152
Int. J. Engineering Management and Economics, Vol. 3, Nos. 1/2, 2012
Optimising project performance: the triangular
trade-off optimisation approach
Baruch Keren
Department of Industrial Engineering and Management,
SCE-Shamoon College of Engineering,
Bialik/Basel Sts., Beer Sheva 84100, Israel
E-mail: [email protected]
and
Department of Management and Economics,
The Open University of Israel,
Ravutzki 108, Raanana 43107, Israel
E-mail: [email protected]
Yuval Cohen*
Department of Management and Economics,
The Open University of Israel,
Ravutzki 108, P.O. Box 808, Raanana, 43107, Israel
Fax: +972-97780668
E-mail: [email protected]
and
Department of Industrial Engineering,
Tel-Aviv Afeka College of Engineering,
Tel-Aviv, 69107 Israel
E-mail: [email protected]
*Corresponding author
Abstract: It is generally accepted that the three major dimensions of project
success are time, budget, and quality. Most of the research in project planning
is focused on the time-cost trade-off, and only a few papers have considered the
three dimensions together. This paper describes the evolution of formulations
for optimising project time-cost and quality, and continues by developing a
new, non-linear optimisation formulation that better reflects the triangular
trade-off structure between time, budget, and quality. In particular, the
Cobb-Douglas formula is adopted and its use is illustrated. The model is
discussed, and the structure of the trade-off is analysed and illustrated.
Keywords: project management; project scheduling; project performance;
project trade-off; Cobb-Douglas function.
Reference to this paper should be made as follows: Keren, B. and Cohen, Y.
(2012) ‘Optimising project performance: the triangular trade-off optimisation
approach’, Int. J. Engineering Management and Economics, Vol. 3, Nos. 1/2,
pp.152–170.
Copyright © 2012 Inderscience Enterprises Ltd.
Optimising project performance
153
Biographical notes: Baruch Keren is a Senior Lecturer in the Industrial
Engineering and Management Department, Shamoon College of Engineering
and Lecturer at the Open University of Israel. He received his BSc, MSc
and PhD (Summa Cum Laude) in Industrial Engineering from Ben-Gurion
University. His professional experience includes 13 years in the Israel
Chemicals Ltd. and its subsidiaries in the areas of industrial engineering,
economics and auditing. His current research interests are decision-making
under uncertainty, production planning, project management and operations
research.
Yuval Cohen is the Head of the Industrial Engineering programme at the
Open University of Israel and Senior Lecturer at Tel-Aviv Afeka college of
Engineering. His areas of specialty are production planning, industrial learning,
design of manufacturing control systems and logistics management. He has
published many papers in these areas. He received his PhD from the University
of Pittsburgh (USA), his MSc from the Technion – Israel Institute of
Technology and BSc from Ben-Gurion University. He is a Fellow of the
Institute of Industrial Engineers (IIE) and a full member of the Institute for
Operations Research and Management Sciences (INFORMS).
1
Introduction
Planning and executing projects (and their goals) always involves elements of time
(due-date), cost (budget) and quality (specifications). There is a natural trade-off
between time, cost and quality, but these three critical objectives are not independent,
but rather intricately related. The time-cost-quality trade-off problem (TCQP) is a
multiple objective optimisation problem, which mainly focuses on selecting options with
corresponding time, cost and quality to complete an activity in order to minimise project
duration and cost, while maximising project quality (see Rwelamila and Hall, 1995;
Wang and Feng, 2008).
Obviously, the value and performance of a project increase as time and cost decrease,
and as quality increases. While the importance of project quality has never been disputed,
modern project management research began with emphasis on minimising the time
needed for a project. The first two major project planning techniques focused on project
duration (and its relationship to costs):
1
the critical path method (CPM) (Kelley, 1961)
2
project and evaluation review technique (PERT) (Malcolm et al., 1959).
Research on the time-cost trade-off followed using linear programming (LP) crashing
techniques (Charnes and Cooper, 1962; Fulkerson, 1961; Prager, 1963). These crashing
techniques were deterministic models of activities characterised by linear (or piece-wise
linear) cost functions with respect to their duration. In particular, some models were
presented both as network-flow problems and as the network-flow dual problems
(Moder and Phillips, 1970). Other models focused on improving the algorithm
efficiency (Goyal, 1975; Siemens, 1971). The research has continued in several
directions:
154
B. Keren and Y. Cohen
1.1 Non-linear and/or non-continuous trade-off functions
Kapur (1973) presented an algorithm for project cost-duration analysis problems with
quadratic and convex cost functions. Falk and Horowitz (1974) presented cost-duration
analysis for critical path problems with concave cost-time functions. Demeulemeester
et al. (1996) describe two algorithms, based on dynamic programming logic, for
optimally solving the discrete time/cost trade-off problem (DTCTP) in deterministic
activity-on-arc (AOA) networks of the CPM type. The costs are not given in a function
form, but rather as discrete values related to certain durations. This structure turns the
problem into a non-polynomial combinatorial problem. Later, Demeulemeester et al.
(2000) presented an improved branch-and-bound technique for solving the DTCTP.
Sunde and Lichtenberg (1995) extended the time-cost trade-off to include net present
value (NPV) calculations.
1.2 Trade-off structure
Berman (1964) studied the effect of resource allocation. Pulat and Horn (1996) extended
the time-cost trade-off to the time-resource trade-off, making in possible to deal with
duration taken as a function of several resources. Liberatore and Pollack-Johnson
(2006) suggested extending the trade-off by removing precedence relationships and
task-streaming. Another direction was to minimise project cost: Wu and Li (1994)
proposed a new technique for minimising the project cost based on a cut set parallel
difference method. Their work attracted some related feedback and corrections, such as
Kamburowski (1995). Drezet and Billaut (2008) proposed a scheduling solution
technique for projects with labour constraints and time-dependent activities.
