Robust balancing of straight assembly lines with interval task times ✩ Evgeny Gurevsky, ¨ Onc¨ u Hazır, Olga Batta¨ ıa, Alexandre Dolgui LIMOS, UMR CNRS 6158, Henri Fayol Institute ´ Ecole Nationale Sup´ erieure des Mines de Saint- ´ Etienne 158, cours Fauriel, 42023 Saint- ´ Etienne C´ edex 2, France Abstract This paper addresses the balancing problem for straight assembly lines where task times are not known exactly but given by intervals of their possible values. The objective is to assign the tasks to workstations minimizing the number of workstations while respecting precedence and cycle time constraints. An adaptable robust optimization model is proposed to hedge against the worst-case scenario for task times. To find the optimal solution(s), a breadth first search procedure is developed and evaluated on benchmark instances. The results obtained are analyzed and some practical recommendations are given. Keywords: Assembly line balancing, Uncertainty, Robust optimization, Branch and bound algorithm 1. Introduction 1 A straight assembly line represents a sequence of worksta- 2 tions that function simultaneously. A product item is released 3 at the first workstation and then visits all workstations in a lin- 4 ear order. The transfer of a product item from a workstation 5 to the next is synchronized via an automatic material handling 6 system. 7 The problem here deals with assigning a given set of elemen- 8 tary tasks V = {1, 2,..., n}, required for completing a prod- 9 uct, to workstations of the line. The objective is to minimize 10 the number of workstations required taking into account prece- 11 dence and exclusion constraints among tasks as well as a cycle 12 time constraint. 13 The precedence constraints define non-strict partial order re- 14 lations among tasks and can be presented by an acyclic direct 15 graph G = (V , A). An arc (i, j) belongs to A iff task j must be 16 assigned to a workstation that does not precede the workstation 17 where task i is assigned. However, this partial order relation 18 does not exclude assigning tasks i and j to the same worksta- 19 tion. 20 The exclusion constraints define the pairs of tasks that can- 21 not be assigned to the same workstation because of their tech- 22 nological incompatibility. These constraints are represented by 23 a family E of pairs of V such that the tasks of a subset e ∈ E 24 cannot be assigned to the same workstation. 25 The cycle time constraint requires that the time spend by a 26 product item at a workstation of the line must be less than a 27 given value T 0 calculated on the basis of the line throughput 28 required. 29 ✩ This research was financially supported by Saint- ´ Etienne Metropole gov- ernment and the European Project AmePLM. Email addresses: evgeny.gurevsky@gmail.com (Evgeny Gurevsky), hazir@emse.fr ( ¨ Onc¨ u Hazır), battaia@emse.fr (Olga Batta¨ ıa), dolgui@emse.fr (Alexandre Dolgui) When exclusion constraints do not exist, this problem cor- 30 responds to the Simple Assembly Line Balancing Problem of 31 type 1 (SALBP-1, see Battini et al., 2007; Baybars , 1986). 32 Contrary to previous publications, where the task times are 33 assumed to be given and fixed, we consider that their exact 34 values cannot be known a priori and even can vary during the 35 life cycle of the line. This is a broader and more realistic as- 36 sumption. For example, we have some experience working 37 with a company producing car door locks for different auto- 38 motive groups. This company uses automatic assembly lines. 39 One of their problems is the micro-stoppages of workstations 40 (from 10 to 30 seconds) that is especially caused by an error 41 in automatic part positioning (or blocking). In such cases, an 42 operator has to manually open the workstation and unblock 43 or reposition the part. The task times are reliable and fixed 44 but when a stoppage occurs the corresponding task time is in- 45 creased. The micro-stoppages are unforeseeable but concern 46 only certain tasks. Therefore, an attentive management is able 47 to give the list of tasks subject to micro-stoppages and evalu- 48 ate the minimum and maximum stoppage time. This is one of 49 many examples where we need to take into account this and 50 other forms of uncertainties at the preliminary design stage of 51 assembly lines. 52 In the literature, to model the variability of task times, au- 53 thors used either normally distributed independent random vari- 54 ables with known means and variances (A˘ gpak and G¨ okc ¸en, 55 2007; Baykaso ˘ glu and ¨ Ozbakır, 2007; Chiang and Urban, 2006; 56 Erel et al., 2005; Gamberini et al., 2009; ¨ Ozcan, 2010; Urban 57 and Chiang, 2006), (see also Dolgui and Proth, 2010, chap. 8) 58 or fuzzy numbers with given membership functions (Gen et al., 59 1996; Hop, 2006; Tsujimura et al., 1995). However, the appli- 60 cation of such models in practice is often impossible because 61 of insufficient information known a priori in order to deduct the 62 required probability or possibility distribution functions. As a 63 consequence, we assume that the task times are given by in- 64