We will develop a mathematical model for the integration of lot sizing and flow shop scheduling with lot streaming. We will develop a mixed-integer linear model for multiple products lot sizing and lot streaming problems. Mixed-integer programming formulation is presented which will enable the user to find optimal production quantities, optimal inventory levels, optimal sublot sizes, and optimal sequence simultaneously. We will use numerical example to show practicality of the proposed model. We test eight different lot streaming problems: (1) consistent sublots with intermingling, (2) consistent sublots and no intermingling between sublots of the products (without intermingling), (3) equal sublots with intermingling, (4) equal sublots without intermingling, (5) no-wait consistent sublots with intermingling, (6) no-wait equal sublots with intermingling, (7) no-wait consistent sublots without intermingling, and (8) no-wait equal sublots without intermingling. We showed that the best makespan can be achieved through the consistent sublots with intermingling case.
In the manufacturing industries, the commonly used planning and scheduling decision-making strategy generally follows a hierarchical approach, in which the planning problem is solved first to define the production targets, and the scheduling problem is solved next to meet these targets [
Lundrigan [
In the following section, we summarize research on lot streaming problems and focus on the flow shop environment.
Trietsch and Baker [
Brucker et al. revealed that there exists a polynomial algorithm for any regular optimization criterion in the case of two jobs while the problem with three jobs is All lots are available at time zero. The machine configuration considered constitutes a flow shop. Any breakdowns and scheduled maintenance are not allowed. Set-up times between operations are negligible or include processing times. There are no precedence constraints among the products. The demand is always satisfied (no backlogging). There is an external demand for finished products (processed by last machine). All machines have capacity constraints. Planning horizon is a single period (i.e., a day). All programming parameters are deterministic and there is no randomness. An idle time may be present between the processing of two successive sublots of a lot on a machine (intermittent idling). Consistent and equal sublots are considered (no variable sublots). The number of sublots for all lots is known in advance.
This problem with above-mentioned assumptions can be formulated as follows.
Consider,
This setting might be advantageous if the setup costs for one or more products are high. A quick approach for this setting is to use the model formulations (
In order to measure this model’s performance, we use the model to test the following randomly generated problem: we have three types of products being processed on four machines. The number of sublots per product is three. Demands are 20, 20, and 15 for products 1 to 3, respectively. Production costs are 10, 15, and 12 for products 1 to 3, respectively. Holding costs are 3, 4, and 3 for products 1 to 3. The maximum available capacity of machines is 400 time units for machines 1 to 4. The beginning inventory is zero. Cost per unit time
Processing times of jobs on machines.
Product | Machine number | |||
---|---|---|---|---|
1 | 2 | 3 | 4 | |
1 | 2 | 1 | 2 | 2 |
2 | 2 | 4 | 1 | 1 |
3 | 4 | 2 | 2 | 3 |
LINGO solver defined the model of example as a mixed-integer linear problem (MILP) and used the branch and bound (B-and-B) method to solve it. The resulting formulation has a total of 169 variables and 691 constraints for consistent sublots with intermingling case. The solution was achieved after running the solver for 146 seconds. The results of the consistent sublots with intermingling case are as follows. Total costs are 3570, and makespan is equal to 170. Sublot sizes are as follows:
Sublot completion times on different machines in consistent sublots with intermingling setting.
Machine number |
|
|
|
|
|
|
|
|
|
---|---|---|---|---|---|---|---|---|---|
1 | 10 | 68 | 110 | 26 | 82 | 140 | 50 | 98 | 130 |
2 | 15 | 82 | 126 | 58 | 110 | 160 | 70 | 118 | 140 |
3 | 25 | 100 | 138 | 66 | 117 | 165 | 82 | 126 | 150 |
4 | 35 | 119 | 150 | 74 | 126 |
|
100 | 138 | 165 |
Sublot completion times on machines for different cases.
