Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol68.Issue1.Oct2000:

Int. J. Production Economics 68 (2000) 95}106

Greedy heuristic algorithms for manpower shift planning
A.G. Lagodimos!,*, V. Leopoulos"
!University of Piraeus, Department of Business Administration, 80 Karaoli and Dimitriou Street, GR-185 34 Piraeus, Greece
"National Technical University of Athens, Department of Mechanical Engineering, GR-157 80 Zographou, Athens, Greece
Received 31 March 1998; accepted 24 June 1999

Abstract
Consideration is given to a particular personnel planning problem faced by a food manufacturing company. This
problem, referred to herein as manpower shift planning (MSP), seeks for the minimum workforce needed to work in each
available workday shift over a given planning horizon in order to complete predetermined production objectives
associated with individual production lines. We formulate MSP as an integer linear program, whose structure allows us
to conjecture that it is an NP-Complete problem. We then propose two greedy heuristic algorithms for solving MSP. One
for single and another for multiple workday shifts. Using results from a standard ILP optimiser together with a lower
bound developed for the MSP solution, we test the performance of the multi-shift heuristic for a variety of operating
environments. The results demonstrate very satisfactory performance in terms of both solution time and quality. The
paper is concluded with a discussion on the results and proposals for further research. ( 2000 Elsevier Science B.V.
All rights reserved.
Keywords: Production planning; Manpower planning; Heuristics

1. Introduction
In several companies, particularly those in the
foods and pharmaceuticals industrial sectors, other
than some initial processing phases, production
largely consists of simple packaging operations.
Packaging is accomplished on dedicated packing
lines usually requiring one skilled operator and
several (practically) unskilled workers. The latter
are often hired under short term contracts to work
on a particular workday shift (i.e. day, evening or
night shift respectively). At the end of their contract,
often of one month duration, workers are "red to

* Corresponding author.

be recalled later, if needed. Workers cannot be "red
before their contract expires.
In manufacturing environments as the above,
total work requirements for a particular month are
usually determined by a master production schedule, which gives the total production load of each

packing line for all the products it produces. However, the production capacity of each packing line
directly depends on the number of unskilled
workers available. Therefore, a common planning
problem faced by management is to decide how
many unskilled workers to employ in every daily
work-shift of some particular month of the master
schedule so as to complete the production load of
each packing line, while maximising workforce utilisation. It should be noted that the objective of

0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 0 9 9 - 7

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A.G. Lagodimos, V. Leopoulos / Int. J. Production Economics 68 (2000) 95}106

maximising workforce utilisation is in most cases
equivalent with minimising the total workforce to
be employed.
We will refer to this particular personnel and

capacity planning problem as the manpower shift
planning (MSP) problem. It is with the formulation
and the solution of MSP, which to our knowledge
has never been formally addressed in the past, that
this paper is concerned. In fact, as described here,
the MSP problem has been the subject of an industrial research project carried out for the Greek
a$liate of a multinational company operating in
the processed foods and beverages sector.
Since the "rst published work by Dantzig [1],
there has been an abundance of research related
with personnel planning problems. This traditional
paradigm has been primarily concerned with the
determination of the workforce to be assigned to
each one of a given set of work-time patterns so as
to minimise total labour costs, while satisfying
given labour requirements associated with each period of a planning horizon. In its most general form
where one practically considers any work-time patterns, this problem is known as the tour scheduling
problem. However, other simpler variations of this
problem have also been studied. Jarrah et al. [2]
provides a recent brief review of these variations,

while more comprehensive, but somewhat dated,
reviews are given in [3}5].
All variations of tour scheduling have invariably
been formulated as integer linear programs. Since
these have been shown to be NP-Complete [6],
most associated research concentrated in developing e$cient heuristic solutions. Although it is not
our intention to provide a comprehensive description of the related work, it is worth stating that
these heuristics have been based on a wide range of
di!erent methodologies such as simulated annealing [7], LP-based approaches [8}10] and goal
programming [11].
There are two major di!erences between MSP
and the above traditional personnel planning paradigm. Firstly, traditional research is concerned with
deriving directly applicable detailed labour schedules. In this context, related models attempt to
encapsulate all possible work speci"cations and
minute details (such as meal breaks, days-o! and
part-time work) in order to realistically represent

the respective operating environments. In contrast,
MSP e!ectively tackles an aggregate capacity
planning problem under general production targets

and constraints. Secondly, traditional research assumes that workforce requirements for all time
periods of the planning horizon are known in
advance. In contrast, it is the goal of MSP to
determine these workforce requirements for each
individual period of the planning horizon so as to
ensure that speci"c production objectives (using
equipment of given manning needs) are completely
satis"ed.
The remainder of the paper is organised as follows. Section 2 provides a formal de"nition of the
MSP problem and formulates it as an integer optimisation program. It also discusses the problem
computational complexity in comparison with
other known problems and presents a lower bound
for its solution. Section 3 presents two heuristic
algorithms for solving MSP together with information concerning their order of convergence. Section 4
describes an example of using one of the algorithms
for the solution of a particular application problem.
Section 5 reports computational results for testing
the performance of the heuristics for a variety of
operating conditions. Finally, Section 6 discusses
possible MSP extensions and concludes with directions for further research.

