Within this section we propose a linear integer programming (IP) model for decision making in assembly line feeding systems taking into account space constraints at the assembly stations.
As described in the previous section, the assembly line feeding problem assigns every part to a line feeding policy. We extend this approach by additionally assigning it to a discrete locationl.
By this, we generate a better accuracy when determining assembly operator times which affect the objective function in terms of costs. Furthermore, it also determines the exact design of the BoL. In the proposed model, every part (index: i) is needed at a certain station (index: s) and therefore assigned to a corresponding location that can be used for that stationl ∈ Ls. In this model, five line feeding policies (index: p), i.e., line stocking (p =L), boxed-supply (p =B), sequencing (p=S), stationary (p=K), and traveling kitting (p=T), are included.
For a better understanding, some important assumptions of the model are summarized in the following:
Assumptions
1. Demand is assumed to be deterministic.
2. The size of the BoL of a station may vary as locations can be used for multiple stations.
3. Overall shop floor space is limited by the sum over all locations available at the stations.
4. For every line feeding policy, a generic load carrier is used, meaning that, e.g., all kitted parts are transported in cart containers with a specific volume and weight constraint.
5. Part families (index: f) contain multiple parts, that are substituting each other.
6. At most one traveling kit can be used for one final product.
7. Parts are assumed to be used for one task only.
While assumption 1, 5, and 6 should be self-explanatory, we want to give some explanation for the remaining assumptions. Assumption 3 states that the length of the assembly line is fixed and cannot be changed. This assumption is taken since the size of a station usually depends on the size of a product and thus the overall length of the line depends on the number of stations and cannot be changed. Assumption 2 states, however, that stations can borrow some space from adjacent stations if this is beneficial. Lastly, assumption 4 states that for every line feeding policy a standard container is used. For example, all pallets are assumed to have the same footprint being, e.g., 0.8m x 1.2m. Similarly, kits are transported in containers of a fixed size, e.g., 0.8 x 1.2 x 1.4m. Every container in turn contains multiple kits, e.g., 3 such that the available volume is equally split up between all kits in that container. Lastly, we assume that each part is only used for one task at a particular station, which may be different in practice. Not adjusting the model accordingly may result in suboptimal solutions. That is because the kitting cells would be larger in the model than needed as using the same part for two different tasks which are both served by a traveling kit would not require to store the part twice which would be the case in model presented.
The notation used throughout the model can be found in Table 3.3.
Table 3.3Notation for the IP model Sets
F Set of part families Fl Set of families that may be assigned to location l
Fm Set of families that are used for modelm Fp Set of families for which line feeding policiesp can be used
Fs Set of families that are needed at stations I Set of parts
If Set of parts that are in familyf Il Set of parts that might be placed at locationl Im Set of parts that are used in modelm Ip Set of parts for which line feeding policies p
can be used
Is Set of parts that are used at stations L Set of locations to store parts Li Set of locations that can be used to store part
i
Lf Set of locations that can be used to store parts of familyf
Ls Set of locations that can be used for stations M Set of models to be assembled Mf Set of models that use familyf Mi Set of models that use parti
P Set of line feeding policies P0 Set of line feeding policies excluding traveling kitting
Pi Set of line feeding policies usable for parti Pi0 Set of line feeding policies excluding traveling kitting usable for parti
S Set of assembly stations Sl Set of stations that can use locationl Binary variables
um Using a traveling kit for modelm xilp Assigning partito locationland line feeding policyp∈P0
yi Assigning partito a traveling kit µlm Using a stationary kit at locationlfor model m
τf Assigning familyf to a traveling kit χlsp Using locationlfor stationsand line feeding policyp∈ P0
ψf lp Assigning part familyf to locationland line feeding policyp∈ {S, K}
Parameters
cilp Costs for providing parti to locationl∈ Li
with line feeding policyp∈Pi0
ci Costs to provide parti∈ IT with a traveling kit
cmK Fixed costs for a stationary kit for modelm cmT Fixed costs for a traveling kit for modelm
fi Family of parti M Large number
r Rack capacity, i.e., number of boxes that fit in a rack used for boxed-supply
si Station at which partiis used
vf Volume of part familyf vi Volume of parti
Vp Volume of a container/box used for policyp wf Weight of part familyf
wi Weight of parti Wp Weight limit of container used for policyp
In the following, we give a generic linear integer model that solves the assembly line feeding problem by unambiguously assigning every part to a line feeding policy and a location.