Only a few papers include a direct treatment of the trade-off between quality, time
and budget together: Babu and Suresh (1996) suggested that project quality may be
affected by project crashing and developed LP models to study the trade-offs between
time, cost, and quality. Each of the three models developed optimises one of these entities
by assigning desired bounds to the other two. Their model was validated by Khang and
Myint (1999) using an illustrative case study for showing its practical value. Tareghian
and Taheri (2006) developed three inter-related integer programming models for discrete
values such that each model optimises one of the given entities (time, cost, or quality) by
assigning desired bounds on the other two. In order to solve these models, they later used
a scatter search technique (Tareghian and Taheri, 2007). El-Rayes and Kandil (2005)
designed a multi-objective genetic algorithm (GA) model to transform the traditional
two-dimensional time-cost trade-off analysis to a three-dimensional time-cost-quality
trade-off analysis.
1.3 Project trade-off under stochastic scheduling
Research considering the stochastic nature of activities has been dealing with the effect of
resource limitations and extending the deterministic models described above in
Section 1.2, such as Pulat and Horn (1996). Some of the papers in this field are:
Golenko-Ginzburg and Gonik (1998), Lova et al. (2009), Van Dorp and Duffey (1999)
and Van de Vonder et al. (2005). Al-Fawzan and Haouari (2005) proposed a bi-objective
model for robust resource-constrained project scheduling.
Optimising project performance
155
Deckro et al. (1995) and Deckro and Hebert (2002) presented a quadratic formulation
based on minimising costs of normal durations plus costs related to quadratic deviations
from these normal durations. Laslo (2003) proposed a stochastic extension of the CPM
time-cost trade-off model which included four fundamental formulations that represent
different assumptions of the effect of the changing performance speed on the distribution
parameters of the activity duration, as well as the effect of the random activity duration
on the activity cost. Cohen et al. (2007) suggested a model to provide a robust policy
rather than optimal values.
1.4 Evolutionary computing methods for optimising the trade-off
Due to the computational difficulty of solving the time-cost-quality trade-off problem as
a combinatorial problem, researchers have been trying to apply evolutionary methods
characterised by global search capability to solve the problem. These methods are: GA
(e.g., Lambrechts et al., 2008; Xingfu et al., 2007), ant colony optimisation (Afshar et al.,
2007), particle swarm optimisation algorithm (Zhiyong et al., 2007) and hierarchical
subpopulation particle swarm optimisation algorithm (Wang and Feng, 2008). Kilic et al.
(2008) used GA for minimising risk of two criteria.
The quantifying of project quality is one of the major obstacles for the TCQP
formulation. The questions that must be considered when doing so are:
1
How to measure or estimate the quality of each activity in a project when each one
might have different properties and type?
2
How does allocation of more (less) time, budget and resources to an activity affects
its quality?
3
How to quantify the overall project quality given that the quality of each component
activity was determined?
4
How to combine time, cost and quality into one model?
El-Rayes and Kandil (2005) proposed a model that deals with some of those issues. For a
discrete and given level of resources n, they suggested quantifying the overall project
l
quality by the weighted average formula
K
∑ wt ∑ wt
i ,k
i
i =1
× Qin,k , where Qin,k is the
k =1
performance of quality indicator (k) in activity (i) using resource utilisation (n); wti and
wti,k are the weight of activity (i) and the weight of quality indicator respectively.
The model of El-Rayes and Kandil (2005) requires entering input for the quality of all
activities in every feasible scenario. Their method also requires planners to identify two
types of weights: wti that represents the importance and contribution of the quality of this
activity to the overall quality of the project; and wti,k the weight of quality indicator in
activity (i) that indicates the relative importance of this indicator to others being used to
measure the quality of the activity. When solving their combinatorial TCQP formulation,
El-Rayes and Kandil (2005) used an evolutionary method that, by definition, cannot
guaranty optimality.
In this paper, we use continuous functions for cost, quality, and time. This enables us
to formulate the TCQP as mathematical programming problem and to solve it optimally.
The Cobb-Douglas function, validated by Pendharkar et al. (2008) is the key that makes
156
B. Keren and Y. Cohen
this continuous trade-off formulation between time, cost and quality possible. The rest of
this paper is organised as follows: Section 2 presents the evolution of terminology and
related formulations. Section 3 develops a linear formulation that includes a quality-time
trade-off. Section 4 presents the Cobb-Douglas function and its relationship to the
activity trade-off. Section 5 presents the proposed formulation, discusses it, and displays
graphs to illustrate the triangular trade-off. Section 6 concludes the paper.
2
Terminology and basic related formulations
2.1 Fundamentals
Many of the crashing problem formulations are presented as activities on arcs (AOA)
networks. An AOA network is defined as a directed acyclic graph G(V,E), where V is a
set of vertices (nodes) and E is a set of edges (arcs) connecting certain pairs of these
nodes. The shortest completion time of an acyclic graph problem (AOA network) can be
formulated and solved by a variety of methods. Formulation 1 (below) is a typical LP
formulation for finding the shortest completion time of a graph.
The beginning time of the project at node 1 is set to be V1 ≥ 0. In general we define Vi
to be the time-length of the longest path from node 1 to the node i. We define ti,j as the
time length of the arc from the node i to node j. For any node Vj, we assume that
Vj ≥ Vi + ti,j, for all the nodes Vi which immediately (directly) precede Vj. The nodes are
numbered such that the last node (Vn) occurs at the end of the project.
In this simple CPM problem, the objective is to minimise the makespan of the
project, and is conveniently written as: Min Vn, where Vn is the time of the last node of
the project (sink). Thus, the CPM LP formulation for minimising the makespan is as
follows:
Formulation 1
Min Vn
s.t.
V j − Vi ≥ ti , j j = 2,..., n ; i = j − 1, j − 2,...,1 ; ti , j ∈ E
(1)
Vi ≥ 0 i = 1,.., n.
In formulation (1), an activity cannot begin before all its immediate precedence activities
are completed. All values of ti,j are given constants. The number of the precedence
constraints is the sum of the number of arcs (including the dummy arcs) + number of
nodes n.
2.2 Example and data for a case study
For illustration and comparison, this paper uses a simple project activity network
described in Figure 1. This network and the data in Table 1 are the same data as in Babu
and Suresh (1996). The example and the data in this subsection shall be used throughout
the paper.
Optimising project performance
Figure 1
157
A simple project activity on arcs (AOA) network
1
8
2
6
3
4
10
13
12
11
5
9
7
Table 1
Numerical data
Task
no.
