An optimal model for construction
time-resource tradeoff problem
Xianjia. Wang1, Zhongping. Wan2
1
2
Institute of Systems Engineering, Wuhan University, Wuhan 430072 China.
School of Mathematics and Statistics, Wuhan University, Wuan 430072 China.
Abstract: So-called the time-resource tradeoff problem, it is a specified project content with a set of
activities to determine an efficient project scheduling according to some precedence relationship and
the renewable resource constraints, under the objective of minimizing the project duration and the
total consumed-resources cost. In this paper we propose a new mathematical model with timeresource trade-off problem, in which objective functions with conflict one another is defined as
adaptive and adjustable between the project duration and the total consumed-resources cost in all
period. This adjustment is based on the corresponding Lagrange multiplier, which is viewed as price
information. It helped us to can control and monitor the resources utilized situation at any time. A
satisfied feasible solution can be obtained in our solution procedure by compromising and adjusting
relationship between the project duration and the total consumed-resource cost. A numerical example
is illustrated. In addition, both the project duration and the total consumed-resources cost in the
Lagrangian relaxation form associated with the resource constraints are interpreted as a two-player
game: one player offers prices to p urchase lots of resource available so as to minimize the project
duration; the other sell the redundant resources based on prices to maximize profit (that is the resource
available be wasted as small as possible).
Keywords: time-resource tradeoff problem; resource-constrained; scheduling; Lagragian relaxation;
solution procedure.
1. Introduction
The major objective of the construction project planning and scheduling is usually to determine
the completion times of the project in a way, which minimizes the project completion cost according
to some precedence relationship of project activities and the available resource limited. The cost of

Supported by the National Natural Science Foundation of China (60274048) and the Doctoral Foundation in Ministry of
Education of China(20020486035).
completing an activity of a project is actually an aggregated cost of usage of several resources
required to perform the tasks. Generally, the activity duration can be regarded as a function of
resource availability. A time-cost tradeoff problem generates minimum cost project schedules as a
function of project realization times, and it is one of the most important aspects of construction
planning and control. Many models and solution procedures have been developed for solving timecost tradeoff problems (Leu et al 2001; Li et al 1999; and so on). Simin and Horn (1996) extended the
multiple objective mathematical models with the minimization of project cost and duration (Deckro
and Hebert 1990) to the minimization problem of resource cost for each activity and project duration
for trade-offs in project scheduling. In these models, it is implicitly assumed that resources are
available in infinite amounts. Clearly, it is not actually for this assumption in the engineering practice.
Based on this extended framework, we here present a time-resource tradeoff model with the
precedence relationship and the resource constraints. Our problem is very different from model
presented Simin and Horn, it mainly will show the following several aspects: 1) the activity duration
is treated as a decision variable; 2) the resource is limited in the time-resource tradeoff problem; 3) the
two-objective time-resource tradeoff scheduling problem in which the total consumed-resources cost
is incorporated one. It is evident that the resource-constrained problem must be considered for timeresource tradeoff project scheduling with resource limited, this is almost similar to the resource
constrained time-cost tradeoff project scheduling problem (Brucker et al 1999; Icmeli and Erenguc
1996). Reason is that because the minimization of completion duration of project activities (it is
actually in a sense of PERT- or CPM-path) is impractical, and the resource conflict situation for
which could be taken place at any time in the project scheduling process cannot effectively be
monitored. Therefore that the resource-constrained time-resource tradeoff project scheduling problem
is considered is necessary and becomes practically more appealing, despite it could become more
difficult to solve.