2. Problem formulation
In this section, we present a formal statement of
the MSP problem and develop the corresponding
optimisation model. We also present some initial
observations, which help position the MSP problem in the context of previously identi"ed problems
and provide comments and conjectures on the
problem of computational complexity. We also
present a lower bound for the problem solution,
which is later used for testing the quality of the
proposed heuristics.
2.1. Formal problem dexnition
A machine shop consists of n independent machines and operates P shifts per day. Each machine
i requires a unskilled workers for its operation and
i

A.G. Lagodimos, V. Leopoulos / Int. J. Production Economics 68 (2000) 95}106

has a production load of = shifts/time periods. The
i

MSP problem calls for the determination of the
minimum number of workers to be employed in
each shift in order to complete the production load
of each machine within a predetermined time horizon of D days.
The following operational assumptions are used
throughout the analysis:
1. Workforce is #exible and can operate on any
machine.
2. Workforce is employed for a particular shift and
cannot be moved to another shift.
3. The production load of each machine i covers an
integer number of shifts and can be completed
within the available time horizon (PD*= ).
i
4. There are no precedence or other constraints
between machines.
Although these assumptions appear restrictive, it is
noted that they are entirely compatible with the
operating conditions of the industrial "rm which
instigated this research. However, particular proposal for relaxing these assumptions are presented

in the last section.
2.2. Mathematical model
Based on the formal de"nition given above, the
MSP problem is now formulated as an integer
linear programming (ILP) model. Without loss of
generality, we present this ILP model assuming
three daily shifts (i.e. P"3). It is straightforward to
adopt the model for cases where the number of
daily shifts P takes the value of any positive integer.
Let > be the number of workers employed in
i
shift i and let X be an indicator variable where
tj

G

1 if machine j operates in period t,

X "
tj

0 otherwise.

The period index t indicates a particular day and
shift combination under the following convention:
0(t)D for the day shift,
D(t)2D for the evening shift,
2D(t)3D for the night shift.

97

The workforce needed for the operation of machine
j in period t will be a X . Since the total workforce
j tj
at any period cannot exceed that available for the
respective shift,
n
+ a X )> for all t and i.
j tj
i
j/1

Clearly, in order to complete its targeted production load, each machine j needs to operate for
exactly = periods (possibly spread over all three
j
shifts). Hence:
3D
+ X "= for all j.
tj
j
t/1
The above completes the problem formulation. In
brief
Minimise Z"> #> #>
1
2
3
subject to
n
> ! + a X *0, 0(t)D,
(1)
j tj

1
j/1
n
> ! + a X *0, D(t)2D,
(2)
2
j tj
j/1
n
> ! + a X *0, 2D(t)3D,
(3)
3
j tj
j/1
3D
+ X "= , 1)j)n,
(4)
tj
j
t/1
X 3M0, 1N for all t and j.
(5)
tj
Note that, in order to model the MSP problem for
values of PO3, we simply need to change the
number of variables entering the objective function
and the number of constraints accordingly.
2.3. Initial observations
Given the structure of the proposed MSP model,
we now proceed with some initial observations and
conjectures with respect to its solution. In fact, the
development of the solution algorithms presented
in this paper were strongly in#uenced by these
observations.
As MSP has been formulated, each term > of the
i
objective function physically represents the maximum workforce assigned at any period of a shift i.

