8.5.1 Introduction
In this section, we study the most general problem that is also the most realistic:
the ratios of the product types in the lots to be manufactured, as well as the opera- tion times, are stochastic.
The notations are those used in Section 8.4.3 except that the operation times are stochastic: we assume that the time of an operation i∈{1,L,Op} when belong- ing to the manufacturing process of type j∈{1,L,m} is ruled by a triangular density of probability defined by three parameters ai,j<mi,j<bi,j as explained in Section 8.2.
To check if a given set of operations can be assigned to a certain station, we have to simulate its total operation time, as will be explained in the next section.
Remember that a set can be assigned to a station if the probability that its total op- eration time surpasses the cycle time C is less than ε.
8.5.2 Evaluation of an Operation Time
Assume that the number nj of products of types j∈{1,L,m} in the lot (batch) S to be processed is stochastic and that nj∈{0,L,Nj }. The probability to have nj products of type j in S is pj,nj.
Let i∈{1,L,Op} an operation. If this operation belongs to a process of type
{ m}
j∈ 1,L, , then its manufacturing time is ti,j which is a random variable fol- lowing triangular distribution with ai,j<mi,j<bi,j.
To generate at random the total time of an operation i for lot S in such a manu- facturing system with m possible products, the following algorithm can be used:
Algorithm 8.5. (Tot_Op{i,(ai,j,mi,j,bi,j,Nj,pj,nj)j=1,L,m}) 1. Set TTi =0 (initialization of the operation time).
2. For j=1,L,m do:
2.1. Generate at random x∈[0,1] (density 1 on [0, 1]).
2.2. Compute nj∈{0,L,Nj} such that ∑ ∑
=
−
=
<
≤
j
j n
k k j n
k k
j x p
p
0 , 1
0
, . If nj=0, 0
1
0 , =
∑−
= nj
k k
pj .
300 8 Advanced Line-balancing Approaches and Generalizations 2.3. Compute ti,j=Random_triangular(ai,j,bi,j,mi,j), see Section 8.2.2.
2.4. Compute TTi=TTi+nj×ti,j.
Thus, to evaluate if a set I={i1,L,ir } {⊂ 1,L,Op} of operations can be as- signed to the same station, we apply algorithm Eval_C presented below, knowing that the probability that the total operation time exceeds C should be less than ε. In this algorithm, Z is the number of iterations (20 000, for example).
Algorithm 8.6. (Eval_C{ε,I,(ai,j,mi,j,bi,j,Nj,pj,nj )i∈I,j=1,L,m})
1. Set tot=0 (this variable is the counter of the number of times the total operation time ex- ceeds C).
2. For z=1,L,Z do:
2.1. Set TTI =0 (this variable will contain the total operation time).
2.2. For all i∈I do:
) } {
( Tot_Op TT
TTI = I + i, ai,j, mi,j, bi,j, Nj, pj,nj j=1, …, m . 2.3. If TTI >C, then set tot=tot+1.
3. Compute x=tot/Z.
4. If x>ε, then the set I of operations cannot be assigned to the same station (return value
= 0), otherwise, the assignment can be done (return value = 1).
8.5.3 ALB Algorithm in the Most General Case
This algorithm is close to COMSOAL. It is denoted by COMSOAL-S-2. It is ap- plied a number L of times (1000 times, for instance). The final solution is se- lected depending on the criterion chosen (minimize the number of stations, mini- mize the maximal mean idle time in the stations, etc.).
Algorithm 8.7. (COMSOAL-S-2)
1. Let H={1,L,Op} (H is the set of all operations to be assigned).
2. Initialize N=1 (N is the rank of the station currently under consideration).
3. Set A(N)=∅, where A(N) is the set of operations already assigned to the current station.
4. Compute W that is the set of unassigned operations without predecessors or the predecessors of which have already been assigned to a station.
5. Select i∈W at random.
6. Set H =H\{ }i . 7. Set A1=A(N)∪{ }i .
8. Compute ind=Eval_C{I,ε,(ai,j,mi,j,bi,j,Nj,pj,nj)i∈A1,j=1,L,m}. 9. If ind=0, then do:
9.1. Set N=N+1. 9.2. Set A(N)={ }i . 10. If ind=1, set A(N)=A1. 11. If H ≠∅ then go to 4.
12. Display the contents A(M) of stations M=1,L,N.
8.5.4 Numerical Example
We consider the case of three types of products represented in Figure 8.3. The pa- rameters of the triangular densities are given in Tables 8.12–8.14.