2. Time-resource trade-off model
The time-resource tradeoff for project scheduling problem involves scheduling, resource cost, and
determining the duration of project activities, in such a way that resource (some of which are available
only in limited amounts) and precedence constraints are met. Preemption is not allowed. Our
objective is to minimize several resource costs and completion duration of project activities. Here, we
assume a project in which the duration of each activity is unknown, thus its value might fall between
the normal and crash duration of the activity. The resource cost and the project duration are assumed
2
to have some decreased function relationship, e.g. linear or other function relationship. The following
notations defined will be used in the our model:
N = the number of activities in the project; K = the number of so-called renewable resource
types; f j = the completion time (duration) of activity j ; H = the set of pairs of activities
indicating finish-start precedence relation, i.e. number of events in the project network; S t

= the set of activities in progress during time interval (t  1, t ] = i f i  t  f i  d i  , i.e. the
set of on-going activities at time t ; ND j = the normal duration for activity j . CD j = the
crash duration (the possible lower bound) for activity j . d j = the duration (or processing time)
variable of activity j that takes integer values between ND j and CD j . R kt = the total
availability of resource type k at time t . a kj (d j ) = the cost amount of resource k required
rik (d j ) = the amount of resource type k that required per period by activity i
for activity j . ~
corresponding to duration variable d j .
Without loss of generally, we assume that the activity 1 denotes the start of the project. Let the
activity N  1 , the dummy activity, denotes the completion of the project. The dummy activity does
r( N 1) k (d N 1 )  0  k .
not require any resource nor does it requires any duration, i.e. d N 1  0 and ~
Our optimization model for time-resource tradeoff problem for K resources can be written as
follows:
Min z1  f N 1
Min z 2 
K
(1)
N
 a
k 1 j 1
s.t. f j  f i  d i
k
j
(d j )
(2)
(i, j )  H
(3)
CDi  d i  NDi for activity i
 ~rik (d i )  Rkt for t  1, 2,, f N 1 , k  1,, K .
(4)
(5)
iSt
Expressions (1) and (2) show that the problem is a two-objective programming one, in which is the
minimization of the project completion and the total cost of consuming resources, respectively.
Constraint (3) ensures that precedence relationships are satisfied. Constraint (4) indicates that the
duration variable of activity be bounded. Constraint (5) is a conceptual statement of the resource
constraints, in which the sum of resource requirements does not exceed their respective capacities
availability in any period, i.e. the resource constraints cannot be violated.
Remark 2.1 The problem will become a resource-constrained project scheduling with duration
variable if we delete the second objective function in our model, i.e.
Min z  f N 1
(6)
s.t. (3), (4) and (5).
3
3. Lagrangian relaxation of resource constraints
This section, the Lagrangian relaxation of the resource constraints (5) will be considered. In terms
of the relaxation approach, the resource leveling results would be affected by changing multipliers
associated with resource constraints. Therefore the resource utilization will be monitored at any time.
Let  k be the corresponding to non-negative Lagrange multipliers to the resource constraints (5).
We have the following Lagrangian function:
K
N
K
f N 1
L( y,  )   a kj (d j )     kt ( ~
rik (d i )  Rkt )
k 1 j 1
k 1 t 1
(7)
iS t
For convenience, we consider the following single objective problem
Min z 2 
K
N
 a
k 1 j 1
s.t. f j  f i  d i
k
j
(d j )
(8)
(i, j )  H
CDi  d i  NDi for activity i
 ~rik (d i )  Rkt for t  1, 2,, f N 1 , k  1,, K
iSt
f N 1  T (the absolute due date of project).
It follows that its dual problem corresponding to the programming (8) is
max d ( , R )
(9)
 0
where
K
f N 1
k 1
t 1
d ( , R ) = [ Lk ( )   kt Rkt ]
and
N
f N 1
i 1
t 1 iS t
(10)
Lk ( )  min [ aik (d i )    kt ~
rik (d i )] .
d
(11)
The optimization problem may hight the partial dual problem corresponding to the original twoobjective optimization problem. Obviously, the dual function d ( , R ) is separable in the
contributions to the dual objective of the individual resource type k . Each sub-problem Lk can be
interpreted as a profit maximization problem (i.e. the resource-wasted as less as possible) for resource
type k when it sees the price kt at time t .
We may interpret the dual optimization (9) using the following “two-firm” model. Firm P, a
minimizing resource cost utility (i.e. minimizes the total consumed-resources cost), facing resource
supply R kt at time t (for convenience, the subscript t is omitted), may sell the firm Q the redundant
resource available in the precondition of completing time of the project scheduling is not affected (i.e.
it means selection of the duration of activity, d j ). The firm Q, for its profit (e.g. it minimizes the
completion time of the project), offering firm P the price kt , needs to purchase lots of the resource
available. Firm P’s objective is to minimize its total consumed-resource cost. Consequently it may be
redundant resources available. Firm Q’s problem is to adjust the prices of kt so as to achieve its
4
maximum benefit (i.e. completion duration of the project as early as possible). Hence it will purchase
more redundant resources from the firm P, considering that the firm P will minimize its resource cost.
