Master production scheduling and sequencing at mixed-model assembly lines in the automotive industry

Abstract

The customization of final products in the automotive industry involves a large number of optional parts and leads to a huge variety of operation times at the various stations of the assembly line. The master production scheduling problem (MPS) for high-variant mixed-model assembly lines is to assign the individual customer-defined models of a basic product type to short-term production periods while anticipating the negative impacts of an unbalanced model sequence at the lower planning level. We propose a mathematical model formulation for the MPS and develop heuristic solution procedures that attempt to minimize the workload variability. Specifically, these procedures anticipate decisions on the mixed-model sequence and the resulting work overload at stations which has to be balanced by the assignment of utility workers. Furthermore, an integrated planning approach for solving the MPS and the production sequencing problem is proposed.

Appendices

Appendix A

Notation

Sets
O	The set of orders
S	The set of stations at the assembly line
T	The set of planning periods
$i = \left\{ {1 \ldots N} \right\}$	The intervals of processing times in the PIA heuristic
$\tau = \left\{ {1 \ldots C} \right\}$	The set of production cycles
Parameters
a os	Processing time of order $o \in O$ at station $s \in S$
$A_{st}^{k - 1}$	Workload assigned to station s in Period t until iteration k-1
$AVG_{s} = \frac{{\sum\nolimits_{o \in O} {a_{os} } }}{\left| O \right|}$	Average workload of station $s \in S$ in the order bank
$b_{ios}$	Binary parameter indicating if order o is in interval i at station s
BigM	A sufficiently large number
C	Number of production cycles per period
$c_{ot}^{k} ,\;c_{ot}^{\prime k}$	Cost of assigning order o to period t in iteration k
$e_{s}$	Feasible deviation from the average workload in station $s \in S$
$g^{k}$	Fraction of assigned orders after iteration k
$IB_{ot}^{k}$	Workload imbalance caused by order o in period t in iteration k
$IW_{s}$	Interval width of processing times at station $s \in S$
$l_{s}$	Length of station $s \in S$
$N$	The number of processing time intervals per station and period
$R_{ist}^{k}$	Ratio of orders which have been assigned after iteration k to interval i and station s in period t
$TR_{is}$	Target ratio of orders contained in interval i at station s
$u_{os\tau }^{k}$	Total utility work of order o in production cycle τ in period t in iteration k
$\tilde{u}_{os\tau t}^{k}$	Utility work at station s in production cycle τ in period t if candidate order o has been chosen in iteration k
$v_{os\tau }^{k}$	Total idle time of order o in production cycle τ in period t in iteration k
$\tilde{v}_{os\tau t}^{k}$	Idle time at station s in production cycle τ in period t if candidate order o has been chosen in iteration k
$\tilde{w}_{os\tau t}^{k}$	Worker position at station s at the start of production cycle τ in period t if candidate order o has been chosen in iteration k
$\alpha_{is}$	Weight assigned to interval i and station s
$\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$	Weights assigned to terms of the cost coefficient
λ	Cycle time
Variables
$h_{ist} \ge 0$	Continuous variable which represents the deviation of the actual ratio of orders in station s and interval i from the target ratio in period t
$mw \ge 0$	Continuous variable which represents the maximum workload over all stations and periods
$p_{st} \in \left\{ {0,1} \right\}$	Binary variable which is set to 1 if the workload at station s in period t exceeds the feasible boundaries
$u_{s\tau t} \ge 0$	Continuous variable which represents the work overload at station s in production cycle τ in period t
$w_{s\tau t} \ge 0$	Continuous variable which represents the worker position at station s at the start of production cycle τ in period t
$y_{ot} \in \left\{ {0,1} \right\}$	Binary variable which is set to 1 if and only if order o is assigned to period t
$Y_{o\tau t} \in \left\{ {0,1} \right\}$	Binary variable which is set to 1 if and only if order o is assigned to cycle τ in period t
$MLC_{s} \ge 0$	Continuous variable which represents the maximum workload of station s over all periods
$z_{s} \ge 0$	Continuous variable which represents the gap between the maximal and minimal workload station s

Appendix B: Proof of NP completeness

In the following it is shown that the MPS with the DWB objective (MPS-D-DWB) is NP-complete in the strong sense by reducing it to the well known 3-partition problem.

Given a number B, the decision problem related to the MPS (MPS-D) with an objective function z is to answer the following question:

• Exists a feasible schedule x with an objective value $z(x) \le B$?

Since a given schedule can be evaluated in polynomial time, MPS-D belongs to NP.

The 3-partition can be formulated as follows. Let $A = \left\{ {a_{1} , \ldots ,a_{3m} } \right\}$ be a set of 3m positive integers. The problem is to determine m disjoint subsets, $S_{1} , \ldots ,S_{m}$, each containing exactly 3 elements and with an equal weight: $\sum\nolimits_{{a_{i} \in S_{1} }} {a_{i} } = \ldots = \sum\nolimits_{{a_{i} \in S_{m} }} {a_{i} }$.

The solution is yes if such a partition exists and no otherwise.

To reduce the MPS to the 3-partition problem it is to be proofed that an instance of the MPS can be linearly transformed into an instance of the 3-partition problem and if and only if the answer to the question of the MPS-D is yes the answer to the corresponding 3-partition problem is yes, as well.

For an instance of the 3-partition problem generate an instance of the MPS-D-DWB problem with an assembly line consisting of one single station, an order $o_{i}$ for every integer $a_{i}$ with a processing time of a ₁ time units, m periods with a capacity of 3 Units and set B equal to 0. Obviously, this transformation is linear. Given a feasible schedule of the MPS-D-DWB the subsets $S_{1} \ldots S_{m}$ can easily be generated by identifying every set with a period and assigning the numbers represented by the respective orders. If the answer to the MPS-D is yes, every period has an equal workload. Consequently, the answer to the respective 3-partition problem is yes. Conversely, if a 3-partition exists, the subsets define a feasible solution to the MPS-D-DWB problem. Hence, the MPS-D-DWB problem is as complex as the 3-partition problem which is NP complete in the strong sense (cf. Garey and Johnson 1979).

The preceding proof can easily be adopted according to the IWB, MMW and PIA objectives of the MPS.