A U-line arranges tasks around a U-shaped production line and organizes them into stations that can cross from one side of the line to the other. In addition to improving visibility and communication between operators on the line, which facilitates problem-solving and quality improvement, U-lines can reduce the total number of operators required on the line and make rebalancing the line easier compared to the traditional, straight production line. This paper studies the (type 1) U-line balancing problem when task completion times are stochastic. Stochastic completion times occur when differences between operators cause completion times to vary somewhat and when machine processing times vary. A recursive algorithm is presented for finding the optimal solution when completion times have any distribution function. An equivalent shortest path network is also presented. An improvement for the special case of normally distributed task completion times is given. A computational study to determine the characteristics of instances that can be solved by the algorithms shows that they are able to solve instances of practical size (like the 114 Japanese and U.S. U-lines studied in a literature review paper).

ABSTRACT. We In what follows, R is an associative ring with center C. The set of nilpotent elements is denoted by N; and for a subset S of R, the subring generated by S is denoted by . The term zero divisor will mean a one-sided zero divisor (i.e. not necessarily a two-sided zero divisor), and 0 will be considered a zero divisor. For x e R, the symbols Ar(X A (x), and A(x) denote respectively the right, left, and two-sided annihilators of x. Finally, the symbols Z, Z n, and C(p) denote respectively the ring of integers, the ring of integers mod. n, and the Prefer p-group.We shall frequently have use for direct-sun deccmpositions of R, both as a ring and as an additive group. To make the distinction clear, we use the symbol to denote a ring-theoretic direct sun and the symbol to denote an additive-group direct sum. 2. A FINITENESS RESULT.We begin by discussing rings in which certain subrings of zero divisors are assuned to be finite.

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