4. The solution procedure
To solve our time-resource tradeoff project scheduling problem, in the following solution
procedure given, which could be called a compromise Lagrange relaxation procedure. In this method,
we can adopt the following three-phase structure strategy based on some effective algorithms and
Lagrange relaxation method as well as modification techniques. A satisfied feasible (or near-optimal)
solution to the time-resource tradeoff problem is only obtained by the procedure.
The solution
procedure is in detail described as follows:
Initialization process: Select duration of activity at random in the interval BD j defined by the crash
and normal duration of activity, i.e. BD j = [CD j , ND j ] ; or set all activity duration to normal
durations.
Phase 1: Solve the problem (6) to determine completion duration of the project by means of some
effective algorithms of the resource-constrained project scheduling, e.g. genetic algorithm. Check to
see if the completion duration of the project obtained by Phase 1 algorithm is less than the due date of
the project. If the project duration is less than its due date, go to Phase 2; Otherwise, the duration of
some activities (e.g. the activity of which is that the number (or cost) of resources required times the
duration of the activity is biggish) be decreased such that they are located within the interval BD j ,
back to Phase 1.
Phase 2: Test to see if there is a resource violation at any time in the current solution. If there is no
resource conflict, then go to Phase 3. Otherwise, solve the problem (9) where the duration variable is
limited within the interval BD j to determine the total consumed-resources cost of the project by
using of some primal-dual optimization algorithms in the nonlinear programming with discrete or
continuous variables.
Phase 3: Modify (i.e. reduce) duration of some activities such that there is a proper decreases
nonnegative quantity in the Rkt 
 ~r (d ) .
iSt
ik
i
Verification of the termination condition: One of the following three conditions is met, then
computation terminate. Otherwise, Back to Phase 1.
(1) If Ykt 
 ~r (d )  
iSt
ik
i
kt
for all t and k , where Ykt indicates a supply quantity of given
resource type k at time t and Ykt  Rkt ,  kt be pre-given allowable upper bound (i.e. the
overmuch resources be restricted within the certain limits).
(2) The predetermined number of the Phase iteration is made.
5
(3) For d i j  BD i ( j  1, 2 ) and d i2  d i1  i , if there exists a d i  (d i2 , d i1 ) such that z1 (d )
of satisfied that z1 (d )  z1 (d 1 ) and z1 (d )  z 2 (d 2 ) is the minimum.
5. An illustrative example
To illustrate that this paper proposed model and solving method, a simple project with two type
resources requirements is planned with the network shown in Fig. 1, the normal and crash duration of
all activity in Table 1. For the sake of simplicity, we only discuss that the resource cost and duration
of activity is a linear relationship below:
a kj (d )  b kj  c kj d , for k  1, 2 ; j  1, 2,, 6 .
k
j
(13)
k
j
The nonnegative parameters b , c above generated by random are shown in Table 2. The
resource requirements are R1  10 and R2  6. The relationship between the resource
required per duration and duration for activity are all piecewise linear function, are shown in
Table 3. We now can describe our computation process as follows.
B
C
A
(1
)
(1)
F
D
E
Fig. 1 Project network
progress direction
1. Initialization. Set all activity duration to normal durations. Initial duration and resources required
for activity in Table 4 below.
2. By Phase 1 algorithm, the completion duration of project z1 =20 time units, its scheduling is A-DB-C//E-F, where the symbol C//E denotes parallelity relations of both activity C and E. Note that
the earliest start time of activity B and E are the same.
3. The objective function z 2 = 36.63 units passing through simple computation in Phase 2 algorithm.
4. Proceed to Phase 3 algorithm. By computing, we can select that the duration of the activity and
corresponding to resources required per duration is shown in Table 5 below. Back to 1. Note that
there not exist this qi2 in the I 2  2 and S t2  S t2 1   .