Case name | Machine number |
|
|
|
|
|
|
|
|
|
---|---|---|---|---|---|---|---|---|---|---|
Equal sublots with intermingling | 1 | 48 | 76 | 90 | 14 | 62 | 104 | 34 | 127 | 147 |
2 | 62 | 97 | 104 | 42 | 90 | 132 | 55 | 142 | 157 | |
3 | 90 | 111 | 125 | 55 | 97 | 139 | 65 | 152 | 167 | |
4 | 104 | 125 | 139 | 62 | 111 | 146 | 80 | 167 |
|
|
| ||||||||||
Equal sublots without intermingling | 1 | 116 | 130 | 144 | 14 | 28 | 42 | 62 | 82 | 102 |
2 | 137 | 144 | 151 | 42 | 70 | 98 | 108 | 118 | 128 | |
3 | 163 | 177 | 191 | 49 | 101 | 108 | 118 | 133 | 148 | |
4 | 177 | 191 |
|
101 | 108 | 115 | 133 | 148 | 163 | |
| ||||||||||
No-wait consistent sublots with intermingling | 1 | 12 | 24 | 40 | 54 | 96 | 136 | 82 | 124 | 164 |
2 | 18 | 30 | 48 | 82 | 124 | 160 | 96 | 136 | 168 | |
3 | 30 | 42 | 64 | 89 | 131 | 166 | 110 | 148 | 172 | |
4 | 42 | 54 | 80 | 96 | 138 | 172 | 131 | 166 |
|
|
| ||||||||||
No-wait equal sublots with intermingling | 1 | 56 | 70 | 146 | 14 | 104 | 160 | 42 | 90 | 132 |
2 | 63 | 77 | 153 | 42 | 132 | 188 | 56 | 100 | 142 | |
3 | 77 | 91 | 167 | 49 | 139 | 195 | 62 | 110 | 152 | |
4 | 91 | 105 | 181 | 56 | 146 |
|
77 | 125 | 167 | |
| ||||||||||
No-wait consistent sublots without intermingling | 1 | 12 | 26 | 40 | 50 | 70 | 90 | 130 | 152 | 171 |
2 | 18 | 33 | 47 | 70 | 90 | 130 | 142 | 162 | 179 | |
3 | 30 | 47 | 61 | 75 | 95 | 140 | 154 | 172 | 187 | |
4 | 42 | 61 | 75 | 80 | 100 | 150 | 172 | 187 |
|
|
| ||||||||||
No-wait equal sublots without intermingling | 1 | 74 | 88 | 102 | 116 | 144 | 172 | 20 | 40 | 60 |
2 | 81 | 95 | 109 | 144 | 172 | 200 | 30 | 50 | 70 | |
3 | 95 | 109 | 123 | 151 | 179 | 207 | 40 | 60 | 80 | |
4 | 109 | 123 | 137 | 158 | 186 |
|
55 | 75 | 95 |
Sublot sizes for different cases.
