We will develop a mathematical model for the integration of lot sizing and flow shop scheduling with lot streaming. We will develop a mixedinteger linear model for multiple products lot sizing and lot streaming problems. Mixedinteger 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) nowait consistent sublots with intermingling, (6) nowait equal sublots with intermingling, (7) nowait consistent sublots without intermingling, and (8) nowait 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 decisionmaking 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.
Setup 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 abovementioned 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 mixedinteger linear problem (MILP) and used the branch and bound (BandB) 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  
 
Nowait 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 


 
Nowait 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  
 
Nowait 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 


 
Nowait 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  Nowait consistent sublots with intermingling  Nowait equal sublots with intermingling  Nowait consistent sublots without intermingling  Nowait 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  111213212223313332  170  3570  —  — 
 
Consistent sublots without intermingling  231  189  3665  2/66%  11% 
 
Equal sublots with intermingling  121311222131322333  182  3737  4/67%  7% 
 
Equal sublots without intermingling  231  205  3852  7/9%  20% 
 
Nowait consistent sublots with intermingling  112131121322233233  178  3610  1/12%  4/7% 
 
Nowait equal sublots with intermingling  121311212322333132  202  3837  7/47%  18/8% 
 
Nowait consistent sublots without intermingling  123  199  3715  4/06%  17% 
 
Nowait equal sublots without intermingling  312  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 mixedinteger linear model for multiple products lot sizing and lot streaming problems. Mixedinteger 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.