Table 8.12 Product type 1
A B C D E F G H J K L M N O
a 1 2 3 0 2 0 1 2 2 1 0 0 1 1
m 2 3 4 1 3 2 2 3 3 3 3 4 4 3
b 3 5 6 3 4 3 4 5 4 4 4 6 5 4
Pred. / / / A B, C / D E, F G H J J, K M L, N
Table 8.13 Product type 2
A B C D E G H I J K L M N O
a 3 2 1 1 0 1 0 2 3 2 1 0 0 1
m 4 4 4 3 3 3 3 3 5 5 3 2 3 3
b 5 5 5 5 4 4 4 4 6 6 4 3 4 4
Pred. / / / A B, C D E / G, I G, H J J, K M L, N
Table 8.14 Product type 3
A B D E G H I J K L M N O
a 1 2 0 2 1 3 2 0 1 3 1 2 1
m 3 4 3 4 3 4 4 3 3 4 3 4 3
b 4 5 4 5 4 5 5 5 4 5 5 5 4
Pred. / / A B D D, E / G, I G, H J K M L, N
The number of parts of each product that may appear in a mix and the corre- sponding probabilities are given in Table 8.15.
The “covering” manufacturing process is given in Figure 8.4. COMSOAL-S-2 is applied with the following parameters:
• L=100. L is the number of times COMSOAL-S-2 runs.
• The number of iterations to evaluate the total time of a set I of operations (pa- rameter Z in Algorithm 8.6, i.e., in the procedure Eval_C ), is chosen to be equal to 10 000.
• Cycle time C= 50.
302 8 Advanced Line-balancing Approaches and Generalizations Table 8.15 Mix data 2
Product type 1
Number of parts 0 1 2 Probability 0.2 0.7 0.1
Product type 2
Number of parts 0 1 2 3 Probability 0.1 0.5 0.3 0.1
Product type 3
Number of parts 0 1 2 Probability 0.3 0.5 0.2
We first run the program for ε=0.01. In other words, a set of operations is as- signed to the same station if the probability that the total operation time exceeds
ε is less than 0.01. We obtained 9 solutions with 7 stations each. The numbers provided in Table 8.16 are the ranks of the stations to which operations are as- signed. The solution number 7 is represented in Figure 8.7.
Table 8.16 Case ε=0.01 Operation
Solution
A B C D E F G H I J K L M N O
1 1 2 1 3 3 2 3 4 2 5 4 6 5 6 7
2 2 1 2 3 3 1 4 4 1 5 5 6 6 7 7
3 3 1 1 3 2 1 4 5 2 4 5 6 6 7 7
4 2 1 2 3 3 1 3 4 1 4 5 5 6 6 7
5 1 1 2 2 3 2 3 4 2 5 4 6 5 6 7
6 1 3 1 2 3 2 2 4 2 4 5 5 6 6 7
7 3 2 1 3 2 1 4 4 1 5 6 5 6 7 7
8 2 1 1 3 3 1 4 5 2 4 5 6 6 7 7
9 1 4 2 2 4 1 2 5 1 3 5 3 6 6 7
Then, we run the program for ε=0.1. In other words, a set of operations is as- signed to the same station if the probability that the total operation time exceeds
ε is less than 0.1. We still obtained 9 solutions with 5 stations each. The numbers provided in Table 8.17 are the ranks of the stations to which operations are as- signed.
The number of stations drastically decreases as ε increases. Obviously, if
=1
ε , all the operations can be assigned to the same station. Table 8.18 provides some examples of the number of stations according to ε.
A
B
C
D
E
F
I
G
H
J
K
L
M
O
N
1 2
3
4
5
6 7
Figure 8.7 Representation of solution 7 with ε=0.01
Table 8.17 Case ε=0.1 Operation
Solution
A B C D E F G H I J K L M N O
1 1 2 1 1 3 2 2 3 2 3 4 5 4 4 5
2 1 1 2 2 2 1 3 3 1 3 4 5 4 4 5
3 1 2 1 2 3 1 2 4 1 3 4 3 4 5 5
4 1 2 1 1 3 1 2 3 2 3 4 4 4 5 5
5 1 2 1 2 2 2 3 3 1 4 3 4 4 5 5
6 1 3 1 1 3 1 2 4 2 2 4 3 4 5 5
7 1 2 1 1 3 1 2 4 2 3 4 3 4 5 5
8 1 2 2 1 3 1 1 3 2 3 4 4 4 5 5
9 1 3 1 2 3 1 2 4 1 2 4 3 4 5 5
Table 8.18 Number of stations versus ε
Probability 0.01 0.015 0.05 0.1 0.2 0.3
Number of stations 7 6 5 5 4 4
The solution number 5 of Table 8.17 is presented in Figure 8.8.
Remarks:
• In this chapter, the goal was to minimize the number of stations knowing the cycle time. If the number of stations is given and the goal is to minimize the cycle time, a dichotomy approach can be used.
304 8 Advanced Line-balancing Approaches and Generalizations Assume, for instance, that the number of stations is 4, ε=0.01 and the data are those of the previous example. In Table 8.19, we provide the different steps of a dichotomy approach.
If the computation is stopped at Step 15, we obtain an approximation of the minimal cycle time, which is C= 68.353271. Furthermore, comparing to Step 14, we can see that the maximal error is 68.353271 – 68.347166 = 0.006105.
Thus, the maximal error is less than 0.01%.
• Other strategies are also available for line balancing with random operation times. They consist in using the mean values of the operation times and/or the mix ratios, and to load the stations until a given percentage of C (90%, for in- stance).