Optimization model

The objective function (4.1) minimizes total cost, which includes replenishment costs (cipl), walking costs (c^w), investment costs (c_p), and space costs (c^A_p). Constraint (4.2) is a classical assignment constraint, ensuring that every part is assigned to a feeding policy and a location. As mentioned above, walking distances for all demand patterns d are calculated as the maximum distance for that demand pattern in Constraint (4.3). Next, the assignment of parts to a location is linked to Y_pl in Constraint (4.4) with the big-M being set to |I_pl|. Variable Y_pl indicates if locationl is used for policyp. A kitting cell’s size is determined by its length since its width is

predetermined. Therefore, the use of the last horizontal set of locations is described as using a horizontal row. Three out of eleven possible rows are depicted in Figure 4.3. Constraint (4.5) links the assignment of parts to locations belonging to rowrwith the indicator variableZr with the big-M value being set to |I|. This variable determines the cell’s length as only one row can be selected as the last row (see Constraint (4.6)). Only one row can be used as, e.g., the use of row 3 makes the use of locations in other rows such as 26 impossible (see Figure 4.3).

Furthermore, using row 3 makes other locations such as 3, 4, 13, and 14 inaccessible. This is ensured in Constraint (4.7).

Figure 4.3Using rows to define a cell’s length

Lastly, it is ensured that each location’s capacity is not exceeded (Constraints (4.8), (4.9), and (4.10)). In case boxes are assigned to a location, a rack is used to store them. Such a rack consists of several shelves, available for storing boxes, and one shelve to return empty boxes.

The available space of a rack is determined by the size of its facingsS, calculated by the product of the storage shelves’ width and height (see Figure 4.4). It may further be adjusted by a utilization degree. When storing a pallet, both the available space and the space requirement are set equal to one since one location has exactly the capacity to fit one pallet. However, if a pallet of a part is exceptionally large, adjacent locations may be used to fit this pallet as well.

This is indicated by the binary variableU_l, which is introduced in Constraint (4.9). To clarify this constraint, consider the following example: Assume a large container of part A occupies three locations. Thus, assigning it to location 24 (see Figure (4.3)) requires locations 22 and 23 to be used for this part as well. The parameterfi for this part is set to 2, showing it needs two additional locations. Hence, Constraint (4.9) ensures thatU₂₂ =U₂₃ = 1. Simultaneously, assigning such a part to location 11 is avoided by the definition of the setLip. Lastly, Constraint (4.10) prevents assigning a location to more than one feeding policy. Constraints (4.11)-(4.14) define the variable domains.

Locationl

Avail.space

Figure 4.4Front view of a rack used for smaller containers

4.4.2 Cost calculation

Cost elements can be split into three major aspects: Investment, space, and operational costs.

Table 4.2 presents the notation used for deriving the cost functions.

Table 4.2Parameters used for cost calculations Parameter

b Width of a location

c^I_F Investment costs for a forklift [$/h]

c^I_T Investment costs for a tow train [$/h]

c^O_F Labour cost for a forklift operator [$/h]

c^O_T Labour cost for a tow train operator [$/h]

c^A Space cost [$/m²]

c^O Labour cost for a picking operator [$/h]

d Fixed traveling distance for a tow train in the warehouse d˜ Average distance between two drop-offs

dil Transportation distance to deliver partito locationl d^P Distance from warehouse to flow rack area

d^T Length of a tow train tour

dr Depth of kitting cell when using rowr

nip Quantity of partiwhen delivered using line feeding policyp sip Space of a container used for delivering partiin policyp s^T Available space in the tow train

s^A Additional space needed for traveling in the preparation area

˜

s Average space required to store a box

t^H_F Pick up & drop off time for forklift transportation t^H_T Pick up & drop off time for tow train transportation t^B Time to load and couple tow train wagons

v^F Velocity of a forklift

v^O Walking velocity of an operator v^T Velocity of a tow train

w Width of a kitting cell λi Demand of parti η^F Fill rate of a forklift η^T Fill rate of a tow

^F Utilization rate of a Forklift ^T Utilization rate of a tow

As shown in the previous section, costs incorporated in the model are combined in the parameters c_ipl, c^W, c^A_r, and c_p. Clearly, c_p contains equipment investments costs only. Similarly, c^A_r and c^W exclusively contain space and picking costs, respectively. The former can be calculated based on the depth and the width of a kitting cell with rowrplus some additional spaces^A allowance to permit logistics to and from the cell.

c^A_r = (d_r+s^A)(w+s^A)c^A ∀r∈ R (4.15)

The operator’s velocity and wage determine the latter.

c^W =c^O/v^O (4.16)

In contrast,cipl, contains both vehicle investment and operator costs and constitutes replenish-ment costs.

Replenishment costs consist of operator costs for driving vehicles or carrying parts and vehicle cost. For replenishment of large containers, forklifts are used, and costs are calculated based on a part’s demandλiand the number of parts in a large container niL. The outcome is combined with the transportation distancedil, the vehicle velocityv^F, and its fill rateη^F. Furthermore, handling timest^H_F are considered. Before multiplying this with the investment costc^I_F and the operator costs c^O_F, one might consider the average forklift utilization rate ^F. Combining this data leads to Equation (4.17).

The cost for feeding parts in small containers is calculated similarly. However, parts need to be transported with a tow train and, additionally, need to be replenished to the flow rack area by forklift (see Figure 4.1). Therefore, part of the transportation is done by forklift and another part by tow trains. Furthermore, the time to couple the wagons of a tow traint^B needs to be considered. The length of a trip is estimated based on the distance between different drop-offs and the average amount of boxes that can be transported by a train (see Equation (4.19)). Thus, replenishment costs for small containers are a combination of operational and investment costs for forklifts and tow trains (see Equation (4.18)).