Tasks are numbered in increasing order. The number i assigned to a task is called the task index. We refer to a task either by its name or by its task index. Those tasks which are not succeeded by any other task are called last tasks. The index set of last tasks is denoted by L.
Workstations are also numbered in increasing order. The first workstation is numbered 1 and the last workstation is numbered N. The number j assigned to a workstation is called the workstation index. Workstations are referred in the following by the workstation index. An assumption must be made about the possible number of stations prior to task assignment. The number of stations used in the model is J. That is, J is the number of stations used in the mathematical model, and N is the number of stations used in the actual line.
The assignment of tasks to workstations is expressed with binary decision variable xij. If task i is assigned to workstation j, then xij=1, otherwise xij=0.
In this chapter the following integer linear programming formulation of SALBP-1 is used,
( )
NMin (4.1)
t x T j J
I
i ij
i c 1,...,
1
=
∑
≤=
(4.2)
x i I
J
j
ij 1 1,...,
1
=
∑
== (4.3)
J j
(
x x)
p q Rj
pj
qj − ≥ ∈
∑
⋅=
) , ( ,
0
1
(4.4)
N J
(
j x)
i Lj
ij ∈
⋅
≥
∑
=1 (4.5)
xij =0 j<LTi and j>UTi i=1,...,I (4.6)
The objective of the model is to minimize the number of stations used in the actual system; that is, to minimize the largest index belonging to a station with task assignment.
Table 4.1 Summary of notation of Chapter 4 Subscripts:
i – index of tasks (i=1,…,I), p – index of a subset of tasks, q – index of a subset of tasks, v – index of a subset of tasks, j – index of workstations (j=1,…,J), k – index of skill level (j=1,…,K).
Parameters:
I – number of tasks,
J – number of workstations in the mathematical model, N – actual number of workstations applied,
R – set of pair of indices which belong to tasks with precedence relations, that is, (p;q)∈R , if task p immediate precedes task q,
K – number of skill levels,
ti – time necessary to perform task i (task time),
sj – time necessary to perform all tasks at station j (station time), sj(Q) – station time of station j as a function of production quantity, Tc – cycle time of the assembly line,
T – total available time for production,
LTi – the earliest workstation which can be used as a consequence of preceding tasks of task i,
UTi – the latest workstations which can be used by task i as a consequence of succeeding tasks of task i,
LSk – the earliest workstation which can be used by tasks belonging to skill level k as a consequence of preceding tasks,
USk – the latest workstations which can be used by tasks belonging to skill level k as a consequence of preceding tasks,
Q – production quantity,
Q(j–d, j) – production quantity at which station j–d enters, and station j leaves the bottleneck, cj – capacity utilization of station j,
Wk – limit on special workers with skill level k, z – sufficiently high number (higher than I),
E(Q,N) – efficiency of an assembly line with N workstation at Q production quantity, QMax(N) – maximal production quantity of a line configuration with N stations,
( )
NQMaxOPT – maximal production quantity of the optimal line configuration with N stations, bj – power of the learning curve function at station j,
d – distance of the station index of two stations.
Sets:
L – set of final tasks, that is, i∈L if task i does not precede any other tasks, R – set of the index pairs of immediately preceding tasks,
Pi – index set of those tasks which must be finished before task i is started, Fi – index set of those tasks which cannot be started before task i is finished, Sk – index set of tasks belonging to skill level/type k.
Decision variables:
N – number of workstations applied,
xij – 0-1 variable; if xij=1, then task i is assigned to workstation j, otherwise xij=0, ljk – 0-1 decision variable in case of low-skill constraints; if ljk=1, worker with skill
level k is applied at workstation j, otherwise ljk=0,
hjk – 0-1 decision variable in case of high-skill constraints; if hjk=1, then worker with skill level k is applied at workstation j, otherwise hjk=0,
ejk – 0-1 decision variable in case of exclusive-skill constraints; if ejk=1, then worker belonging to skill type k is applied at workstation j, otherwise ejk=0.