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A.G. Lagodimos, V. Leopoulos / Int. J. Production Economics 68 (2000) 95}106

Therefore, the di!erences represented by the constraint sets in Eqs. (1)}(3) practically correspond to
the non-utilised workforce in each period of the
corresponding shift. Clearly, if it was possible to
equalise workforce between all periods in each shift
at the minimum level possible, while satisfying the
production targets given by constraints (4), we
would have achieved an ideal solution with full
workforce utilisation. Obviously, due to the integrality of the manning needs of each machine as
well as constraints (4)}(5), this is not always a feasible solution to the problem (see below).
Based on the above and concentrating on the
single shift problem (i.e. for P"1), there appears to
be a strong similarity between the MSP problem
and the well-known problem of scheduling parallel
unrelated machines so as to minimise the total
makespan. In the latter problem, an ideal solution
(again not always feasible) is obtained by schedules
which equalise the production load allocated to
each parallel machine. In fact, the usual formulation of the parallel machines scheduling problem
(e.g. [12]) involves an ILP model very similar to the
one developed here for the single-shift MSP problem. The di!erence is that in the model for parallel
machines scheduling the left-hand side of constraints (4) always equals to one. In this respect, the
parallel machines scheduling problem is simpler
than the MSP problem. We have used, however,
the strong similarity between these two problems to
develop the heuristic algorithms presented here.
It is worth stating that, even in its simplest form,
the parallel machines scheduling problem is known
to be an NP-Complete [13]. Based on the similarity of this problem with MSP and considering
the extra structural complexities of the latter, we
may safely conjecture that the MSP problem presented here is also NP-Complete. Although this
remains to be formally proven, in the following we
seek to develop e$cient heuristic algorithms for its
solution.
Another issue concerns the MSP problem solution space. This is important since it corresponds to
the number of feasible variables combinations to be
searched by an (explicit or implicit) enumeration
algorithm (such as branch and bound) in order to
locate the optimal solution. Based on the MSP
model and particularly considering constraints set

(4), it is not hard to see that the maximum number
of feasible variables combinations (and so candidate solutions) can be calculated by the expression

A B

n PD
.
<
=
i/1
i

(6)

Clearly, even for moderate values of the problem
variables, the solution space may grow into very
large numbers. An example of this is given in Section 4, when describing a particular case study in
detail.
We close this section by presenting a lower
bound for the optimal solution of the MSP problem. This bound is given by the following expression (which is valid for any number of shifts):
n
B" + a = /D.
i i
i/1

(7)

The above gives the total workforce-periods necessary to meet all production targets averaged over
the days in the planning horizon. It physically represents the minimum (optimal) workforce needed to
be employed daily (in all shifts), if workforce could
be equalised between the periods of each individual
shift (i.e. the ideal MSP solution). Since this ideal
solution is not attainable (thus feasible) for any
problem setting, the actual optimum for a given
problem can exceed the value given by (7). Note
that an analogous expression is also used as a lower
bound for the optimal solution in the parallel machines scheduling problem discussed earlier
[14,15]. We used the bound in (7), when testing the
performance of one of the heuristic algorithms we
propose.

3. Heuristic solution algorithms
We now present and discuss two heuristic solution algorithms for the MSP problem; namely one
strictly valid for single-shift operating environments and another, being an extension of the former, which explicitly takes into account the
particularities of multi-shift operating environments. We should stress, however, that with the
appropriate adaptation of its parameters, even the

A.G. Lagodimos, V. Leopoulos / Int. J. Production Economics 68 (2000) 95}106

single-shift heuristic can be used to tackle the
multi-shift MSP problem.
Throughout the presentation of the heuristic
algorithms, we have deliberately chosen to
sacri"ce some mathematical rigour (particularly on
detailed data-management activities for the heuristic implementation) in favour of the underlying
logic clarity.
3.1. Single-shift algorithm
The basis for developing the heuristic for the
single-shift operation (with P"1) has been the
excellent performance of the so called greedy heuristic for the parallel machines scheduling problem
discussed earlier (see Baker [15] for a description of
the greedy heuristic in the context of this problem).
The similarity between the two problems allowed
as to adopt the greedy heuristic for solving the
single-shift MSP as follows:
Step 1: List all machines in decreasing manning
requirements.
Step 2: List all periods in the planning horizon.
Step 3: Select (and consider) the "rst machine in
the list.
Step 4: Find the period with least-assigned workforce so far and reserve it for the operation
of this machine.
Erase the reserved period from periods list.
Step 5: Repeat from Step 4 until the periods reserved for the operation of the machine
exactly equals its production load.
Erase the machine from the machine list.
Step 6: Repeat from Step 2 until the machine list is
empty.
Considering one machine at a time, the heuristic
reserves time periods (by allocating the respective
workforce) for the operation of a machine until all
time periods needed for the production of its respective load has been covered. The justi"cation for
the greedy characterisation of the heuristic stems
from the fact that machines are considered in decreasing order of manning. Hence, machines with
high manning needs are given priority (that is time
periods for their operation are reserved "rst), considering machines with less manning later in order

