subsets of P i.e. {1}, {2}. . . {vPv} while the second group is
associated with the non-singleton subsets e.g. {1,2}, {1,3}, etc. Eq. (3) can then be written as:
Maximize hs ˆ Sj2O usj ysj
subject to
Si2I vsi x si ˆ 1
Sj2O usj ypj ÿ Si2I vsi x pi E0
Sj2O usj Yqj ÿ Si2I vsi Xqi E0
usj , vsj e0

s 2 P…P†

p2P
q 2 P…P† ÿ P 0
j 2 O, i 2 I

…4†

where P'={{1}, {2}. . . {vPv}}, the set of all singleton subsets of P.
It is evident that any constraint in the second group of `` E '' constraints is an additive
combination of two or more constraints in the ®rst group. Thus, if the constraints in the ®rst
group are satis®ed, then, so must any constraint in the second group. Therefore, the second
group contains only redundant constraints and can be removed. The basic prioritization model
for composite projects thus becomes:
Maximize hs ˆ Sj2O usj ysj
subject to
Si2I vsi Xsi ˆ 1
Sj2O usj ypj ÿ Si2I vsi x pi E0
usj , vsi e0

s 2 P…P†
…5†
p2P
j 2 O, i 2 I

We now restrict the scope of the index s in the objective function of Eq. (5) by recognizing
that, in this context, interest would be restricted to a particular subset of P(P ). This subset, S,
can be characterized by the following two conditions:
Condition (a). For all s $ S, the constraints on all resources are satis®ed:
8i 2 I Sk2s x ki ELi ,
Condition (b). For all s $ S, no project can be added without violating Condition (a):

90

W.D. Cook, R.H. Green / Socio-Economic Planning Sciences 34 (2000) 85±99

8p 2 P ÿ s 9i 2 I such that Sk2s[f pg x ki > Li :
Condition (a) is an obvious requirement while Condition (b) follows from the observation
that all projects are supportable. Any composite project to which a further project could be
added without violating Condition (a) would be so augmented. Thus, the proposal is to look
only at those composites that absorb at least one of the resources up to its usable limit; i.e. any
amount of that resource left over is not sucient to permit inclusion of another project. Then,
within that subset of composites, one ®nds the composite whose aggregate bene®t to aggregate
cost ratio is maximized.
Rather than generating the set S explicitly, and subsequently evaluating each of its members
via Eq. (5), we do so implicitly by placing Eq. (5) within a mixed-binary non-linear
programming framework Eq. (6), below. Here, ck is 1 if project k is included in the composite
s, and 0 otherwise. Optimization now takes place over ck, uj and vi.
Maximize Sj2O uj …Sk2P ck ykj †
…ck , uj , vi †

subject to
Si2I vi …Sk2P ck x ki † ˆ 1
Sj2O uj ypj ÿ Si2I vi x pi E0
Sk2P ck x ki ‡ li ˆ Li
…1 ÿ ck †x ki ‡ Mck ‡ Mdki eli ‡ 1=M
Si2I dki E j I j ÿ1
ck , dki 2 f0, 1g
uj , vi , li e0
M >> 0

p2P
i2I
k 2 P, i 2 I
k2P
k 2 P, i 2 I
j 2 O, i 2 I

…6†

Before going on to linearize Eq. (6), some explanation is in order. The model now seeks the
best evaluated subset satisfying Conditions (a) and (b), above. Obviously, Condition (a)
translates directly into the constraints:
Sk2P ck x ki ‡ li ˆ Li ,
where li is the slack in resource i. Condition (b) is a little more dicult but is implemented by
the constraints:
…1 ÿ ck †x ki ‡ Mck ‡ Mdki eli ‡ 1=M k 2 P, i 2 I
Si2I dki E j I j ÿ1 k 2 P:
The e€ect here is to require that at least one of the resource slacks, li, be too small to allow
another project into s. For a given s $ S, consider the ®rst of these constraints for some k $ s,
i.e. ck=1. The constraint is obviously satis®ed because of the positive multiple of M on the
left-hand side. Now, consider the situation for some k ( s, i.e. ck=0, and xki E li. The
corresponding ®rst constraint can be satis®ed by setting dki=1, thus achieving a positive
multiple of M on the left-hand side. However, the e€ect of the second constraint is to ensure

