In this section, the concept, basic definitions, and mathematical model of in-plant milk-run especially for better understanding the case study are introduced. At first the concept of in-plant milk-run system is briefly presented. The stages of designing such kind of systems are then explained. At the end of this section, in-plant milk-run with fixed routes or fixed intervals are formulated as optimization problems.
3.1 In-plant milk-run concept
The basic structure of milk run concept for material provision is shown in Figure 1. This concept can also be used for moving finished products away from the production lines. However, it is not the concern of the research. In this paper, there is only one central source for materials, which is normally the warehouse or in some cases the supermarket in the plant. There are more than one destinations or points of use, where the material is required and should be provided. Tugger trains are the vehicles used in the milk-run system. They are driven by the operators to move simultaneously several trailers, on which the materials are loaded. The tugger trains start from the source with unit load and come back to the source with empty loading means, which are in this paper only small load containers (SLCs).
3.2 Definitions in milk-run concept
To better understand the model in the following section, it is necessary to introduce several basic definitions regarding in-plant milk-run. These are route, tour, interval, and tour cycle time.
A route describes a predefined track from the source to different destinations. For the materials at one destination, i.e. the point of use (PoU) they should be moved by a tugger train always along the same route, although the destination can be reached by different routes. In other words, if some destinations are assigned to one route, then they are only supplied by trains on this route. The tugger trains on the other routes do not stop at these destinations for material provision any more. More than one route is normally required in the milk-run system in order to reach all of the destinations.
A tour is a trip of the tugger train on one route. The tour can have a defined start time for a tacted milk-run system. The tour can also just start after the previous tour for untacted milk-run system. The number of tours along one route in one shift from eight hours depends on the capacity of the tugger trains and the requirements of the materials moved by the tours. The steps involved in a milk-run tour or cycle are loading material on means of transport, transporting material to the point of use, unloading material at PoU, loading empties on means of transport, and finally unloading empties at return location.
Route
PoU
PoU
PoU PoU
Source Route White
Tour 1: 06:00 h Tour 2: 08:00 h Tour 3: 10:00 h Tour 4: 12:00 h Tour 5: 14:00 h
PoU
PoU PoU
PoU
Figure 1: Basic structure of in-plant milk-run for material provision.
The interval represents the time between two planned departures of a tugger train on a route. That means, the interval is the time between the starts of two tours on the same route. The interval of the milk-run in Figure 1 equals to two hours. Every two hours, on the same route, one tugger train starts with materials for supplying the workstations on the route.
The tour cycle time is the time needed for one tour, from the beginning of loading the materials till unloading of empties. This time is influenced by the steps or activities to be done by the operators in the milk-run system. In each case, the interval must be bigger than the cycle time.
3.3 Optimization model
Generally an optimization model is composed of four important elements.
They are objectives to be minimized or maximized, constraints, which represent the situations, conditions, or assumptions, and decision variables.
Furthermore, some parameters as input should be carefully defined. These elements can be formulated mathematically through symbols and formulations. Here, they will be just described.
Assumptions
In order to formulate the concerted milk-run problem for the related production environment as optimization problem and establish the mathematical model, some assumptions must be made first as follows:
Milk-run is implemented in a lean production environment.
The layout is not to be changed.
Material provision is pull-based.
Every destination, i.e. PoU corresponds to a material.
The demand of each material is considered to be constant.
There are no traffic problems during the tours.
The same tugger trains are used in the system.
The speed of the tugger trains during the tours is the same and constant.
Parameters
The following parameters or given information of milk-run system should at least be included:
Layout of the production environment.
Required travelling distances.
Used capacity and speed of the tugger trains.
Locations of PoUs or destinations.
Material demand at each PoU.
Safety stock at every PoU.
Time to load, unload, or reload SLCs either loaded or empty.
Loading time at the beginning of the tour at source.
Waiting time at each stop.
Based on the above mentioned information and the conditions or rules, such as ways for only one direction in the two physical ways are identified.
Along the same physical way, two routes are defined as different routes.
When it is necessary, more routes on the same way can be defined. All of these possible routes are considered in the model as predefined routes.
These predefined routes are then handled as variables. Which routes are selected, depends on the contribution of them to the objective.
Furthermore, the assumption of same intervals for all of the routes is also an input of the model. Hence, it is only necessary to define one variable to present the interval for all routes. It provides more transparency from lean production view of point. When the economical aspect is not acceptable any more in some cases, it is also recommended to have different intervals for the routes. In the model, there is then one variable for interval for each route.
In this milk-run system, one tugger train is not only used along one route.
That means, it is not bounded to the route. It can also serve other routes. In other words, different routes can be assigned to the same tugger train for avoiding idle time and economic disadvantages. However, this kind of operation strategy does not influence the formulation of the objectives.
Objective
In this paper, two objectives are concerned. One objective is the number of tours in a specific time period to be minimized. The second objective is the time required for material provision in a specific time period, which should also be minimized.
Constraints
The constraints in this milk-run include:
Each destination is only assigned to one selected route.
A destination is not assigned to a route if it is not on the way of the route.
The total material demand at each destination should be fulfilled.
The capacity of the tugger train is controlled at each assignment of a destination to a route.
Decision variables are either 0-1 binary or integer.
Variables
Decision variables in this milk-run are:
If one route is selected or not.
If one destination is assigned to one route or not.
The interval for all routes