99

to even-out workforce allocation between periods
(i.e. "ne tuning).
Throughout the heuristic, the main criterion for
selecting a particular time period for the operation
of a machine is the least-assigned workforce so far.
In this respect, the heuristic attempts to combine
the operation of machines at particular periods so
as to e!ectively equalise workforce between all periods of the planning horizon. Clearly, when this is
achieved, the solution becomes equal to the lower
bound in (7) and corresponds to an optimal solution for the single-shift MSP problem.
A "nal point worth making is that the above
heuristic is single-pass. As a result, decisions concerning the allocation of periods to particular machines are made once and these are not revised (or
altered) as the application of the heuristic develops.
The implications of this logic on the speed of the
heuristic are discussed in Section 3.3.
3.2. Multi-shift algorithm
The problem with directly applying the above
heuristic to multi-shift operating environments
arises from the fact that it is practically insensitive
to any qualitative segmentation of the planning
horizon (that is into di!erent shifts). However, since
workforce is practically allocated to shifts, the myopic choice of the period with the least-assigned
workforce so far cannot guarantee to be the best
available choice and may actually lead to a decrease in overall workforce utilisation "gures.
This may happen, for example, when the said
period belongs to some shift with an even workforce distribution among its periods (which corresponds to a high utilisation "gure). An allocation
decision to this period could a!ect the workforce
distribution, resulting in a net increase of the workforce planned for this shift (and consequently a decrease of workforce utilisation). This could have
possibly been avoided if we had chosen a period
perhaps with more-assigned workforce so far, but
belonging to a shift with a workforce distribution
that can accommodate this allocation without an
overall workforce increase for this shift.
The rational of the above is that, in order to be
e!ective, a heuristic algorithm for the solution
of the multi-shift MSP must take explicitly into

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A.G. Lagodimos, V. Leopoulos / Int. J. Production Economics 68 (2000) 95}106

account the e!ects that the choice of a particular
period may have on the total workforce utilisation.
Among the possible ways to achieve this, we
have chosen one which was introduced into the
multi-shift MSP heuristic algorithm described below. We stress that the particular multi-shift algorithm is an extension of the single-shift heuristic,
obtained by incorporating a speci"c shift selection
criterion into its operational logic (this is described
below).
Although we have tried to avoid most mathematical formalism, the following additional notation is necessary for presenting the multi-shift
heuristic:
f
workforce assigned so far to period t
t
Z workforce assigned so far to shift i;
i
Z "max
MfN
i
(i~1)DxtxiD t
¸ non-utilised workforce so far in period t (of
t
shift i); ¸ "Z !f
t
i
t
S total non-utilised workforce-periods so far in
i
shift i; +
¸
(i~1)DxtxiD t
Since we are considering one machine at a time, in
the presentation of the heuristic all machine indices
are suppressed. Hence, production load = and
manning a always correspond to the machine currently under consideration.
Step 1: List all machines in decreasing order of
manning requirements.
Step 2: List all periods in the planning horizon
(keeping respective shift information).
Step 3: Select (and consider) "rst machine in the
list.
Set N"=.
Step 4: Locate all ¹ periods for which ¸ *a.
t
Use single-shift heuristic logic to reserve
min(N, ¹) of these periods for operation of
selected machine.
Erase reserved periods from the periods
list.
Set NPN!min(N, ¹).
If N"0 continue from Step 7.
Step 5: Locate the remaining K free periods in
i
each shift i.
Considering each shift separately, use
single-shift heuristic logic to reserve min

(N, K ) periods for the operation of sei
lected machine.
Evaluate total non-utilised workforce in
each shift S resulting from above allocai
tion.
Step 6: Keep the allocation for the shift iH which
has the minimum S .
i
Erase all periods of shift iH from the periods list.
Set NPN!KH and KHP0.
i
i
Step 7: Repeat from Step 5 until the periods reserved for the operation of the machine
exactly equal its production load (i.e.
N"0).
Erase the machine from the machines list.
Step 8: Repeat from Step 2 until the machine list is
empty.
As the single-shift heuristic, the multi-shift algorithm gradually constructs a solution by considering machines in decreasing manning order (thus it
is also a greedy heuristic). First the heuristic reserves (in Step 4) all possible periods which can
accommodate the operation of the selected machine without any workforce increase (i.e. the periods for which ¸ *a). It then starts reserving all the
t
available periods of individual shifts, considering
each shift separately in a predetermined sequence.
By reserving as many periods in a shift as possible,
the heuristic attempts to avoid any unnecessary
splitting of a machine operation between di!erent
shifts, something that could simultaneously increase workforce in several shifts (Steps 5}7). To do
this, a criterion is needed for deciding shift priorities (that is, the order in which shifts are selected for
consideration).
The particular shift priorities selection criterion
incorporated into the heuristic (Step 6) is to select
the shift for which the allocation of the machine
under consideration provides the least non-utilised
workforce periods so far (that is, select the shift with
minimum S ). Clearly, since we ignore the requirei
ments imposed by machines not yet considered (because they are next in the machines list), this
criterion is myopic and cannot guarantee the best
choice in the long run. However, it allows for the
rapid selection of a shift, making use of all information available at the point where the decision is taken.