Task
code
Task
Slope Slope
Precedence Tndays Tcdays CnUSD CcUSD Qn Qc
Ei,j
CT
Q
1
E12
E1,2
-
2
1
2
E23
E2,3
E1,2
4
2
3
E34
E3,4
E2,3
10
7
6,200 7,300
1
0.8
367 0.067 415.4
4
E45
E4,5
E3,4
4
3
4,100 4,900
1
0.8
800 0.200
820
5
E46
E4,6
E3,4
6
4
2,600 3,000
1
0.7
200 0.150
390
6
E47
E4,7
E3,4
7
5
2,100 2,400
1
0.7
150 0.150
315
7
E57
E5,7
E4,5
5
3
1,800 2,200
1
0.3
200 0.350
630
8
E68
E6,8
E4,6
7
4
9,000 9,600
1 0.75 200 0.083
747
800
Pi,j
2,300
1
0.9 1,500 0.100
80
3,200 3,600
1
0.7
480
200 0.150
9
E79
E7,9
E4,7, E5,7
8
6
4,300 4,600
1
0.4
150 0.300
1290
10
E810
E8,10
E4,5, E6,8
9
6
2,000 2,500
1
0.5
167 0.167
334
11
E911
E9,11
E7,9
4
3
1,600 1,800
1
0.9
200 0.100
160
12
E912
E9,12
E7,9
5
3
2,500 3,000
1 0.95 250 0.025
62.5
13
E1013 E10,13
E8,10
2
1
1,000 1,500
1
500 0.100
100
14
E1213 E12,13 E9,11, E9,12
6
1
3,300 4,000
1 0.95 140 0.010
33
44,500
Source: Taken from Babu and Suresh (1996)
0.9
158
B. Keren and Y. Cohen
The critical path and minimal time of the network in Figure 1 has the following LP
formulation:
Example of Formulation 1
Min V13
s.t.
V2 − V1 ≥ t1,2 ;
V3 − V2 ≥ t2,3 ;
V4 − V3 ≥ t3,4 ;
V5 − V4 ≥ t4,5 ;
V6 − V4 ≥ t4,6 ;
V7 − V4 ≥ t4,7 ;
V7 − V5 ≥ t5,7 ;
V8 − V5 ≥ t5,8 ;
V8 − V6 ≥ t6,8 ;
V9 − V7 ≥ t7,9 ;
V10 − V8 ≥ t8,10 ;
V11 − V9 ≥ t9,11 ;
V12 − V9 ≥ t9,12 ;
V12 − V11 ≥ t11,12 ; V13 − V10 ≥ t10,13 ; V13 − V12 ≥ t12,13 ;
(2)
Vi ≥ 0 i = 1, 2,...,13
For the normal durations in Table 1 applied to the activities in Figure 1, formulation (2)
yields project duration of V13 = 44. The variable cost of the project (VC) is the sum of the
normal cost of all activities: VC = USD44,500 (The notation ‘$’ is used in formulae for
simplicity’s sake.) Assume also that a fixed cost of K is required for each day of the
project. Hence the fixed cost is K ⋅ Vn. In our example it is assumed that K = USD400, so
the fixed cost is FC = 400 × 44 = $17,600. Therefore, the total cost of the project in this
solution is: TC = 44,500 + 17,600 = $62,100. The total cost formula for this case is:
TC = K ⋅ Vn +
∑∑ Cn
i, j
i
j
i , j∈E
2.3 Crashing: cost/time trade-off problem and budget consideration
Some variable definitions are required for introducing more data to the example.
Definitions
Tni,j
normal activity duration (i and j are the activity’s start and finish milestones)
Tci,j
crash activity duration
Cni,j
normal activity cost
Cci,j
crash activity cost
Qni,j
normal activity quality
Qci,j
crash activity quality
CTi,j
cost-time’ slope for an activity
QTi,j
quality – time’ slope for an activity
ai,j
amount of activity time reduced from its normal duration (Tni,j)
B
total expediting budget (for shortening activity times).
(3)
Optimising project performance
159
An extension of the previous problem is known as ‘the cost/time trade-off problem’.
For this extension, a limited budget B for reducing the activities of the project is
given. Each activity Ei,j has a given normal duration ti,j and can be accelerated to a
given crash time value TCi,j by spending some expediting budget. A common assumption
is linearity, i.e., in order to reduce the duration of activity Ei,j by ai,j units of time,
a budget value of CTi,j × ai,j must be spent, where CTi,j is a given constant and ai,j
is a decision variable. Since an activity cannot be performed in a shorter time
than its crash value ai,j ≤ ti,j – Tci,j, the general formulation of this problem is the
following:
Formulation 2
Min Vn
s.t.
(
V j − Vi ≥ Tni , j − ai , j
∑∑ CT
i, j
i
)
j = 2,..., n ; i = j − 1, j − 2,...,1 ; Tni , j ∈ E
⋅ ai , j ≤ B ∀i, j ∈ E
(4)
j
ai , j ≤ Tni , j − Tci , j ∀i, j ∈ E
ai , j ≥ 0 ∀i, j ∈ E
Vi ≥ 0 i = 1,.., n
The example project shown in Figure 1, with the corresponding data of Table 1 and a
shortened budget over USD7,552, gives the minimal makespan V13 = 26. However,
reducing the budget below USD7,552 increases the makespan (e.g., to B = USD3002
results in V13 = 32). Comparing the total cost of these two points, we have:
B = $7,552 : V13 = 26
TC = 400 × 26 + 44,500 + 7,552 = $62,452
B = $3, 002 : V13 = 32
TC = 400 × 32 + 44,500 + 3, 002 = $60,302
The difference between the total cost (TC) of these two cases is due to the trade-off
between expediting expenditure and the overhead cost saved by it: since we assumed
that there is a fixed overhead cost of K for each day of the project, the fixed cost is
K ⋅ Vn. The fixed cost can be reduced by performing the project activities faster. But
accelerated (expedited) performance increases the variable cost of the project by
∑∑ CT
i, j
i
⋅ ai , j which may be limited by the budget B. Thus, saving a day reduces the
j
overhead by USD400. The expediting cost is either determined by the budget (B) or by a
marginal cost of USD400 per day. The expediting cost is a piece-wise linear function of
time.