W.D. Cook, R.H. Green / Socio-Economic Planning Sciences 34 (2000) 85±99

91

that at least one of the variables dk1, dk2, . . . dkjIj remain at zero. Hence, 9i $ I such that xki e
li+1/M as required.
Whilst software capable of solving Eq. (6) is available, it can be linearized to bring it within
the capability of more readily available mixed-binary linear programming software. This
linearization involves the following changes of variables: akj=ckuj and bki=ckvi.
Model (6) now becomes:
Maximize

…ck , akj , bki , uj , vi †

S…k2P,

j2O† akj ykj

subject to
S…k2P, i2I † bki x ki ˆ 1
Sj2O uj ypj ÿ Si2I vi x pi E0
Sk2P ck x ki ‡ li ELi
…1 ÿ ck †x ki ‡ Mck ‡ Mdki eli ‡ 1=M
Si2I dki E j I j ÿ1
ck , dki 2 f0, 1g


akj e0

akj EMck k 2 P, j 2 O
uj eakj 

p2P
i2I
k 2 P, i 2 I
k2P
k 2 P, i 2 I
(7)



bki e0


bki EMck
k 2 P, i 2 I

vi ebki

vi Ebki ‡ M…1 ÿ ck † 
M >> 0
where the two sets of constraints highlighted by the vertical bars serve to connect the new
variables akj, bki to the original variables ck, uj and vi.
Before applying this model in the next section, it is important to note that the fairness in
evaluation implicit in Eq. (2) is retained in our prioritization model. Each project proposal
thus has an equal right to participate in the de®nition of the `production technology' as well as
to combine with other projects to be evaluated against said technology and to be selected. This
process depends only on the data for the projects relative to each other and on the available
resources.

4. An example application
OKL [1] demonstrate their approach to collective evaluation and selection in an application
relating to the Turkish iron and steel industry. Here, 37 projects are available each of which is
predicted to provide bene®ts under ®ve headings:

92

W.D. Cook, R.H. Green / Socio-Economic Planning Sciences 34 (2000) 85±99

. direct economic contribution to the iron and steel sector through improved quality and
productivity, cost reductions, etc.
. indirect economic contribution to sectors depending on the iron and steel sector through
better quality, lower prices, etc.
. technological contribution through better use of imported technology, etc.
. scienti®c contribution in the sense of better use of existing scienti®c knowledge, advancing the
body of scienti®c knowledge, etc.
. social contribution in terms of job creation, better working conditions, higher living
standards, etc.
In achieving bene®ts under these headings, each project would require a budget allocation from
a single monetary resource. The data for the projects are shown in Table 1.
The total resource available to the selected projects is 1000.00 units. The average resource
requirement over the 37 projects is 67.99. It could therefore be expected that s would contain
in the order of 15 projects.
On solving Eq. (7), using AMPL/CPLEX [14], with the data summarized in Table 1, s={1,
6, 14, 15, 16, 17, 18, 23, 26, 27, 31, 32, 34, 35, 36, 37} with a collective rating (hs ) of 0.700 and
a total resource use of 962.8 units. For purposes of comparison, OKL's selected subset is
identical except our projects 6 and 32 are replaced by 21 and 29, with a resource use of 964.7.
We thus agree on 14 of the 16 projects selected. Their selected subset has a collective rating of
0.690 when evaluated by Eq. (5).
It should be emphasized that the weights uj, vi do not re¯ect any a priori judgements
concerning their absolute or relative values. If it is considered important to re¯ect such
judgements within the prioritization process, further constraints can be added in the manner of
Thompson et al.'s [15] `assurance region' extension to the CCR DEA model. (See also [16±19]
for further discussion on this form of extension to CCR DEA.)
In general, we can consider Eq. (7) as augmented with a possibly empty set of constraints
AR (uj, vi ). These represent any restrictions on the weights and their inter-relationships that
the decision maker(s) deem appropriate.
By way of sensitivity analysis for our solution to OKL's problem, we have experimented
with various forms of assurance region augmentation to Eq. (7). As an illustration, AR (uj, vi )s
of the form:
uj 1euj 2euj 3euj 4euj 5,
where j 1, . . .,j 5, a permutation of the integers {1,. . . ,5}, re¯ect weak orderings of the weights uj.
Taking all such weak orderings into account identi®es a robust `core' of 13 projects {1, 14, 16,
17, 18, 23, 26, 27, 31, 34, 35, 36, 37} which is invariably selected. It also identi®es a `margin' of
6 projects {6, 11, 15, 21, 29, 32} which are selected in various groups of 2 or 3 according to the
speci®c ordering imposed on the weights.

