In this section we determine the size of our solution space, as this may affect the design of a suitable method. During the load assignment of orders, we need to make several choices on which orders go when, in which compartments. In Section 3.3 we defined the requirements of the model. From these requirements, we identify five parameters that affect the size of the solution space. Table 4.1 depicts these parameters together with the number of possible settings.
Parameters Number of settings
Resources 1, 2
Compartments 1...6(max. 3 per resource)
Orders 1...n Baffle settings 1...m Compartment types 1, 2, 3
Table 4.1: The five parameters of the solution space. Three of the five parameters have a pre-defined
number of settings. Two of them have no maximum, but typical it does not exceed n = 25 and m = 10.
When a sequence of orders gets validated, we need to decide in which resource to place the order, followed by the decision in which compartment to place it. The number of possibilities equals cn, with c as the total number of compartments and n as the number of orders. Note that we take the sum of the compartments over all resources, so the number of resources are not part of the equation. However, as we have a maximum of three compartments per resource, the number of resources affects the number of compartments and therefore the solution space. For example, a sequence of 25 orders is placed in a
resource combination with one resource of three compartments and one resource of two compartments. The total amount of arrangements is therefore(3 + 2)25, which equals to298∗1015.
However, this is only the arrangement of orders over a given configuration of compartments. We have two more things that influence the solution space. The compartment types of the compartments together with the size of the compartments (baffle settings). First, we have the ability to predefine the compartment type of a compartment. We can also leave the compartment type blank, meaning it can be either one of the three. It is also possible to exclude one compartment type, so a compartment can be either one out of two types. In the hardest case, we do not predefine any compartment type to a compartment, so all of the compartments are blank. This means that there are tc number of possibilities, with t as the number of compartment types and c as the number of compartments. In our example, this would be three compartment types (frozen, chilled, and ambient) to the power of five compartments, which equals
243. Second, we need to multiply this number with the number of baffle settings. We approximate the number of settings, by taking the number of baffle places to the power of the number of baffles. This number also contains duplicate configurations, so we divide it with the factorial of the number of baffles. A resource would have around eight to ten different locations to secure the baffle, so in our example we assume we would have nine baffle places. Our example has one resource with three compartments and one resource with two compartments, resulting in one resource with two baffles and one resource with one baffle respectively. So we would have (92/2!)∗(91/1!) ≈365 configurations. Multiplying the total
amount of arrangements with the number of compartment configurations and the baffle settings, we have a solution space of264∗1020. If we take the same example, but take a sequence of only ten orders, the
solution space will still be866∗109.
Given this example, we can state that even for smaller instances, the solution space is tremendous. However, this solution space can be reduced with some simplifications. First, we can remove the number of baffle settings from the equation, because we can deduct the baffles settings directly from the required volume of the compartments. A compartment only needs to be large enough for the orders assigned to that compartment, taking into account the number of rows required by the orders. For example, if we have one resource with 2 compartments, we do not have to validate all baffle settings of these 2 compartments. We only validate if compartment 1 and 2 have enough room to accommodate the assigned orders, and if both compartment still fit inside the resource. If the resource has a width of three roll cages, and we place ten roll cages in the first compartment, we know that we need to place the first baffle after four rows of roll cages.
This way, we can also deduct the compartment type based on the assigned orders. When a batch of orders are assigned to a compartment, we do not have to check for every compartment type if the batch is valid. We know that orders with different product groups are not allowed together, and that orders are not allowed inside a compartment with an incompatible compartment type. So instead of validating for a given compartment type if all orders are compatible, we reverse it. Given a batch of orders, we search for a compatible compartment type. This way, we only validate all possible order arrangements over the compartments.
To conclude, even with the simplifications of removing the baffle settings and compartment types from the equations, we keep a solution space ofcn. The following sections describe our developed algorithm, and show that even for this large solution space, we can determine feasibility relatively fast. It becomes clear that it is unnecessary to validate every possible arrangement or orders, because we can exclude certain subsets of a sequence of orders successfully.
Chapter 4. Load allocation algorithm 32