In this chapter, the routing, scheduling, and loading problems will be investigated together in parallel to minimize the number of trains, inventory costs, and system variability as shown in the right side of figure 6.1. The three problems and their interrelationships which will be explained later are shown in figure 6.2. During working on these problems, four limitations, which are tugger train capacity, line-side inventory limit, time buffers, and routing time, as shown in the corners of figure 6.2 are considered.
To achieve the objectives, the general procedures of the study are shown in the left hand side of figure 6.1 where there are 5 major steps. The first step which is computing the demand in bins unit was explained above in the example. The rest of the steps are explained in the next sections. Analytical equations are used in the second step, and the DP is used in the third and fourth steps. Step 5 is the longest one and it contains a lot of calculations. It needs, besides others, DP and IP, and it will be explained later in figure 6.4.
85 Figure 6.1 Major steps and objectives
Figure 6.2 General problems, their interrelationships, and constrictions
Objectives: minimizing: 1. Number of trains 2. Inventory costs Normal loading Early loading 3. System variability Route length Train loading 1 3 2 1. Compute demand per station
cycle (number of bins)
2. TCT and initial feasible space
3. Compute minimum number of trains
4. A new feasible space based on optimal number of trains
5. Finding optimal solution
General Procedures
Analytical equations
DP
DP
Scheduling
Inventory cost and CV
Tugger train capacity Line-side area capacity
Time buffer among train cycles Route time
Feasible space TCTs First train cycles start Routing Loading No. of trains Feasible routing Optimal routing Early loading decision Early start decision Early loading Normal loading TCT ranges
86 6.3 TCT determination
It is assumed that if a tugger train feeds some stations in the first train cycle, the same train will also feed the same stations in every next train cycle. It is assumed in this chapter that the TCTs are the same during the shift, that is, the train feeds some certain stations every 30 minutes for example. Therefore, if the train makes its first cycle in just 20 minutes, it must wait another 10 minutes until it starts the next cycle.
Train capacity and line-side area capacity are two restrictions that push TCT not to be too long. On the other hand, the TCT must consider the time needed for the train to move from the decentralized supermarket to the stations, from a station to the next one, and from the stations back to the supermarket including all the times of loading and unloading of full and empty bins. To find the TCT for each cell, at first the maximum possible TCT for the train with a cell from station si to station sj including the station ‘s’ is computed. To do that, at first, the minimum number of train cycles is computed based on line-side area capacity and the train capacity. Minimum number of train cycles in the shift for the cell from si to sj, MNCij, can be estimated based on equation (6.1)
Where TDSij is the total cell demand during the shift, and Qs which computes the number of train cycles based on MLSI can be estimated using equation (6.2)
Where is the lower rounded integer value for the variable X, and is the upper rounded integer value for the variable X. Moreover, dSs is the total station demand during the shift for station ‘s’. KLS is the line-side area capacity which is the maximum capacity of the area beside a station for the inventory for immediate use. In this study, KLS is assumed to be the same for all stations in the cell. It is assumed that any bin from which some parts were consumed by the station is not considered in the calculation of the line-side inventory. The maximum TCT, MAXCij, can be estimated using equation (6.3)
87
Where г is the number of the station cycles in the shift. The time unit for MAXCij is number of station cycles. The minimum TCT which is MINCij is computed based on the time buffer, the Routing Time Inside the cell (RTIij) and Routing Time Outside the Cell (RTOC)
In the example above, RTIij is estimated to be sj-si +1 since the loading and unloading of bins are assumed to take 1 station cycle for each station inside the cell. RTOC was assumed to be the same for all cells and equal to 2.
The feasible TCT Range (CRij) can be computed using equation (6.5)
In the case that MAXCij-MINCij is less than zero, the solution is not feasible. So in this case and based on the previous constrictions, there can be no cell starting from station si to station sj. After finding the feasible space, the optimal TCT for the cell containing the stations from si to sj is set to be the minimum since it gives minimum inventory costs, and this coincides with the principle of JIT in which frequent small replenishments of materials are done. So the maximum TCT shown above was only computed to find the feasible solution space. As stated before, the scheduling problem consists of two parts: finding the value of TCT, and finding the point of time at which the first movement of the train is started. So far, the first part was done, but the second part is interrelated to the loading problem, as will be shown later.