5. By computing, the completion duration of project z1 =19 time units, its scheduling is the same
with result in Step 2 above and, the objective function z 2 =38.32.
6
Such-and-such progressing, a satisfied result, i.e. we can accept, must be obtained.
Remark. In studying of the time-resource tradeoff problem with the resource constraints, we
sometimes shall discover that the project duration is not decreased while the duration of activity is
reduced. For example, we set the duration of the activity to as follows: A=3, B=3 or 4, C=3, D=6,
E=4, F=5. By simple computation, the project duration z1 is 21 or 22 time units, its scheduling is AD-B-C-E-F or A-D-B-E-C-F. The total resources cost is 44.18 and 40.95 respectively. Obviously,
these project durations are all larger than the project duration before computation. This problem
indicates that the project duration is not always decreased with the activity reduction (crashing) under
the resource constraints.
6. Conclusions
In this paper, we proposed a new mathematical model for time-resource tradeoff problem. This
model is different from the other models (e.g. Simin and Horn 1996), it mainly shows the following
several aspects: 1) the activity duration is treated as a decision variable; 2) the resource constraints are
considered; 3) the two-objective functions with conflict one another, the project duration and the total
consumed-resources cost, which is defined as adaptive and adjustable between them in all period. It is
no surprise that the time-resource tradeoff problem is a difficult problem to solve. The objective is to
determinate duration of each activity so that the project duration and the total consumed-resources
cost are as minimized as possible under the some precedence and the resource constraints.
We presented a compromise Lagrangian relaxation procedure based on the three-phase structure
strategy (i.e. Scheduling-Feasibility-Modification), which attempts to solve our time-resource tradeoff
problem. A most satisfied feasible solution to the model could be obtained by using this compromise
Lagrangian relaxation procedure for the continuation. In addition, both the project duration and the
total consumed-resources cost in the Lagrangian relaxation form associated with the resource
constraints are interpreted as a two-player game: one player offers prices to purchase lots of resource
available so as to minimize the project duration; the other sell the redundant resources based on prices
to maximize profit (that is the resource available be wasted as small as possible).
Further research may be undertaken to develop (1) more efficient solution procedure for solving
the time-resource tradeoff problem, (2) the time-resource tradeoff problem under uncertain
environment, (3) the time-resource tradeoff problem with variable resources, and so on.
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Appendix: Date-Tables
Table 1: Bounds of activity duration
Table 2: Resource cost and activity duration
Activity
Crash (CD)
Normal (ND)
A (1)
1
3
B (2)
2
4
C (3)
3
3
D (4)
4
6
E (5)
3
5
F (6)
2
2
Activity
K=1
K=2
Num.
b
c
b
c
A (1)
6.09
2.43
6.58
2.00
B (2)
8.01
1.48
9.80
1.75
C (3)
9.50
2.30
7.20
2.86
D (4)
10.95
1.19
12.65
0.50
E (5)
9.55
1.98
12.14
2.34
F (6)
12.05
1.96
14.88
2.71
Table 3: Resource required per duration for activity
 8  d1 1  d1  2
~
r11 (d1 )  
14  4d1 2  d1  3
~
r1 2  7  d1
1  d1  3
13  2d 2 2  d 2  3
~
r21  
 10  d 2 3  d 2  4
~
r22  7  d 2
2  d2  4
~
r31 (d 3 )  5
~
r3 2 ( d 3 )  4
~
r41 (d 4 )  13  d 4
13  2d 5
~
r51 (d 5 )  
9  d5
~
r 1 (d )  4
6
~
r42 (d 4 )  8  d 4
4  d4  6
3  d5  4
4  d4  6
3  d5  4
 8  d5
~
r52 (d 5 )  
12  2d 5 4  d 5  5
~
r 2 (d )  2
4  d5  5
6
6
6
Table 4. Initial duration and resources requirement
Act.
A
B
C
D
E
F
Dur.
3
4
3
6
5
2
~
r1
~
r2
2
4
6
3
5
4
7
2
4
2
4
2
Table 5 Selecting duration and resources requirement
Act.
A
B
C
D
E
F
Dur.
3
4
3
5
5
2
~
r1
~
r2
2
4
6
3
5
4
8
3
4
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