5. Choice of DEA model
The choice of the CCR DEA model as the basis for our prioritization model implies that its

Research and development
project

Indirect economic
contribution

Direct economic
contribution

Technical
contribution

Social
contribution

Scienti®c
contribution

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

67.53
58.94
22.27
47.32
48.96
58.88
50.10
47.46
55.26
52.40
55.13
32.09
27.49
77.17
72.00
39.74
38.50
41.23

70.82
62.86
19.68
47.05
48.48
77.16
58.20
49.54
61.09
55.09
55.54
34.04
39.00
83.35
68.32
34.54
28.65
47.18

62.64
57.47
6.73
21.75
34.90
35.42
36.12
46.89
38.93
53.45
55.13
33.57
34.51
60.01
25.84
38.01
51.18
40.01

44.91
42.84
10.99
20.82
32.73
29.11
32.46
24.54
47.71
19.52
23.36
10.60
21.25
41.37
36.64
15.79
59.59
10.18

46.28
45.64
5.92
19.64
26.21
26.08
18.90
36.35
29.47
46.57
46.31
29.36
25.74
51.91
25.84
33.06
48.82
38.86
(continued on

Project
cost
84.20
90.00
50.20
67.50
75.40
90.00
87.40
88.80
95.90
77.50
76.50
47.50
58.50
95.00
83.80
35.40
32.10
46.70
next page)

W.D. Cook, R.H. Green / Socio-Economic Planning Sciences 34 (2000) 85±99

Table 1
Data on 37 research and development projects relating to their expected performance on ®ve criteria and their costsa

93

94

Research and development
project

Indirect economic
contribution

Direct economic
contribution

Technical
contribution

Social
contribution

Scienti®c
contribution

Project
cost

19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37

53.02
19.91
50.96
53.36
61.60
52.56
31.22
54.64
50.40
30.76
48.97
59.68
48.28
39.78
24.93
22.32
48.83
61.45
57.78

51.34
18.98
53.56
46.47
66.59
55.11
29.84
58.05
53.58
32.45
54.97
63.78
55.58
51.69
29.72
33.12
53.41
70.22
72.10

42.48
25.49
55.47
49.72
64.54
57.58
33.08
60.03
53.06
36.63
51.52
54.80
53.30
35.10
28.72
18.94
40.82
58.26
43.83

17.42
8.66
30.23
36.53
39.10
39.69
13.27
31.16
26.68
25.45
23.02
15.94
7.61
5.30
8.38
4.03
10.45
19.53
16.14

46.30
27.04
54.72
50.44
51.12
56.49
36.75
46.71
48.85
34.79
45.75
44.04
36.74
29.57
23.45
9.58
33.72
49.33
31.32

78.60
54.10
74.40
82.10
75.60
92.30
68.50
69.30
57.10
80.00
72.00
82.90
44.60
54.50
52.70
28.00
36.00
64.10
66.40

a

Source: Oral et al. [1].

W.D. Cook, R.H. Green / Socio-Economic Planning Sciences 34 (2000) 85±99

Table 1 (continued )

W.D. Cook, R.H. Green / Socio-Economic Planning Sciences 34 (2000) 85±99

95

underlying empirical `reference technology' or `production possibility set' [20±22] is suitable to
our purposes.
An (empirical) production possibility set is a declaration of the totality of potential
production possibilities that might plausibly be observed. In our case, this is based on the
evidence of the ®nite collection of production possibilities that are to be observed. In a
situation where there is a set D={1. . . d. . . vDv} of decision making units (DMUs) where DMU
d has produced a vector of outputs Yd ˆ …Yd1 . . . Ydj . . . YdjOj † from a vector of inputs
Xd ˆ …Xd1 . . . Xdi . . . XdjIj †, then the CCR production possibility set, T CCR (D ), can be
represented as:
o
n
j
S
l
X
EX,
S
l
Y
eY,
l
e0,
8d
2
D
…8†
…X, Y † 2 RjIj‡jOj
d2D
d
d
d2D
d
d
d
‡
We then identify the set D of DMUs in Eq. (8) with our set of composite projects P(P ),
where the latter obviously contains the individual projects as the singleton subsets {1}, {2}, . . .
{vPv} of P. It can then be immediately observed that T CCR (P(P )) contains non-negatively
scaled versions of all projects, individual and composite. This is implicit in Eq. (3), above,
which is the starting point for our prioritization model.
The argument behind the derivation of model (5) from Eq. (3) can now be seen. Imagine the
set of input/output vectors corresponding to the composites in P(P ) divided into two subsets:
those associated with the vPv singleton subsets {1}, {2}, . . . {vPv} (i.e. the individual projects
themselves), and the remainder associated with the non-singleton subsets (i.e. the composites).
We can therefore write the vector (Sp$P(P )lpXp, Sp$P(P )lpYp ) in Eq. (8) as