By minor modifications, the cost/time trade-off problem can be formulated as a LP
problem that computes the optimal acceleration (expediting) budget and its allocation
among the activities and the optimal project makespan. The general formulation of this
problem is:
160
B. Keren and Y. Cohen
Formulation 3
Min KVn +
∑∑ CT
i, j
i
⋅ ai , j
j
s.t.
(
V j − Vi ≥ ti , j − ai , j
)
j = 2,..., n ; i = j − 1, j − 2,...,1 ; ti , j ∈ E
(5)
ai , j ≤ ti , j − Tci , j ∀i, j ∈ E
ai , j ≥ 0 ∀i, j ∈ E
Vi ≥ 0 i = 1,.., n
The major difference between this formulation and previous formulations is that the
objective function reflects cost and not the makespan. Applying the data of Figure 1 and
Table 1 to Formulation 3, we get a solution to this formulation (with K = 400) that yields
an optimal project-makespan of V13 = 30 days, a budget for expediting: B = $3,802 and
the optimal shortening plan: a2,3 = 2, a4,6 = 2, a5,7 = 2, a7,9 = 2, a3,4 = 3, a8,10 = 3, a12,13 = 3.
The optimal total cost of the project for this case is TC = 400 × 30 + 44,500 + 3,802
= $60,302 (this must be equal to or better than the TC of previous formulations). In our
case any project makespan in the range [30, 32] gives the same optimal cost. Therefore,
we must choose a minimum cost preferred solution according to secondary criteria
(minimum makespan, minimum expediting cost, etc.).
3
Adding quality to the cost-time trade-off problem
This section presents a linear formulation that is consistent with the approach
presented in Babu and Suresh (1996). Babu and Suresh (1996) and later Khang
and Myint (1999) claimed that the quality of an activity decreases when its duration
is reduced. When the duration of activity Ei,j is reduced by ai,j units of time and more
money is consumed, there is also a loss of quality. In the formulation suggested in this
section the amount of this quality loss is QTi,j ⋅ ai,j, where QTi,j is a given constant of
quality/time trade-off and ai,j is a decision variable. The quality of an activity Ei,j is
defined as Qi,j. The maximum quality of an activity is defined to be Qi,j = 1 (100%)
and is achieved at the end of its normal duration. The actual quality of an accelerated
activity is Qi,j = (1 – QTi,j ⋅ ai,j). However, the quality cannot be set to less than some
limits since it may risk the entire project. Therefore, we must set constraints of the form
Qci,j ≤ {Qi,j = (1 – QTi,j ⋅ ai,j)} ≤ 1. However, even if a project with low quality were
acceptable, its value should decrease. We calculate the weighted loss of quality of the
Pi , j QTi , j ⋅ ai , j , where Pi,j is the activity penalty for lack of quality for
project by
∑∑
i
j
the entire project. The cost of lost quality in that project is:
∑∑ P
i , j QTi , j
i
we obtain,
j
⋅ ai , j . Thus
Optimising project performance
161
Formulation 4
Min KVn +
∑∑ CT
i , j ai , j
i
j
+
∑∑ P
i, j
i
⋅ QTi , j ⋅ ai , j
j
s.t.
(
V j − Vi ≥ ti , j − ai , j
)
j = 2,..., n ; i = j − 1, j − 2,...,1 ; ti , j ∈ E
ai , j ≤ ti , j − Tci , j ∀i, j ∈ E
ai , j ≤
1 − Qci , j
QTi , j
(6)
∀i, j ∈ E
ai , j ≥ 0 ∀i, j ∈ E
Vi ≥ 0 i = 1,.., n.
The objective function in (6) can be written as KVn +
∑∑ (CT
i, j
i
+ Pi , j ⋅ QTi , j )ai , j . The
j
conclusion is that the Babu and Suresh (1996) model includes two cost components that
increase linearly while expediting an activity: the expedition cost and the loss-of-quality
cost. A practical method for estimating the value of Pi,j is to assume that it is proportional
to activity Ei,j normal cost, e.g., Pi,j = Cni,j. According to this assumption, the cost of lost
of quality in a project is:
Cni , j ⋅ QTi , j ⋅ ai , j .
∑∑
i
j
Assuming that Pi,j = Cni,j, K = $400 and applying the other data for the example
project shown in Figure 1 with the corresponding data of Table 1 to (6), we find: the
optimal project makespan is V13 = 40 days; the optimal budget for acceleration is
B = $560; the cost of quality loss is USD132; and the optimal duration shortening plan is:
a12,13 = 4. The optimal total cost of the project for this case is TC = 400 × 40 + 44,500 +
560 +13 = $61,192. This time the cost of quality-loss is also added, resulting in higher
total cost (TC) than the one found for the Formulation 3. As expected (to avoid large
quality loss), the optimal duration is much higher (and the acceleration budget much
smaller) than the one found for Formulation 3. However, Formulation 4 does not consider
the trade-off between cost and quality, and therefore ignores part of the triangular
trade-off. For this purpose, we introduce a new approach, which uses the Cobb-Douglas
function, in Section 4.
4
Cobb-Douglas function and its relationship to the activity trade-off
The validity of the Cobb-Douglas function form for projects was tested and proved by
Pendharkar et al. (2008). They empirically tested Cobb-Douglas functional form with
respect to team size and software size for a real-world data set containing more than
500 software projects. Their results indicated that the hypothesised Cobb-Douglas
function form is true.
The Cobb-Douglas functional form of production functions is widely used to
represent the relationship of inputs to an output (Felipe and Adams, 2005). In their
162
B. Keren and Y. Cohen
seminal paper, Cobb and Douglas (1928) expressed the relationship (i.e., trade-off
formula) between:
1
output denoted by Q
2
labour denoted by L
3
capital denoted by B, as:
Q = ALα ⋅ B β
(7)
where A, α, β are coefficients, determined by technology and traditionally estimated
by linear regression over logarithmic transformation of (7) that gives (8):
ln(Q) = ln( A) + α ln( L) + β ln( B )
(8)
For estimating the values A, α, β for the linear equation (8) the company has to acquire
database of previous projects and operations so that a reliable regression could generate
trustworthy values. The correlation coefficient of the regression gives an indication of the
estimations’ reliability level. The database should include the throughput, the labour and
the budget. The throughput could be measured in monetary values or quantities produced
and labour by total labour cost or by invested hours.