A.G. Lagodimos, V. Leopoulos / Int. J. Production Economics 68 (2000) 95}106

3.3. Algorithmic convergence
We now turn to determine the rate according to
which the proposed heuristic algorithms converge
to a "nal solution. Although the analysis presented
here is strictly valid for the single-shift algorithm, it
approximately holds for the multi-shift heuristic
also.
As described in the previous section, both algorithms mainly consist of a series of sorting operations (one sorting operation for each available
machine) followed by some subsequent decisions
taken on the basis of the sorted data. It is well
known that the operation of sorting k objects according to a given criterion may be e!ectively carried out at a rate k ln(k) (see, for example, [14]).
Introducing k"PD (for the total number of time
periods in all shifts) and considering all available
machines n, we can easily determine the convergence rate of the heuristics to be of the order:
Order of Convergence"O[nPD ln(PD)].
The above logarithmic order of convergence clearly
demonstrates that the proposed algorithms can
reach a solution practically very fast. Therefore,
they may be expected to be successfully applied for
the solution of large MSP problems. Some further
comments concerning the computational time required for running the multi-shift heuristic are
given in Section 5.

4. Application example
We now present and discuss an application
example demonstrating the actual use of the multishift heuristic algorithm. Most data came out of the
industrial case study which instigated this research
and concerns the Greek subsidiary of a multinational foods and beverages producer. Table 1 shows
the data concerning the production load targets and
the manning needs for each machine used in the
example. We wish to determine the minimum workforce required to accomplish the given targets in an
environment operating three daily shifts for a planning horizon of 10 days (i.e. P"3 and D"10).
Before demonstrating the application of the
multi-shift heuristic, some initial comments related

101

Table 1
Data used for application example
Machine

Production load

Manning

1
2
3
4
5

23
21
12
19
5

10
9
6
5
4

to the structure of the problem under consideration
are granted. Despite its apparently fairly limited
size, the solution space of the problem is very large.
In fact, using expression (6), we can determine the
number of feasible parameter combinations to be
circa 3.27]1033, which clearly demonstrates the
need for an e$cient solution heuristic. We can also
determine a lower bound for the expected solution.
Using expression (7) for the data of this problem,
we obtain: B"606/10"60.6P61 workers per
day. Recall that we do not really know whether this
is a feasible solution.
Table 2 shows the results obtained by applying
the multi-shift heuristic algorithm. As it has been
constructed, each line of the table represents a particular allocation decision for the operation of the
corresponding machine. Table entries represent the
workforce allocated in each period, presented in the
order in which the relevant decisions were made.
The last column of the table gives the exact step of
the multi-shift heuristic during which the particular
decision was made. The "nal rows of the table
present the solution, giving the workforce allocated
in each period (by summing all corresponding
column entries) and each shift (the maximum
workforce allocated in all periods of the shift)
respectively.
As shown in Table 2, the solution arrived at by
the multi-shift heuristic algorithm is Z"62 worker
per day (obtained as the sum of the workforce
allocated in each shift). This solution provides
100% workforce utilisation for the "rst shift as well
as high utilisation "gures for the second and third
shifts (93.4% and 99.3% respectively). Compared
now with the lower bound above, the solution
obtained by the multi-shift heuristic di!ers only
marginally (1 worker per day or 1.63%). We have

23

15

24 24 24 24 24 24 24 24 24 24 23 23 23 23 20 21 21 21 21 21 15 15 15 14 15 15 15 15 15 15

