An optimal procedure for the single-model deterministic assem- bly line balancing problem F. VAN ASSCHE Control Data Belgium N. V.. Brussels, Belgium W.S. HERROELEN Department of Applied Economic Sciences, Katholieke UniversiteitLeuven, Leuven, Belgium Received 15 March 1978 Revised 12 June 1978 The purpose of this paper is to develop an optimal solu- t.ion method for the single-model assemblyline balancing problem with deterministicwork element durations. The procedure we present is a branch-and-bound algorithm which concentrates on the specialstructure of the problem. The method is equipped with dominancerules, bounding argu- ments and reliable branchingheuristics.Computational results are given, indicating the method to be more than competitive to ones previouslyreported. 1. Introduction The single-model deterministic assembly line balancing problem involves the assignment of work elements of a fixed duration to a minimum number of work stations along an assembly line, without exceeding the cycle time for each station and satisfying the precedence relations between the elements [6]. Numerous exact and heuristic methods have been proposed for the solution of the problem (see [!,2,3, 6,8]). Exact methods for obtaining optimal solutions to the line balancing problem are recently showing promise. It would appear that among these exact methods, the 0-1 programming approach is the most efficient [11,13]. The purpose of this paper is to © North-HollandFublishing Company European Journal of Operations Research 3 (1978) 142-149. present a branch-and-bound method incorporating bounding arguments, dominance rules and reliable search heuristics, which will guarantee the optimality of the solutions obtained. The branch-and-bound procedure is described in Section 2. A problem ex- ample, illustrating the mechanics of the procedure, is given in Section 3. Extensive computational results appear in Section 4. 2. The branch-and-bound method Essentially the branch-and-bound method pre- sented is a tree search procedure. Each iteration of this procedure begins with a node representing the assign- ment of work elements to a single work station. The problem associated with the node - the assignment of the remaining work elements to the remaining work stations - is inspected to see if it will yield an immediate solution. If it is determined that no im- mediate solution can be found, the procedure branches into a number of descendant nodes corresponding to the feasible undominated next station assignments and computes for each such node a lower bound. The program then chooses the node with the smallest lower bound for the next iteration. If an immediate solution is found for any node in the tree, then a solution is guaranteed for all its ancestors. In the following four sections the basic steps of the procedure will be clarified leading to the overall aJgo- rithmic procedure of Section 2.5. 2.1. Generation of work station assignments The generation of work station assignments (nodes in the search tree) is best explained using the prece- dence diagram of Fig. 1. Symbols inside a circle repre- sent work elements, numbers outside a circle repre- sent e;ement durations. A work element is eligible if (1) it has not yet been assigned to a work station, and (2) all of its immediate predecessors have been already assigned. The generation of work station assignments starts with an empty work station (see Table 1, row 1). We move down an order-free list of all eligible work ele- ment:. (the PEND-list) until a work element A 1 is foun~ which when added to the work station will not cause the station time to exceed the cycle time. A ~ is assigned to the work station and a pointer is moved to the position of A~ in PEND. If the station time is smaller than the cycle time, all of the work 142