The variable A is known as ‘total-factor productivity (TFP)’ and reflects the ratio of
outputs to inputs related to the relevant state of technology. The value of A is computed
via the multiple regression of equation (8).
α and β are coefficients that reflect the change of the throughput related to L and B
respectively. In other words, α and β indicate the marginal throughput increase as result
of increasing L and B by one unit respectively. The partial marginal throughput change
can be increasing for values α or β greater than 1, decreasing for values between zero to
1, and for negative values of α or β the throughput decreases as resources are added.
Empirically (e.g., Felipe and Adams, 2005) α and β are statistically significant,
non-negative and α + β ≈ 1. So that:
Q = ALα ⋅ B1−α
(9)
For an activity in a project Ei,j, we propose replacing the labour/work-load (L) by activity
duration (ti,j) times the number of workers (Wi,j), and defining A as a conversion constant
between output and quality. The capital (B) is translated into the investment in equipment
and materials, or the activity cost, and instead of capital we will use the term budget. As
mentioned, Cobb-Douglas production function is widely used in economic applications to
represent the relationship of inputs to output. Since budget and time can be considered as
inputs of a project while its quality (or performance) can be considered as an output, it is
natural to extend Cobb-Douglas to project management. Moreover, Cobb-Douglas has
three parameters A, α, β which provide flexibility for shaping the trade-off curves close
to their empirical nature. The accurate values of Cobb-Douglas’s parameters can be
estimated by linear regression for any project/activity based on analysis of historical data.
The correlation coefficient which is computed in any linear regression can be a means for
validating the trade-off correctness.
Since the model uses the AOA structure, activities will be identified by their pair of
origin-destination nodes (i, j). The actual quality (performance index) of each activity
(i, j), in the project is expressed by Qi,j. Defining the upper boundary for the quality level
Optimising project performance
163
as 100%, the value range of Qi,j is: 0 ≤ Qi,j ≤ 1. The single activity adjusted formula
becomes:
αi , j
Qi , j = Ai , j (Wi , j ⋅ ti , j )
1−α i , j
⋅ Bi , j
(10)
Equation (10) holds for each project operation (i, j) and expresses its quality.
Alternatively, by several simple computing manipulations the work-load and the budget
can be expressed as:
1−α i , j
1
αi , j
(Wi, j ⋅ ti, j )
=
Ai , j ⋅ Bi , j
⇒ (Wi , j ⋅ ti , j )
Qi , j
−α i , j
1−α i , j
=
Ai , j ⋅ Bi , j
Qi , j
(11)
That is,
1−α i , j
Ai , j ⋅ Bi , j
Wi , j ⋅ ti , j
(
)
Wi , j ⋅ ti , j
⎛
Qi , j
=⎜
⎜ Ai , j ⋅ Bi , j1−α i , j
⎝
−α i , j
=
(
⇒ Wi , j ⋅ ti , j
Qi , j
)
αi , j
=
Qi , j
1−α i , j
Ai , j ⋅ Bi , j
(12)
and
1
⎞α i , j
⎟
⎟
⎠
(13)
In the same manner:
Bi , j
⎛
Qi , j
=⎜
⎜ Ai , j ⋅ (Wi , j ⋅ ti , j )αi , j
⎝
1
⎞1−α i , j
⎟
⎟
⎠
(14)
It is reasonable to assume that during the planning phase of a project, the planner sets for
every work package of the work breakdown structure (WBS), the necessary workforce
and time Wi,j ⋅ ti,j, and appropriate equipment, materials and budget Bi,j in order to achieve
the required performance level Qi,j. Any lack of budget, time, or workers for the activity
would decrease the performance level Qi,j.
5
Minimum cost formulation for the triangular trade-off
This section presents our proposed formulation. For that purpose, the formulation in
Section 3 will be converted to a Cobb-Douglass oriented formulation. For this purpose,
we define the Cobb-Douglass somewhat differently:
α
(1−αi , j )
Qi , j = Ai , j ⋅ ti , ij, j ⋅ Bi , j
(15)
In equation (15), Bi,j is the budget of activity Ei,j, ti,j is its time and Qi,j is its quality
(performance level). According to equation (15), we can simultaneously set planning
values for time, budget (cost), and quality for all project activities. Now time, budget, and
quality are decision variables which get their values by optimisation.
164
B. Keren and Y. Cohen
The cost minimisation of the following Formulation 5 includes the fixed cost (K ⋅ Vn),
the cost of carrying out all activities
Bi , j and the penalty for deviations from the
∑∑
normal quality:
i
∑∑ P
i , j (1 − Qi , j ).
i
j
The precedence constraints of the AOA network
j
preserve their main structure while using the above Cobb-Douglass formulation for tradeα
(1−α i , j )
offs. Since we assume that Qi , j = Ai , j ⋅ ti , ij, j ⋅ Bi , j
for all activities, we can also
conclude that:
1
ti , j
⎛
⎞α i , j
Qi , j
⎜
⎟
=⎜
1−α i , j ) ⎟
(
⎜ Ai , j ⋅ Bi , j
⎟
⎝
⎠
(16)
Thus, the new form of Formulation 5 is:
Formulation 5
Min K ⋅ Vn +
∑∑ P (1 − Q )
∑∑ B
i, j +
i
i, j
j
i
i, j
j
s.t.