52 The right-hand side of constraint (4.5) determines the index of those workstations which perform last tasks. The highest such index must be minimized. If each of these indices is smaller than or equal to N, and N is minimized, then the index of the final workstation, and consequently the number of workstations, is minimized.
Cycle time constraints are expressed by constraints (4.2). For each workstation the sum of task times of the assigned tasks is not allowed to exceed the cycle time. As a consequence of constraints (4.3) each task is assigned to one of the workstations.
Constraints (4.4) express the precedence constraints. If task p must be performed before task q, the difference in the bracket is equal to -1, 0 or 1 for each workstation. Since task p must be assigned to an earlier or to the same workstation as task q; the weighted sum of these differences is always greater than or equal to 0, if the weights are the indices of the corresponding workstations.
Finally, the number of variables is reduced by constraints (4.6). Some tasks cannot be assigned to very early workstations because of preceding tasks. For example, if in the problem indicated by Figure 4.1, the required cycle time is 25 minutes then the earliest station for task B is the second station. On the first station, the sum of task times of tasks A and B (11+17= 28 minutes) would violate the cycle time constraint. The earliest workstations which can be used by task i is determined by LTi. LTi is a lower limit of the feasible station indices of task i, and its value is calculated as follows,
Some tasks cannot be assigned to very late workstations because of succeeding tasks. For example, if in the problem indicated by Figure 4.1, the required cycle time is 35 minutes then the latest station for task C is the last but one station. On the last station, the sum of task times of task C and the succeeding tasks (F, E, H, G) would violate the cycle time constraint. The latest workstation which can be used by task i is determined by UTi. UTi is an upper limit of the feasible station indices of task i, and its value is calculated as follows,
number of binary variables can be determined using the LTi and UTi values with the following formula,
We note that model (4.1)-(4.8) is slightly different from the models used in the literature.
Most models formulate the problem for a single last task, that is, only one index belongs to L (see for example White, 1961). If several final tasks exist (see the sample problem in Figure 4.1) then a dummy task is used which directly succeeds the real final tasks. This dummy task increases the number of 0-1 variables; because in that case I+1 task must be assigned to J workstations. In formulation (4.1) to (4.8), however, instead of the dummy task, the index of the final workstation is used. This way only one new variable (N) is required. The value of N must be integer, but as a consequence of the integer lower bound and of the minimization objective, N can be considered as continuous variable.
SALBP-2 minimizes the cycle time for a given number of workstations (N), that is, the objective function is as follows,
Tc
Min (4.10)
Cycle time constraint (4.2), constraints for the performance of each operation (4.3) and precedence constraints (4.4) are the same as in SALBP-1. That is, SALBP-2 is determined by objective function (4.10) and constraints (4.2)-(4.4). The limit on the number of variables in this case can be determined by using an estimate of the upper bound of the cycle time (UB
( )
Tc ). A trivial upper bound of Tc is the sum of task times, however, generally more efficient approximations can be found. For example, the cycle time of the optimal solution of a corresponding SALBP-1 can be used to determine an upper bound for Tc.The earliest workstations which can be used by task i is now the following,
( )
The latest workstation which can be used by task i is now calculated as follows,
( )
Pastor and Ferrer (2009) published an improved method for the calculation of the feasible lower and upper workstation indices. Their method increases computational efficiency in case of large problems. In the problems presented in the following chapters, however, the estimate of the feasible workstation indices with formula (4.7), (4.8), (4.11) and (4.12) is sufficient, because computation time is insignificant.
Consequently SALBM-1 is defined by constraints (4.1)-(4.8) and SALBM-2 is defined by constraints (4.2)-(4.4), (4.6) and (4.10)-(4.12). These models are summarized in the first row of Table 4.2.
In the following chapter the basic SALBP-1 and SALBP-2 models will be completed with constraints expressing work force skill requirements.