ÿ
…9†
Sp2P lp x p ‡ Sq2…P…P †ÿP 0 † lq Xq , Sp2P lp yp ‡ Sq2…P…P †ÿP 0 † lq Yq
where P '={{1}, {2}, . . . {vPv}}. Now, with the convention that an index q identifying a
composite in Eq. (9) also identi®es the subset of projects comprising that composite, any lq $0
in Eq. (9) can be set to zero by the algorithm:
8p 2 q lp 4lp ‡ lq :
This follows from the way that input/output vectors corresponding to composites are
constructed, i.e., by addition of the input/output vectors of the projects themselves. Repeated
application of the above algorithm for all lq $ 0 would serve to drive all lq to zero. T CCR
(P(P )) is essentially equivalent to T CCR (P ). The latter is thus capable of representing the
former.
A key feature of the CCR reference technology, by virtue of the unbounded (from above)
multipliers lp, is that constant returns to scale are assumed. We regard this as appropriate here
for two main reasons. Firstly, there seems little merit in rewarding projects for being relatively
ecient technically but at an inappropriate scale [20] in terms of their proposed conversion of
inputs into outputs. Secondly, the e€ect of the resource constraints will ensure that the
composites in S will operate at a similar scale, subject only to the granularity inevitable in
combining discrete projects.
However, if it were required to take scale aspects into account, this would typically be done
via the Banker, Charnes and Cooper (BCC) [20] production possibility set, T BCC (P(P )). This

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W.D. Cook, R.H. Green / Socio-Economic Planning Sciences 34 (2000) 85±99

di€ers from T CCR (P(P )) in that, as well as the lower bounds on the multipliers lp, there is a
`convexity' constraint in the form of Sp2P…P †lp ˆ 1 on their sum. It can easily be seen that
T BCC (P(P )) and T BCC (P ) are not equivalent; the composite project equivalent to the sum of
all the individual projects is in the former, by de®nition, but not in the latter.
Importantly, Kao [23] shows that T BCC (P(P )) is equivalent to the Koopmans production
possibilty set [21] T Koopmans (P ) which, itself, di€ers from T CCR (P ) by the incorporation of
upper bounds lp E 1 on the multipliers. It is a straightforward matter to modify Eq. (5) and,
hence, Eqs. (6) and (7), to implement T Koopmans (P ) rather than T CCR (P ) if desired.

6. Further extensions to the prioritization model
The previous sections have presented a basic prioritization model, Eq. (7) +AR (uj, vi ). In
the present section, some useful extensions will be sketched out in the context of prioritizing
highway safety retro®t projects. In this context, the projects comprise speci®c sections of
highway that are being considered for improvement from the viewpoint of accident potential or
hazard. It is advantageous to use this alternative application setting, which itself instigated our
work, as this will not only lend some perspective on the breadth of applicability of our basic
model, but will also serve as an example of the necessity for, and ease of, its extension. These
extensions will relax the assumptions concerning the independence of the projects and in so
doing will exploit the binary structure in Eq. (7) to model details such as mutual exclusion/
inclusion within subsets of projects.
Identifying hazardous sections of highway and prioritizing measures to improve them, in
terms of reducing potential accidents, is a major consideration in all highway departments. A
signi®cant literature exists on the characterization of hazardous locations. The subject of
interest here consists of two inter-related aspects. Firstly, there is a concern with the prediction
of accident rates and their severity in terms of explanatory factors such as trac levels, road
geometrics and so on. This research has focused on the use of multiple regression as a
mechanism for obtaining appropriate predictions [e.g. 24, 25]. A second component of research
in the road safety and accident analysis arena involves accident reduction factors; speci®cally
the improvement in safety that will be achieved if a segment of highway network is modi®ed in
some way [e.g. 26±28]. This corpus of work, together with appropriate expert judgement,
enables the prediction of bene®ts achievable consequent on the allocation of retro®t funds to
speci®c project proposals.
With regard to prioritizing identi®ed hazardous locations for treatment, the practice in most
jurisdictions has been to rank these locations by either total accident frequency (e.g. using the
total number of accidents on the road section over the past 3 years), or accident rate (e.g.
accidents per million vehicle kilometers). A number of jurisdictions have recognized the multicriteria nature of the prioritization problem. Thus, in looking at accident reductions, total
accidents should properly be broken down into di€erent severity classes such as fatal, major
injury, minor injury, and property damage. In Kentucky, for instance, numerical weights are
applied to various accident types to re¯ect costs to the public. Troxel [29] discusses a number
of severity models for combining fatal and injury accidents into an overall ®gure. There are, of
course, di€erent views in di€erent jurisdictions of what the weights on di€erent accident types