⎛
Qi , j
⎜
V j − Vi − ⎜
(1−αi, j )
⎜
⎝ Ai , j ⋅ Bi , j
V j − Vi ≥ 0
1
⎞αi , j
⎟
≥0
⎟
⎟
⎠
∀i, j ∈ E and known Ai , j , α i , j
for any dummy activity Ei , j
(17)
1
⎛
⎞αi , j
Qi , j
⎜
⎟
− ti , j (min) ≥ 0
⎜
⎟
1−αi , j ) ⎟
(
⎜
⎝ Ai , j ⋅ Bi , j
⎠
Qci , j ≤ Qi , j ≤ 1
∀i, j ∈ E
∀i, j ∈ E and known Ai , j , α i , j
Vi ≥ 0 i = 1,.., n
In practice, any activity has limited ranges for quality, duration and cost (budget). In
order to avoid out-of-range or feasibility problems, it is important to verify by
equation (15) that each activity Ei,j would at least obtain its minimum acceptable quality
Qi,j when its maximum duration ti,j and maximum budget Bi,j are set. Another
recommended check is to verify that both the minimum time duration and minimum
budget for each activity, provide a quality level such that Qi,j ≤ 1. The budget constraints
Bi , j (min) ≤ Bi , j ≤ Bi , j (max) and time constraints can be added to formulation (5). These
constraints also facilitate the process of deriving an initial feasible solution used to solve
the mathematical programming problem in (17).
The numerical values of Ai,j and αi,j for our example are contained in Table 2. Since
we have only two points for each activity with values for time, cost (budget) and quality,
Optimising project performance
165
the values of Ai,j and αi,j for each activity were calculated by equation (18) and the rest of
the data is as given in Table 1. For any case where equation (18) for activity Ei,j, yields:
1 – αi,j = 0, a constraint that Bi,j ≥ Bi,j(min) must be set. Otherwise, the optimisation process
would drive the activity cost to zero (Bi,j = 0).
α i, j
Table 2
⎧
⎪
⎪
0
if
⎪
⎪
⎪
⎪
⎛ Bn ⎞
⎪ ⎛ Qn ⎞
⎪ ln ⎜ Q ⎟ − ln ⎜ B ⎟
⎪
⎝ C ⎠ if
=⎨ ⎝ C ⎠
⎛
⎞
⎛
T
⎪ ln n − ln Bn ⎞
⎜
⎟
⎪ ⎜⎝ TC ⎟⎠
BC ⎠
⎝
⎪
⎪
⎪
⎪
1
if
⎪
⎪
⎪⎩
⎛ ⎛ Qn ⎞
⎛ Bn ⎞ ⎞
⎜ ln ⎜
⎟ − ln ⎜
⎟⎟
QC ⎠
BC ⎠ ⎟
⎝
⎝
⎜
0≤
⎜ ⎛T ⎞
⎛B ⎞⎟
⎜ ln ⎜ n ⎟ − ln ⎜ n ⎟ ⎟
⎜
⎟
⎝ BC ⎠ ⎠
⎝ ⎝ TC ⎠
⎛ ⎛ Qn ⎞
⎛ Bn ⎞ ⎞
⎜ ln ⎜
⎟ − ln ⎜
⎟⎟
Q
⎝ BC ⎠ ⎟ < 1
0<⎜ ⎝ C ⎠
⎜ ⎛T ⎞
⎛B ⎞⎟
⎜ ln ⎜ n ⎟ − ln ⎜ n ⎟ ⎟
⎜
⎟
⎝ BC ⎠ ⎠
⎝ ⎝ TC ⎠
(18)
⎛ ⎛ Qn ⎞
⎛ Bn ⎞ ⎞
⎜ ln ⎜
⎟ − ln ⎜
⎟⎟
Q
⎝ BC ⎠ ⎟
1≥ ⎜ ⎝ C ⎠
⎜ ⎛T ⎞
⎛B ⎞⎟
⎜ ln ⎜ n ⎟ − ln ⎜ n ⎟ ⎟
⎜
⎟
⎝ BC ⎠ ⎠
⎝ ⎝ TC ⎠
Numerical values for Ai,j and αi,j for the example
Task
no.
Task
Ei,j
Tndays
Tcdays
CnUSD
CcUSD
Qn
Qc
αi,j
1 – αi,j
Ai,j
1
2
3
4
5
6
7
8
9
10
E1,2
E2,3
E3,4
E4,5
E4,6
E4,7
E5,7
E6,8
E7,9
E8,10
2
4
10
4
6
7
5
7
8
9
1
2
7
3
4
5
3
4
6
6
800
3,200
6,200
4,100
2,600
2,100
1,800
9,000
4,300
2,000
2,300
3,600
7,300
4,900
3,000
2,400
2,200
9,600
4,600
2,500
1
1
1
1
1
1
1
1
1
1
0.9
0.7
0.8
0.8
0.7
0.7
0.3
0.75
0.4
0.5
0.6640
0.5851
0.7432
0.8615
0.9111
1.0000
1.0000
0.5643
1.0000
1.0000
0.3360
0.4149
0.2568
0.1385
0.0889
0.0000
0.0000
0.4357
0.0000
0.0000
0.0668
0.0156
0.0192
0.0957
0.0971
0.1429
0.2000
0.0063
0.1250
0.1111
11
E9,11
4
3
1,600
1,800
1
0.9
0.5503
0.4497
0.0169
12
E9,12
5
3
2,500
3,000
1
0.95
0.3370
0.6630
0.0032
13
E10,13
2
1
1,000
1,500
1
0.9
0.4650
0.5350
0.0180
14
E12,13
6
1
3,300
4,000
1
0.95
0.1228
0.8772
0.0007
As in Section 3, we assume that Pi,j = Cni,j (proportionality of the normal cost). Running
formulation (5) using the LINGO-11 software package for the numerical values of
Tables 1 and 2 yields the following results: The project total cost is TC = $43,662.8 and
its makespan is V13 = 37 days. All the data for the optimal triangular trade-off solution is
given in Table 3. It can be seen that money can be saved due to downgraded quality for
some activities.