5

24

5
5
5
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
4

3

Solution

5
5

6
6
6
6
6
6
9
9
9
9
9
9
9
9
9
9
9
9
9
9
2

10 10 10 10 10 10 10 10 10 10
1

Total

6
6
5

5

5

10 10 10
10 10 10 10 10 10 10 10 10 10

Shift 3
Shift 2
Shift 1
Machine

Table 2
Results obtained for application example data

5

9

9

9

9

6

9

6

9

6

9

6

6
6
6
4
6
6
4
6
4
6
6
6

A.G. Lagodimos, V. Leopoulos / Int. J. Production Economics 68 (2000) 95}106

Step

102

not been able to locate (by using extensive local
neighbourhood search) any better solution than the
above, something that might indicate that the heuristic solution is actually optimal in this case.
We also solved the above problem using the
LINGO optimisation package (which employs
a branch and bound implicit enumeration algorithm) and the single-shift heuristic algorithm. The
best solution we arrived at with the LINGO package was Z"64 (using a PC-586 at 90 MHz),
obtained after consideration of circa 2.15]108
distinct solution nodes during 238 hours of computer run-time as de"ned by the LINGO optimiser
[16]. Note that, when applied to this problem, the
single-shift heuristic provided a solution Z"65,
marginally worse than the LINGO solution, but
obtained faster (at a speed slightly higher than that
of the multi-shift heuristic).
In order to apply the single-shift algorithm to
this multi-shift problem, we have initially ignored
any correspondence between time periods and
shifts and applied directly the single-shift heuristic
to solve the resulting transformed single-shift problem (i.e. P"1 and D"30). Based on the conventional correspondence between periods and shifts
introduced in Section 2.2, we have then grouped
individual periods into three groups (of ten periods), each representing a particular shift. From
then on, we evaluated the workforce to be employed in every shift (and in total) in exactly the
same manner as with the multi-shift algorithm outlined earlier in this section.

5. Computational results
Numerical computations were performed in
order to test the performance of the multi-shift
heuristic algorithm. In the absence of a de"nite
knowledge of the optimal solution, the heuristic
results were tested against two di!erent surrogate
measures of e$ciency; namely, the lower bound in
expression (7) and the results provided by the
LINGO optimizer.
Heuristic solutions were obtained by running
a computer code, specially developed using the
Visual Basic for Applications software by Microsoft. LINGO solutions were obtained using the

A.G. Lagodimos, V. Leopoulos / Int. J. Production Economics 68 (2000) 95}106

standard setting of the package, with a stopping
limit for ending the branch and bound search set at
106 solution nodes (which corresponds to circa
65 min computer run-time as indicated by the
package).
5.1. Test problems
The multi-shift heuristic was tested using 40 distinct test problems, representing a variety of di!erent operating environments. For all problems, we
kept the same number of machines, daily shifts and
the planning horizon (i.e. n"8, P"3 and D"10)
and varied the respective machine manning and
production load. The reason for this was that (as
some preliminary results have indicated) these are
the main parameters which in#uence the quality of
the problem solution.
All 40 di!erent problems were constructed as the
result of combining 5 machine manning with 8 production load pro"les. These pro"les are shown in
Table 3. Machine manning pro"les vary in the
range (1, 10) and were chosen to represent several
di!erent operating environments. In this context,
pro"les M and M represent cases with nearly
1
2
even manning needs for all machines, while pro"les
M }M allow for more extreme manning distribu3 6
tions. In the same way, load pro"les vary in the
range (3, 25) and include approximately even (P ),
1
entirely random (P ), parabolic (P }P ) and
2
3 4
skewed pro"les (P }P ). Observe that load pro"le
6 8
pairs P }P and P }P are reversed images of each
6 5
7 8
other.

103

5.2. Numerical results
The results for all test problems are presented in
Table 4. Except for the cells in the last column and
row of the table which give corresponding averages
over all production load (PAV) and manning
(MAV) pro"les respectively, each other cell of the
table corresponds to a speci"c test problem (indicated as the particular pro"les combination). In
each cell we show the following results: the heuristic
solution (HS), the lower bound B (rounded to the
closest larger integer) from (7), the LINGO solution
(LS), the e$ciency of the heuristic relatively to
lower bound (EB) and the LINGO solution (EL)
(obtained as the ratio of the heuristic solution over
B and LS respectively).
Observe that some results of the table are
italicised. These indicate values we know with
certainty they represent optimal solutions for
particular test problems. In most problems, we
identi"ed this optimal solution when the result obtained by either the heuristic or LINGO equalled
the lower bound. For a few test problems, however,
the optimal solution was identi"ed even when this
did not occur (observe row M ). In fact, for these
1
particular problems, since the manning for all machines was set to 10, the optimal solution could
only be the integer multiple of 10 closest to the
lower bound. Its is possible that other solutions
from either the heuristic or LINGO are also optimal (particularly those very close to the lower
bound) but we have no means of verifying this
statement.

