Stability radii of optimal assembly line balances with a fixed workstation set
Introduction
The assembly line consists of m workstations, which are linked by a conveyor belt (or another equipment) moving an in-process product from one workstation to the next at a constant pace. The set V of n assembly operations is fixed. Each workstation needs to perform a specific subset of the operations from the set V within the line's cycle-time. All the m workstations start simultaneously to process their own operations. A partial order on the operation set V arises due to technological and economical considerations, which are represented by the precedence digraph with the set A of arcs. A Simple Assembly Line Balancing Problem is to find an optimal assembly line balance, i.e. an assignment of the operations V to the m workstations such that the cycle-time is minimal. The abbreviation SALBP-2 for denoting this problem has been introduced by Baybars (1986). The problem SALBP-2 is NP-hard (Gutjahr and Nemhauser, 1964, Wee and Manoj, 1982) since the bin-packing problem is NP-hard and is a special case of the problem SALBP-2, where in the bin-packing problem, the digraph has no arcs, .
Throughout this paper, it is assumed that the set V consists of two specific subsets of the assembly operations. The non-empty subset includes all the manual operations and the subset includes all the automated operations. The initial vector of the operation durations is known before solving the problem SALBP-2. However, for the subset of the manual operations , each duration tj may vary due to different factors such as the operator skill, motivation, learning effect, etc. In contrast to the manual operations, the duration ti of each automated operation is fixed. We assume that and , . The vectors of the operation durations are denoted as follows: , , . Let a subset of the set V be assigned to the workstation Sk, where . The assignment of the operations V to the ordered workstations , , , is called a line balance, if the following two conditions hold. Condition 1 The assignment br does not violate the partial order given on the set V by the precedence digraph , i.e. each arc implies that operation is assigned to workstation Sk and operation is assigned to workstation Sl in a way such that . Condition 2 The assignment br uses all the m workstations, i.e. the subset is not empty for each workstation Sk, .
Let B(G) denote the set of all assignments br satisfying Condition 1. The subset of the set B(G) consists of all line balances. The cycle-time for the line balance br with the vector of the operation durations is determined as , where the sum is a workstation time. The line balance is optimal with the operation durations if it achieves a minimal cycle-time c as follows: Condition 3 .
Note that Condition 2 allows us to restrict a set of the line balances since the set contains the optimal line balance without fail. Let denote a set of all the optimal line balances, , with the vector of the operation durations. If operation i belongs to the set , its duration ti is fixed. Without loss of generality, we assume that for each automated operation since the automated operation with the fixed zero duration has no influence on a solution to the problem SALBP-2. The initial duration ti is a strictly positive real number for each operation . A value of the duration of the manual operation can vary during the assembly line lifespan. The varied duration may be even equal to zero, which means that the manual operation j from the setis processed by an additional operator in parallel with the processing of other operations assigned to workstation Sk. Due to the additional operator, the processing of the manual operation j does not increase the workstation time, i.e.where indicates the modified vector , for which the workstation time is calculated. The second equality in (2) is valid because of holding equality . We summarize the above in the following remark. Remark 1 The initial duration ti is a strictly positive real number for each operation . A value of the duration of the manual operation can vary during the assembly line lifespan. The varied duration may be equal to zero: .
The aim of this paper is to investigate the stability of the optimal line balance with respect to variations of the operation durations. The stability radius of the optimal line balance b0 is interpreted as a maximum of simultaneous and independent variations of the durations of operations without violating the optimality of the line balance b0, i.e. . A formal definition of the stability radius is given in Section 2.1 along with a sufficient and necessary condition for a zero stability radius. In Section 3, it is shown that the stability radius may be infinitely large, . Formulas for calculating the stability radius for the line balance are given in Section 4.1. The calculation of the stability radius is illustrated in 2.2 Example 1, 3.3 Example 2, 4.3 Example 3. In Section 4.2, it is shown on how to restrict a subset of the set , which must be compared with the line balance for calculating the stability radius . An algorithm for calculating the stability radius is presented in Section 5. Section 6 reports the computational results for the stability analysis of the benchmark instances from the old dataset and the recent one (Otto et al., 2013) tested in Morrison et al., 2014, Otto and Otto, 2014). In Section 7, the managerial implications are spelled out on how to use the stability results in the assembly industry. Concluding remarks and perspectives are discussed in Section 8.
Section snippets
Contributions of this work, previous results, and related literature
The assembly lines are widely used in a mass production for assembling components into final products. An effectively balanced assembly line allows a factory to increase its efficiency via reducing a production cost. Since the production conditions may change over time, the need of a re-balancing of the assembly line may arise from time to time in order to serve customer demands in the competitive market environment. The assembly re-balancing is tedious procedures requiring significant costs
An infinite stability radius of the optimal line balance
Theorem 1 gives sufficient and necessary condition for a zero stability radius. To present a criterion for the infinitely large stability radius, we need to introduce the simple assembly line balancing problem SALBP*-2, which has the same input data as the original problem SALBP-2, except that the initial vector t* of the operation durations: . The line balance is optimal for the problem SALBP*-2, if .
The calculation of stability radii for the optimal line balances
Theorem 1, Theorem 2 give criteria for the extreme values of the stability radii: or . In 4.1 Formulas for calculating the stability radius, 4.2 The redundant line balances for calculating the stability radius, we show how to calculate the stability radius if .
An algorithm for the calculation of stability radii for the optimal line balances
If the set is constructed, it is not difficult to find the set , having chosen the line balances from the set with the minimal cycle-time. Using Theorem 1, one can choose all unstable optimal line balances . Using Theorem 2, one can choose all optimal line balances with infinite stability radii if they exist. Next, we present an algorithm for constructing sets , , , , and calculating the stability radii for all
Computational results
Algorithm RAD was implemented in C++ and tested on the benchmark instances available on http://www.assembly-line-balancing.de The computational experiments were run on a laptop with the following characteristics: Intel(R), Pentium(R), CPU 2020 M @2.40 GHz 2.40 GHz, 4.00 GB Internal Memory. The main characteristics of the benchmark instances, which determine their complexity, are presented in Table 1 for the benchmark dataset given in Scholl (1999), and in Table 2 for the benchmark dataset given in
The managerial implications of the stability results
We answer the following three questions: 1. Why and how are our stability results useful? 2. How does one obtain a stable optimal line balance if it exists? 3. How does one overcome a large size of the instance SALBP-2 for making the stability analysis either for all optimal line balances or for one of them? First of all, one can observe that an exact algorithm used for solving the NP-hard problem SALBP-2, traditionally, terminates once an optimal line balance is constructed, which may make
Conclusion
Since the durations of manual operations are variable, it is impossible to construct the optimal line balance, which will be the best for all variable operation durations. Nevertheless traditionally, the solution procedure for the problem SALBP-2 ends when the first optimal line balance is constructed. Due to the computational results presented in Table 3, Table 4, Table 5, Table 6, we can argue that for the problem SALBP-2 there is a high probability that the first constructed
Acknowledgment
The authors are grateful to the anonymous referees for their useful suggestions on the early draft of this paper. The first and second authors acknowledge the research grant from Ministry of Science and Technology, Taiwan.
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