166
B. Keren and Y. Cohen
Table 3
The optimal values for the Cobb-Douglas trade-off example
Time for
nodes
V1
V2
V3
V4
V5
V6
V7
V8
V9
V10
V11
V12
V13
Total:
Time for activities
ti,j
0.0
1.0
3.3
13.9
19.3
19.0
22.3
23.0
28.3
32.5
31.3
31.3
37.3
t1,2
t2,3
t3,4
t4,5
t4,6
t4,7
t5,7
t6,8
t7,9
t8,10
t9,11
t9,12
t10,13
t12,13
1.0
2.3
10.6
5.4
5.1
8.4
3.0
4.0
6.0
9.5
3.0
3.0
4.7
6.0
Budget for activities
Bi,j – USD
B1,2
B2,3
B3,4
B4,5
B4,6
B4,7
B5,7
B6,8
B7,9
B8,10
B9,11
B9,12
B10,13
B12,13
2,217.9
2,994.2
7,306.1
2,252.7
238.7
210.0
180.0
9,614.3
430.0
200.0
1,800.1
2,998.0
939.2
3,286.9
B
34,668.1
TC
43,662.8
Quality
Qi,j
Q1,2
Q2,3
Q3,4
Q4,5
Q4,6
Q4,7
Q5,7
Q6,8
Q7,9
Q8,10
Q9,11
Q9,12
Q10,13
Q12,13
0.900
0.700
0.800
0.920
0.700
0.756
0.600
0.750
0.750
0.667
0.900
0.950
0.900
1.000
While we used the Cobb-Douglass formula for the activities trade-offs, it could also be
used for describing a whole project trade-off. However, Cobb-Douglass trade-off
structure is not easy to visualise. Figure 2 gives an example for such a trade-off and helps
visualise the trade-off structure for one activity. In Figure 2, the quality is measured in
percentage and thus is between zero and one. Note that time (in days) replaces labour,
and budget remains in its monetary value. Figure 3 depicts equal quality lines. Each line
illustrates the trade-off between cost and time while quality remains the same.
Figures 2 and 3 illustrate the real view of the project trade-offs, and help us to see that the
traditional linear (or piece-wise linear) trade-offs are only an approximation.
Figure 2
An example of the quality-cost-time Cobb-Douglas trade-off structure
Optimising project performance
Figure 3
6
167
Equal quality trade-off lines of cost and time for Figure 2
Conclusions
This paper proposes an optimisation model for the triangular trade-off structure of
projects between
1
time
2
cost
3
quality/performance.
The formulation is based on the Cobb-Douglas formula form as a natural extension of the
existing project trade-off models. The concept is illustrated using a simple project and its
implications are discussed. Formulations are developed for various practical situations
illustrating the use of the proposed model. This research can help practitioners and
project managers to find the optimal trade-off for a given project. The values for the
Cobb-Douglas parameters in a real world problem can be concluded by multiple linear
regression of previous existing data.
Analysis of the values of αi,j and βi,j can provide insights related to the performance
and the risks in a given activity. We anticipate to find that 0 < αi,j < 1 and 0 < βi,j < 1, i.e.,
increase in activity duration and budget would increase its quality. This is our normal
assumption during the planning phase.
While this paper focuses on the planning phase some other situations may sometimes
arise during the execution phase: if αi,j < 0 then increase in the activity duration would
decrease its quality, and if βi,j < 0 then an increase in the activity budget would decrease
168
B. Keren and Y. Cohen
its quality. In such anomaly, extra time or budget would actually decrease the quality.
Unfortunately, there are many real life projects with significant deviations from their
budget and their time, which also have poor quality. Therefore, it would not be surprising
to find αi,j < 0 and/or βi,j < 0 in historical data. Planners that identify those phenomena in
historical data should use extra care in planning similar projects to allow more time
and/or budget to achieve a desired quality level.
Future research in this field could be directed towards treating the uncertainty of
activity durations, the uncertainty of activity costs, and the uncertainty of
quality/performance.
References
Afshar, A., Kaveh, A. and Shoghli, O.R. (2007) ‘Multi-objective optimization of time-cost-quality
using multi-colony ant algorithm’, Asian Journal of Civil Engineering (Building and
Housing), Vol. 8, No. 2, pp.113–124.
Al-Fawzan, M.A. and Haouari, M. (2005) ‘A bi-objective model for robust resource-constrained
project scheduling’, International Journal of Production Economics, Vol. 96, No. 2,
pp.175–187.
Babu, A.J.G. and Suresh, N. (1996) ‘Project management with time, cost, and quality
considerations’, European Journal of Operational Research, Vol. 88, No. 2, pp.320–327.
Berman, E.B. (1964) ‘Resource allocation in a PERT network under continuous activity time-cost
function’, Management Science, Vol. 10, No. 4, pp.734–745.
Charnes, A. and Cooper, W.W. (1962) ‘A network interpretation and a directed sub-dual algorithm
for critical path scheduling’, Journal of Industrial Engineering, Vol. 13, No. 4, pp.213–218.
Cobb, C.W. and Douglas, P.H. (1928) ‘A theory of production’, American Economic Review,
No. 1, pp.139–165.
Cohen, I., Golany, B. and Shtub, A. (2007) ‘The stochastic time-cost trade-off problem: a robust
optimization approach’, Networks, Vol. 49, No. 2, pp.175–188.
Deckro, R.F. and Hebert, J.E. (2002) ‘Modeling diminishing returns in project resource planning’,
Computers & Industrial Engineering, Vol. 44, No. 1, pp.19–33.
Deckro, R.F., Hebert, J.E., Verdini, W.A., Grimsrud, P.H., and Venkateshwar, S. (1995) ‘Nonlinear
time/cost trade-off models in project management’, Computers and Industrial Engineering,
Vol. 28, No. 2, pp.219–229.
Demeulemeester, E., De-Reyck, B. and Herroelen, W. (2000) ‘The discrete time/resource trade-off
problem in project networks: a branch-and-bound approach’, IIE Transactions, Vol. 32,
No. 11, pp.1059–1069.
Demeulemeester, E., Herroelen, W.S. and Elmaghraby, S.E. (1996) ‘Optimal procedures for the
discrete time/cost trade-off problem in project networks’, European Journal of Operational
Research, Vol. 88, No. 1, pp.50–68.
Drezet, L.E. and Billaut, J.C. (2008) ‘A project scheduling problem with labour constraints and
time-dependent activities requirements’, International Journal of Production Economics,
Vol. 112, No. 1, pp.217–225.
El–Rayes, K. and Kandil, A. (2005) ‘Time-cost-quality trade-off analysis for highway
construction’, Journal of Construction, Vol. 131, No. 4, pp.477–486.