Table 3
Manning and production load pro"les used for test problems
Manning

Production load

M
1

M
2

M
3

M
4

M
5

P
1

P
2

P
3

P
4

P
5

P
6

P
7

P
8

10
10
10
10
10
10
10
10

10
10
10
9
9
9
9
8

10
9
8
7
6
5
4
3

10
9
7
5
4
3
2
1

10
10
10
3
3
2
1
1

20
21
19
19
20
22
21
19

25
3
11
21
6
4
13
15

5
15
20
25
25
20
15
5

25
20
15
5
5
15
20
25

24
21
18
15
12
9
6
3

3
6
9
12
15
18
21
24

22
22
22
3
3
3
3
3

3
3
3
3
3
22
22
22

104

A.G. Lagodimos, V. Leopoulos / Int. J. Production Economics 68 (2000) 95}106

Table 4
Results obtained for all test problems
Pro"le

P
1

P
2

P
3

P
4

P
5

P
6

P
7

P
8

PAV

M
1

LB
HS
LS
EB
EL

161
170
170
1.056
1.000

98
100
100
1.020
1.000

130
130
130
1.000
1.000

130
130
130
1.000
1.000

108
110
110
1.019
1.000

108
110
110
1.019
1.000

81
90
90
1.111
1.000

81
90
90
1.111
1.000

112.1
116.2
116.2
1.042
1.000

M
2

LB
HS
LS
EB
EL

149
157
159
1.054
0.987

91
93
95
1.022
0.979

121
121
123
1.000
0.984

121
121
124
1.000
0.976

104
105
106
1.010
0.991

97
99
101
1.021
0.980

80
86
85
1.075
1.012

72
79
82
1.097
0.963

104.4
107.6
109.4
1.035
0.984

M
3

LB
HS
LS
EB
EL

105
108
107
1.029
1.009

67
68
69
1.015
0.986

85
85
86
1.000
0.988

85
85
89
1.000
0.955

83
84
85
1.012
0.988

58
59
61
1.017
0.967

67
69
69
1.030
1.000

39
39
42
1.000
0.929

73.6
74.6
76.0
1.013
0.978

M
4

LB
HS
LS
EB
EL

83
84
83
1.012
1.012

54
55
55
1.019
1.000

65
65
66
1.000
0.985

69
69
72
1.000
0.958

72
73
73
1.014
1.000

39
40
40
1.026
1.000

62
63
64
1.016
0.984

24
24
24
1.000
1.000

58.5
59.1
59.6
1.011
0.992

M
5

LB
HS
LS
EB
EL

81
81
82
1.000
0.988

51
51
57
1.000
0.895

61
61
72
1.000
0.847

71
71
81
1.000
0.877

74
76
76
1.027
1.000

35
35
35
1.000
1.000

69
70
70
1.014
1.000

20
20
20
1.000
1.000

57.8
58.1
61.6
1.005
0.951

MAV

LB
HS
LS
EB
EL

115.8
120.0
120.2
1.030
0.999

72.2
73.4
75.2
1.015
0.972

92.4
92.4
95.4
1.000
0.961

95.2
95.2
99.2
1.000
0.953

88.2
89.6
90.0
1.016
0.996

67.4
68.6
69.4
1.016
0.989

71.8
75.6
75.6
1.049
0.999

42.2
50.4
51.6
1.042
0.978

81.3
83.2
84.6
1.021
0.981

From Table 4, we observe that the heuristic has
found a known optimal solution for 22 problem
settings (55% of test problems). More important,
however, is the fact that, other than the 4 problems
for which the heuristic solution di!ers more (but
always less than 10%), for all other problems (90%
of test problems) the heuristic solution deviated less
than 3% from the lower bound. Given the nature of
the bound, this indicates a very satisfactory overall
quality of the heuristic solution at least for the
problems set tested.
Turning to the e!ects of the parameters setting
on the heuristic solution quality, these are not entirely uniform for all cases. Although a more de-

tailed investigation of this issue may be granted, it
appears that (as expected) the form of the manning
and load pro"les may be factors a!ecting the solution quality. In general, solution quality improves
when the manning pro"les used allow for the combined machines operation in particular periods.
Observe the increased optimal solution occurrences for such pro"les (as M and M ). Similar e!ects,
1
5
appear for the load pro"les (as P and P ). How3
4
ever, by not knowing the actual optimal solution
for all problems, these are just initial observations
which need further study.
Comparing the heuristic with the LINGO solutions in Table 4, the heuristic outperforms the latter