W.D. Cook, R.H. Green / Socio-Economic Planning Sciences 34 (2000) 85±99

97

should be. Persaud et al. [28] present a multicriteria methodology for determining appropriate
weights to attach to di€erent classes of accident in evaluating the relative importance of a set
of retro®t measures. Further to these considerations, bene®ts can go beyond accident reduction
and may also include improved road serviceability and trac ¯ow.
Thus, on the bene®t (or output) side, the prioritization problem is clearly multi-dimensional.
On the cost (or input) side, the multi-dimensional nature is also apparent. Obviously, the
monetary expenditure required to implement a particular safety improvement, vis a vis the
overall retro®t budget for the planning period, is the primary input at issue. Other factors may
also impinge: availabilty of labour, plant, materials, and design oce time, for example.
In this multi-output/multi-input setting, a model of the form dealt with in the previous
sections, i.e Eq. (7) +AR (uj, vi ), clearly has potential as a tool for assisting in the selection of
a subset of projects from a larger collection of proposals. However, hitherto, we have assumed
that projects are essentially independent, an assumption which now must be modi®ed.
Whilst in many prioritization settings the issue to be addressed is whether or not to
undertake a given project, in the highway safety project prioritization problem there is at least
one additional dimension; namely, the treatment or design choice. Speci®cally, there can be
alternative ways to take corrective action at a particular hazardous site. For example, run o€
road accidents may be preventable or can be reduced either through shoulder upgrading
(paving or widening), installation of guard rails, or even corrections to the geometry of the
roadway. Each option has di€erent associated outputs in terms of reductions in the various
accident types and roadway serviceability as well as di€erent calls on resources. Thus, there is
a mutually exclusive set of treatments that may be applied for each hazardous site being
considered. Model (7) can be easily modi®ed to cater for this situation. Firstly, denote the set
of distinct hazardous sites under consideration as Q and the set of (mutually exclusive)
treatments being considered as T. The index set of all project variants under consideration thus
becomes the cartesian product of sets Q and T:
P ˆ Q  T:
Subscripts referring to this index set, such as k and p, are now ordered pairs hq,ti where q $ Q
and t $ T. To ensure that no more than one project variant at a site q is selected, the following
constraints are needed:
St2T chq, ti E1 for q 2 Q
A second practical modeling requirement is that of specifying commonality of treatment,
whereby a group of potential project sites in some geographic area should receive the same
treatment. If, for example, shoulder widening is applied in a particular location to prevent
accidents, it would normally be the case that this treatment would be implemented throughout
the surrounding area. Thus, if A (subset of Q ) is a set of sites to be so considered, we proceed
as follows:
1. de®ne binary variables gAt for t $ T,
2. include the constraint St$TgAt E1
3. Include the constraints

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W.D. Cook, R.H. Green / Socio-Economic Planning Sciences 34 (2000) 85±99

cha,

ti

ˆ gAt

a 2 A, t 2 T

or
cha, ti EgAt

a 2 A, t 2 T:

The ®rst set of constraints in (iii) implies that all (or none) of the sites in A be selected and
treated identically, whereas the second version allows some of the sites to remain untreated.

7. Concluding remarks
The sketches of the previous section exemplify the manner in which exclusivity and
inclusivity relationships amongst projects may be incorporated into Eq. (7) +AR (uj, vi ). It
should now be apparent that the structure implicit in Eq. (7) +AR (uj, vi ) is actually ¯exible
enough to accommodate rather complex inter-relationships quite straightforwardly. In
particular, the various practical considerations arising in the highway safety retro®t setting
were seen above as easily implemented.
Whilst our original intention was to adapt OKL's [1] approach for our purposes, upon
further investigation it became clear that modeling inter-relationships amongst the projects
could not be conveniently managed by their selection model. This is not to be taken as a
criticism of their approach if, for no other reason, it was not designed with this scenario in
mind. Indeed, it otherwise appears well thought out. However, the mixed-binary linear
program at the heart of our approach is considerably smaller than theirs. This may be an
advantage even in situations where modeling inter-relationships amongst project proposals is
not a requirement.

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