The assembly line feeding problem

To define the assembly line feeding problem formally, parts and part families need to be defined.

Parts are physical items which are describable by a single part number or stock-keeping-unit (SKU), and are denoted by the set I, containing all distinct physical partsi used for assembly which may be used at every station in the set of stations S. Part families on the other hand describe a set of parts, including different part variants with the same functionality, each iden-tifiable by a single part number. The set of part families is denoted by F, and a part family f contains a subset of parts If ⊆ I. All part families f and g are distinct from each other If∩I_g:g6=f =∅, and the union of all part families equalsI. Every model that is produced on an assembly line may require parts from various part families, whereas a single part and the corresponding family may be used for one or multiple models. Demand of parts is denoted by λis∈N0∀i∈I ∀s∈S. This notation is summarized in the Table 2.2.

Theassembly line feeding problemdescribes an unambiguous assignment of every part to a single line feeding policy. If a part is used at multiple stations it may be assigned to different line feeding policies (one assignment per station). This decision making process is usually performed once whenever a new product is planned to be assembled on an assembly line and is done before the

start of production (Sternatz, 2015). As the decisions taken in the ALFP affect the performance of an assembly line’s feeding system for the entire product life cycle, we define this problem as tactical (see Section 2.2.4). Furthermore, decision making in line feeding is heavily depending on an assembly system’s layout and the available resources in a system. These may not be known up-front as they also depend on the decisions taken in the assembly line feeding problem. Therefore, other tactical decisions like the layout of storage areas or the amount of workers hired may also be incorporated in the assembly line feeding problem. For example, it might be infeasible to feed all parts in boxed-supply due to the insufficient space in the area where those parts needed to be prepared. In this case the layout is a limiting factor. Contrary, it is also possible to imagine an assembly system, where no such area has yet been used. Then it might be worthwhile to investigate where to place such an area and how to dimension it. Both, limiting factors, but also design decisions may be incorporated in the assembly line feeding problem without altering the main purpose, namely the assignment of parts to line feeding policies.

In the following, we give a basic model describing the assignment of parts to line feeding policies.

χ_isp = 1 denotes the assignment of part i at station s to a feeding policy p∈ P, P being the set of policies, whileχisp = 0 denotes no assignment to that policy. The ALFP is solved when P

p∈Pχ_isp = 1 for all parts and stations where λ_is > 0. The auxiliary variables ψ_sp and Ω_p indicate if a policy is used at a stations, or in the overall assembly system, respectively. Using an integer linearisable programming formulation, the basic problem formulation is the following:

Minimize

In the objective function (Equation (2.1)), the sum of all line feeding costs is minimized. Note, costs may describe actual costs, but also effort or time. We distinguish variable costsc^v_isp that are different for every possible combination of parts, stations and line feeding policies and fixed costs such asc^f_spandc^f_p. These fixed costs depend on the assignment decisions taken as one can see in equations (2.3) and (2.4). Obviously, both constraints are non-linear, as the maximal value

over a set has to be chosen. However, this can be linearized easily, either by big-M constraints or by adding a constraint for every element from the domain of the maximum value function.

The latter may be done as follows:

χisp≤ψsp ∀i∈I ∀s∈S ∀p∈P (2.8)

χisp≤Ωp ∀i∈I ∀s∈S ∀p∈P (2.9)

An example for such a fixed cost is the consideration of investment costs for hoists if a certain policy for a certain part is chosen at a station. Similarly, it might be necessary to invest in a supermarket when any part is assigned to boxed-supply, sequencing or kitting. Another example are the fixed costs incurred for preparing and transporting a stationary kit which may trigger a

’free-riding-effect’ (parts that are only kitted due to empty space in a kit) as reported by Lim`ere et al. (2012).

Constraint (2.2) assures that every part is assigned to exactly one line feeding policy at station sif the demand for that partiat stationsis greater than 0. Constraints (2.5)-(2.7) describe the binary characteristic of the decision variables.

Modeling the ALFP as such only holds true for the very simple case where neither limitations on available resources are given nor additional tactical decisions are considered. For that reason, to well represent practical problems, a decision model should be more complex, incorporating constraints like a space constraint at the BoL, a capacity constraint for vehicles, limitations on the preparation area, or others. These can, e.g., be included by knapsack constraints of which two specific examples are given in equations (2.10) and (2.11).

X

Equation (2.10) reflects a space limitation. Here, aip is the required area for storing a load carrier containing parti fed with line feeding policyp, whereasA_s describes the available space at stations. Equation (2.11) gives an example for constraints regarding the volume of containers used for stationary kits (p=Ks) where kitting containers have a certain volume VKs which is not allowed to be exceeded by the sum over the largest part volumes vi within a part family.

In that sum only parts that are decided to be fed in this stationary kit should be considered.

Again, this constraint is non-linear but can easily be linearised.

Although these constraints might seem intuitive, they cannot be found in every research article.

We will explain two possible valid reasons to neglect those constraints. Firstly, neglecting space constraints is reasonable if the problem at hand is not limited by space constraints (cf. Caputo and Pelagagge (2011)). This is the case when facilities are planned from scratch and the available space at assembly stations can still be altered in order to meet space requirements (see above).

In such cases, the implementation of additional costs for space usage might be a useful approach.

Secondly, the type of constraint regarding the volume of containers might not explicitly be

covered, when kit containers are customized for every station after deciding on the line feeding policy as we observed during company visits. In summary, we decided not to include these constraints in the basic ALFP since we believe they are not appropriate and necessary in every situation. That way, we describe the fundamental problem which can be extended by including additional aspects as those discussed in Section 2.3. Next, we will shortly discuss the complexity of the ALFP.

Theorem 2.2.1. The ALFP, described by formulas (2.1)-(2.7)is an N P-hard problem.

Proof. To verify this theorem, we will show that the problem can be restricted to the uncapac-itated facility location problem (UFLP) which is proven to be N P-hard by Cornu´ejols et al.

(1990). We follow the problem definition given by Korte and Vygen (2012)

UFLP Input: A set of facilitiesJ, a set of clients K, costscij ∈R⁺ for serving client ifrom facilityj, and a cost cj∈R+ for opening facility j.

The taskis to find a subset of facilitiesX to be opened and an assignmentσ:K→X of clients to open facilities minimizing costsP

j∈Xcj+P

i∈Kcσ(i)i

Given this, we restrict the ALFP to instances with the set of partsI equal to the set of clients K, the set of policies P equal to the set of facilitiesJ,c^v_isp =cij ∀s∈S, andc^f_sp =cj ∀s∈S.

Furthermore, we restrict the instances by setting c^f_p = 0 and only considering a single station

|S|= 1 which allows us to neglect the indexsfor the sake of clarity. This transformation can be done in polynomial time and we obtain instances equal to the ones described for the UFLP.

Clearly, any solution for the UFLP directly corresponds with a solution for the restricted ALFP, and vice versa, since opening a facility in the UFLP is equivalent to activating a policy in the ALFP. In the UFLP clients can only be served from opened facilities. This is equivalent to the ALFP where parts can only be assigned to a policy if the policy is activated. In addition, the cost of a solution for one problem is equivalent to the cost of the transformed solution since the variable costs for serving a client from a facility are identical to the variable costs for assigning a policy to a part, and the fixed costs for opening a facility are identical to the fixed cost for activating a policy.