Falk, J.E. and Horowitz, J.L. (1974) ‘Critical path problems with concave cost-time functions’,
Management Science, Vol. 19, No. 4, pp.446–455.
Felipe, J. and Adams, F.G. (2005) ‘A theory of production: the estimation of the Cobb-Douglass
function: a retrospective view’, Eastern Economic Journal, Vol. 31, No. 3, pp.427–444.
Fulkerson, D.R. (1961) ‘A network flow computation for project cost curves’, Management
Science, Vol. 7, No. 2, pp.167–179.
Optimising project performance
169
Golenko-Ginzburg, D. and Gonik, A. (1998) ‘A heuristic for network project scheduling with
random activity durations depending on the resource allocation’, International Journal of
Production Economics, Vol. 55, No. 2, pp.149–162.
Goyal, S.K. (1975) ‘A note on a simple CPM time-cost trade-off algorithm’, Management Science,
Vol. 21, No. 6, pp.718–722.
Kamburowski, J. (1995) ‘On the minimum cost project schedule’, Omega, Vol. 23, No. 4,
pp.463–465.
Kapur, K.C. (1973) ‘An algorithm for project cost-duration analysis problems with quadratic and
convex cost functions’, AIIE Transactions, Vol. 5, No. 4, pp.314–322.
Kelley, J. (1961) ‘Critical path planning and scheduling: mathematical basis’, Operations
Research, Vol. 9, No. 3, pp.296–321.
Khang, D.B. and Myint, Y.M. (1999) ‘Time, cost and quality trade-off in project management: a
case study’, International Journal of Project Management, Vol. 17, No. 4, pp.249–256.
Kilic, M., Ulusoy, G. and Serifoglu, F.S. (2008) ‘A bi–objective genetic algorithm approach to risk
mitigation in project scheduling’, International Journal of Production Economics, Vol. 112,
No. 1, pp.202–216.
Lambrechts, O., Demeulemeester, E. and Herroelen, W. (2008) ‘A tabu search procedure for
developing robust predictive project schedules’, International Journal of Production
Economics, Vol. 111, No. 2, pp.493–508.
Laslo, Z. (2003) ‘Activity time-cost tradeoffs under time and cost chance constraints’, Computers
& Industrial Engineering, Vol. 44, No. 3, pp.365–384.
Liberatore, J.M. and Pollack–Johnson, B. (2006) ‘Extending project time-cost analysis by removing
precedence relationships and task streaming’, International Journal of Project Management,
Vol. 24, No. 3, pp.529–535.
Lova, A., Tormos, P., Cervantes, M. and Barber, F. (2009) ‘An efficient hybrid genetic algorithm
for scheduling projects with resource constraints and multiple execution modes’, International
Journal of Production Economics, Vol. 90, No. 1, pp.17–25.
Malcolm, D.G., Roseboom, J.H., Clark, C.E. and Fazar, W. (1959) ‘Application of a technique
for research and development evaluation program’, Operations Research, Vol. 7, No.5,
pp.646–669.
Moder, J.J. and Phillips, R.C. (1970) Project Management with CPM and PERT, 2nd ed.,
Van Nostran Reinhold, Litton Educational Publishing, New York, NY.
Pendharkar, P.C., Rodger J.A. and Subramanian G.H. (2008) ‘An empirical study of the
Cobb-Douglas production function properties of software development effort’, Information
and Software Technology, Vol. 50, No. 12, pp.1181–1188.
Prager, W.A. (1963) ‘Structural method of computing project cost polygons’, Management Science,
Vol. 9, No. 3, pp.394–404.
Pulat, P.S. and Horn, S.J. (1996) ‘Time-resource trade-off problem’, IEEE Transactions on
Engineering Management, Vol. 43, No. 4, pp.411–417.
Rwelamila, P.D. and Hall, K.A. (1995) ‘Total systems intervention: an integrated approach to
time, cost and quality management’, Construction Management and Economics, Vol. 13,
No. 3, pp.235–241.
Siemens, N. (1971) ‘A simple CPM time-cost trade-off algorithm’, Management Science, Vol. 17,
No. 6, pp.B354–B363.
Sunde, L. and Lichtenberg, S. (1995) ‘Net-present-value cost/time trade-off’, International Journal
of Project management, Vol. 13, No. 1, pp.45–49.
Tareghian, H.R. and Taheri, S.H. (2006) ‘On the discrete time, cost and quality trade-off problem’,
Applied Mathematics and Computation, Vol. 181, No. 2, pp.1305–1312.
Tareghian, H.R. and Taheri, S.H. (2007) ‘A solution procedure for the discrete time, cost and
quality trade-off problem using electromagnetic scatter search’, Applied Mathematics and
Computation, Vol. 190, No. 2, pp.1136–1145.
170
B. Keren and Y. Cohen
Van de Vonder, S., Demeulemeester, E., Herroelen, W. and Leus, R. (2005) ‘The use of buffers in
project management: the trade-off between stability and makespan’, International Journal of
Production Economics, Vol. 97, No. 2, pp.227–240.
Van Dorp, J. R. and Duffey, M. R. (1999) ‘Statistical dependence in risk analysis for project
networks using Monte Carlo methods’, International Journal of Production Economics,
Vol. 58, No. 1, pp.17–29.
Wang, W. and Feng Q. (2008) ‘Multi–objective optimization in construction project based on a
hierarchical subpopulation particle swarm optimization algorithm’, in Intelligent Information
Technology Application, IITA ‘08. Second International Symposium on Intelligent Information
Technology Application, Vol. 1, pp.746–750.
Wu, Y. and Li, C. (1994) ‘Minimal cost project networks: the cut set parallel difference method’,
Omega, Vol. 22, No. 4, pp.401–407.
Xingfu, G., Chengshun, H. and Denghua, Z. (2007) ‘Study synthesis optimization of
time-cost-quality in project management’, Systems Engineering Theory & Practice, Vol. 27,
No. 10, pp.112–117.
Zhiyong, C., Zhida, D. and Hua, Z. (2007) ‘Research on the unlimited resource levelling
optimization with PSO’, China Civil Engineering Journal, Vol. 40, No. 2, pp.93–96.