A.G. Lagodimos, V. Leopoulos / Int. J. Production Economics 68 (2000) 95}106

in most cases. In particular, of a total of 23 problems for which the two approaches gave di!erent
solutions, the proposed heuristic solution was better in 20 problems (that is for 87% of these
problems and 50% of all problems). Moreover,
LINGO gave a known optimal solution for only 12
test problems (30% of all problems compared with
55% for the heuristic). Note, however, that other
than a few problems where LINGO solutions were
indeed very poor (see cases M }P and M }P ),
5 3
5 4
these did not di!er dramatically from the heuristic
ones. Additionally, it is possible that the LINGO
solutions could improve if the stopping condition
set for the branch and bound search were relaxed.
A "nal issue concerns the computer run-time
required for arriving at a heuristic solution. In all
test problems the solution time varied between 12
and 16 seconds computer run-time. This is considered as extremely fast, given that the implementation software used is not dedicated for the
solution of such combinatorial problems. We can
further support this statement by comparison with
the solution time required by the LINGO optimiser. In fact, other than the few cases where
LINGO reached a known optimal solution (which
took circa 2 minutes run-time) in all other cases the
solution was obtained after the stopping condition
was activated at circa 65 minutes computer runtime.

6. Discussion and conclusions
This paper has dealt with the formulation and
the heuristic solution of a not previously studied
personnel planning problem, we code-named MSP.
The problem arises naturally during the aggregate
production planning stages of particular types of
industries where the available production capacity
strongly depends on the workforce employed. In
fact, one of the heuristics developed has now been
integrated as a regular computerised decision support tool within the planning procedures of
a manufacturing company.
We formulated MSP as an ILP model and demonstrated its strong resemblance to the well known
parallel independent machines scheduling problem
for minimising makespan. Although it still remains

105

to be formally established, on the basis of the structural similarity between the two models we conjectured that MSP (even in its simplest single-shift
variation) is an NP-Complete problem. We also
proposed an expression for evaluating a lower
bound of the MSP solution and developed heuristic
algorithms for tackling single and multi-shift problems. The multi-shift algorithm e!ectively extends
the single-shift algorithm by incorporating a particular shift priorities selection criterion in its operating logic.
Both heuristics, which may be classi"ed as
greedy algorithms, construct the solution gradually
by considering one machine at a time in one single
pass. This operating logic makes both heuristics
extremely fast (having logarithmic order of convergence), something that allowed industrial implementation using standard commercial software and
hardware tools. Moreover, as demonstrated by
computational results for multi-shift environments,
the heuristic solutions quality is very satisfactory.
Heuristic solutions were better than those obtained
by the LINGO optimiser in 50% of all test problems and deviated not more than 3% from the
lower bound in most cases examined. This is not
surprising considering the good performance generally reported for greedy algorithms in relation
with various machine scheduling problems [14,17].
Since this is the "rst formal study of the MSP
problem, there is still much ground for further
research with respect to both the problem formulation per se and the re"nement of the heuristic
algorithms. Considering "rst the MSP formulation,
we may attempt its generalisation by relaxing the
underlying assumptions. In this context, extra features could be incorporated such as the use of less
#exible workforce, limits to the workforce employed in each shift, constraints restricting particular machines to operate simultaneously or daily
shifts di!ering in duration. Moreover, we might
introduce a cost structure which would allow to
di!erentiate workforce costs per shift.
Turning to research issues related to the proposed heuristic algorithms, it might be interesting
to experiment with di!erent (and perhaps less myopic) shift selection criteria which might improve
the multi-shift heuristic e$ciency. More important,
however, appears the adaptation of the heuristics

106

A.G. Lagodimos, V. Leopoulos / Int. J. Production Economics 68 (2000) 95}106

for the solution of generalised versions of MSP such
as those discussed above. Noticeably, due to their
simple operating logic, the heuristics proposed may
be easily adapted to deal with most of these variations. It remains, however, to be seen whether the
heuristic performance will still remain satisfactory
when dealing with these more complex problems.
A "nal related issue concerns the comparison of
the proposed heuristic algorithms with other approaches. As review papers indicate [18,19], several
researchers have proposed heuristics for dealing
with particular aspects of the parallel machines
scheduling problem with the objective of minimising makespan. Others, in the context of di!erent
applications, have proposed heuristics for dealing
with problems e!ectively identical to parallel machines scheduling (see, for example, [20,21] in relation with the modi"ed bin-packing problem). Due to
the strong similarity between these problems and
MSP, it seems that with minor modi"cations some
of these heuristics could be easily adapted for the
solution of MSP. It would be interesting, therefore,
"rst to extend some of these heuristics to cover MSP
and then to compare their performance with that of
the heuristics we have proposed in this paper.

Acknowledgements
This research was partially supported by the
Greek Development Ministry under research grant
PABE-96-BE140.

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