Sublot name | Equal sublots with intermingling | Equal sublots without intermingling | No-wait consistent sublots with intermingling | No-wait equal sublots with intermingling | No-wait consistent sublots without intermingling | No-wait equal sublots without intermingling |
---|---|---|---|---|---|---|
|
7 | 7 | 6 | 7 | 6 | 7 |
|
7 | 7 | 6 | 7 | 6 | 7 |
|
7 | 7 | 6 | 7 | 6 | 7 |
|
7 | 7 | 6 | 7 | 6 | 7 |
|
7 | 7 | 6 | 7 | 7 | 7 |
|
7 | 7 | 6 | 7 | 7 | 7 |
|
7 | 7 | 6 | 7 | 7 | 7 |
|
7 | 7 | 6 | 7 | 7 | 7 |
|
7 | 7 | 8 | 7 | 7 | 7 |
|
7 | 7 | 8 | 7 | 7 | 7 |
|
7 | 7 | 8 | 7 | 7 | 7 |
|
7 | 7 | 8 | 7 | 7 | 7 |
|
7 | 7 | 7 | 7 | 5 | 7 |
|
7 | 7 | 7 | 7 | 5 | 7 |
|
7 | 7 | 7 | 7 | 5 | 7 |
|
7 | 7 | 7 | 7 | 5 | 7 |
|
7 | 7 | 7 | 7 | 5 | 7 |
|
7 | 7 | 7 | 7 | 5 | 7 |
|
7 | 7 | 7 | 7 | 5 | 7 |
|
7 | 7 | 7 | 7 | 5 | 7 |
|
7 | 7 | 6 | 7 | 10 | 7 |
|
7 | 7 | 6 | 7 | 10 | 7 |
|
7 | 7 | 6 | 7 | 10 | 7 |
|
7 | 7 | 6 | 7 | 10 | 7 |
|
5 | 5 | 7 | 5 | 6 | 5 |
|
5 | 5 | 7 | 5 | 6 | 5 |
|
5 | 5 | 7 | 5 | 6 | 5 |
|
5 | 5 | 7 | 5 | 6 | 5 |
|
5 | 5 | 6 | 5 | 5 | 5 |
|
5 | 5 | 6 | 5 | 5 | 5 |
|
5 | 5 | 6 | 5 | 5 | 5 |
|
5 | 5 | 6 | 5 | 5 | 5 |
|
5 | 5 | 2 | 5 | 4 | 5 |
|
5 | 5 | 2 | 5 | 4 | 5 |
|
5 | 5 | 2 | 5 | 4 | 5 |
|
5 | 5 | 2 | 5 | 4 | 5 |
Results of these eight different lot streaming problems.
Classification | Optimal sequence | Makespan | Objective function ( |
Comparison of total cost ( |
Comparison of makespan |
---|---|---|---|---|---|
Consistent sublots with intermingling | 11-12-13-21-22-23-31-33-32 | 170 | 3570 | — | — |
| |||||
Consistent sublots without intermingling | 2-3-1 | 189 | 3665 | 2/66% | 11% |
| |||||
Equal sublots with intermingling | 12-13-11-22-21-31-32-23-33 | 182 | 3737 | 4/67% | 7% |
| |||||
Equal sublots without intermingling | 2-3-1 | 205 | 3852 | 7/9% | 20% |
| |||||
No-wait consistent sublots with intermingling | 11-21-31-12-13-22-23-32-33 | 178 | 3610 | 1/12% | 4/7% |
| |||||
No-wait equal sublots with intermingling | 12-13-11-21-23-22-33-31-32 | 202 | 3837 | 7/47% | 18/8% |
| |||||
No-wait consistent sublots without intermingling | 1-2-3 | 199 | 3715 | 4/06% | 17% |
| |||||
No-wait equal sublots without intermingling | 3-1-2 | 214 | 3897 | 9/15% | 25/8% |
Optimal solutions of example with intermingling integer consistent sublots.
Columns 5 and 6 of Table
For instance, the makespan of consistent sublots with intermingling case is 11% better than makespan of consistent sublots without intermingling case. In equal sublots with and without intermingling cases, the production quantity and inventory will be
Figures
Optimal solutions of example without intermingling integer consistent sublots.
Optimal solutions of example with intermingling integer equal sublots.
Optimal solutions of example without intermingling integer equal sublots.
In this research, we developed the first mathematical model for integration of lot sizing and flow shop scheduling with lot streaming. We developed a mixed-integer linear model for multiple products lot sizing and lot streaming problems. Mixed-integer programming formulation was presented which enabled the user to find optimal production quantities, optimal inventory levels, and optimal sublot sizes, as well as optimal sequence simultaneously. We used a numerical example to show the practicality of the proposed model. We tested eight different lot streaming problems:
The authors certify that there is no conflict of interests (considering both financial and nonfinancial gains) with any organization regarding the material discussed in the paper.
The authors would like to thank the anonymous referees for their invaluable comments and suggestions on an earlier draft of this paper.