Minimize:
rX In case it is required or desired to feed all parts of a family using the same line feeding policy, the following two constraints have to be added. Please note that w.l.o.g. it is assumed that parts and their families are assumed to be sorted in increasing order w.r.t. their indices.
|Ifi|yi= X
The objective function (3.14) aims to minimize four types of costs: costs for feeding parti at locationl with line feeding policyp, denoted bycilp; costs for feeding partiwith a traveling kit, ci; fixed costs for using stationary,cmK, or traveling kits,cmT, for model m. The calculation of these costs has been discussed before in Subsection 3.3.2.
Within the first constraint (3.15), it is assured that every part is unambiguously assigned to a location and a line feeding policy (xilp), unless it is fed in a traveling kit (yi) which does not require a location assignment. Next, in Equation (3.16), we linkχlspvariables toxilp variables
using a big-M constraint. The χlsp variables indicate whether a location is used for a certain line feeding policy and station. Obviously, every location can only by used for one station, and one line feeding policy (see Equation (3.17)). Equation (3.18) links another auxiliary variable ψf lp to thexilp variables. It indicates if any part of familyf is assigned to locationl using line feeding policyp.
From Equation (3.19) on, we define specific rules for individual line feeding policies, starting with a limit on line stocked parts and sequenced families per location (Equation (3.19)): A location can contain at most a single line stocked part or one sequenced family. Analogously, there is also a limit r on the number of parts that fit into a rack used for boxed-supply (Equation (3.20)).
As the Border of Line should not be plastered with racks, that are not needed, Equation (3.21) ensures, that no rack can be removed without making the solution infeasible. This equation might be neglected if investment costs for equipment such as racks are considered as an optimization will balance investment costs and savings in walking costs. Equations (3.22) and (3.23) ensure that volume vf and weight wf of part families in a stationary kit do not exceed the respective container limitsVK andWK. Simultaneouslyµlm is introduced, indicating whether a stationary kit for modelmis placed at location l. The binary decision variableτf, introduced in Equation (3.24), indicates if any part of family f is assigned to a traveling kit. This variable is used to enforce volumeVT and weightWT constraints of traveling kits in the following two constraints (3.25) and (3.26). Those constraints also link the variableum to the variableτf indicating if a traveling kit is used for modelm. Lastly, constraint (3.27) ensures that if a locationlis used for station s, no preceding locationk :k < l can be used for a subsequent station q:s < q. This constraint is only active if space borrowing is allowed. A visualization of this constraint is also shown in Figure 3.1. In this simple example the line consists of 2 stations (left: station 1, right:
station 2). Each station may use some locations Ls which is indicated in the top-most line by the set of stations Sl that may use location l. Here, locations 1-5 and 7 are used by station 1 and locations 8-16 are used by station 2. This means, station 2 borrows location 8 from station 1 as location 8 is within the boundaries of station 1. As one can see, location 6 is not used, and constraint (3.27) avoids that station 2 uses it as long as station 1 uses location 7, even if it reduced costs. This prevents crossing of workers. Furthermore, it avoids confusion about the parts used at a station. In practice one might use, for instance, fluorescent tape to indicate that locations 8-16 contain parts for station 2.
Equations (3.28) - (3.34) define the domains of all decision variables used in this model. As mentioned before, Equations (3.35) and (3.36) may be added when the solution should contain equal assignments for all parts of one family. For parts in a traveling kit this is ensured by the first constraint and for parts in other line feeding policies by the second constraint. This might be done to avoid confusion of assembly workers about the exact part to use for a product. We refer to the model without constraints (3.35) and (3.36) as thepart level model, and to thefamily level model if the constraints are included. This is due to the fact that parts can be assigned to every line feeding policy in the former formulation. In contrast, parts of the same family have to be assigned to the same policy in the latter formulation.