Bahram Alidaee

Two parallel machine sequencing problems involving controllable job processing times

In this note we consider a problem of scheduling n single-operation jobs on m non-identical machines where the sequencing of the jobs and their processing times are decision variables. It is assumed that the cost of performing a job is a linear function of its processing time. The scheduling cost to be minimized is: (A) the total processing cost plus total flow time, (B) the total processing cost plus total weighted earliness and weighted tardiness. We reduce each problem to a transportation problem that can be solved by a polynomial time algorithm.

0-1 Quadratic programming approach for optimum solutions of two scheduling problems

Two scheduling problems are considered: (1) scheduling n jobs non-preemptively on a single machine to minimize total weighted earliness and tardiness (WET); (2) scheduling n jobs non-preemptively on two parallel identical processors to minimize weighted mean flow time. In the second problem, a pre-ordering of the jobs is assumed that must be satisfied for any set of jobs scheduled on each specific machine. Both problems are known to be NP-complete. A 0-1 quadratic assignment formulation of the problems is presented. An equivalent 0-1 mixed integer linear programming approach for the problems are considered and a numerical example is given. The formulations presented enable one to use optimal and heuristic available algorithms of 0-1 quadratic assignment for the problems considered here.

Single machine scheduling with nonlinear cost functions

Abstract The problem of scheduling n jobs with a single machine where each job has a differentiable cost function and the total cost being minimized is studied. A heuristic solution procedure is proposed. The algorithm is also applicable when the cost functions are piecewise diflerentiable. Running time of the algorithm is O ( n 2 ). The effectiveness of the algorithm is evaluated and is shown that in many cases the algorithm picks an optimal order. The algorithm proposed here empirically is compared with linearized algorithm by Fisher and Krieger [1].

Optimal assignment of slack due-date and sequencing in a single machine shop

We consider the problem of optimal assignment of NOP due-dates ton jobs and sequencing them on a single machine to minimize a penalty function depending on the values of assigned constant waiting allowance and maximum job tardiness. It is shown that the earliest due date (EDD) order is an optimal sequence. For finding optimal constant waiting allowance, we reduce the problem to a multiple objective piecewise linear programming with single variable. An efficient algorithm for optimal solution of the problem is given.

Schedule of n jobs on two identical machines to minimize weighted mean flow time

Abstract In this note we consider the problem of scheduling n jobs on two identical processors to minimize weighted flow time (WMFT). Similarities between this problem and weighted earliness and tardiness (WET) on a single machine [6] is discussed. Based on results on recent developments on WET problem, we give a dynamic programming algorithm for the WMFT problem on two identical processors.

Proof of permutationally convexity of MCSF games

Granot and Huberman (1982) showed that minimum cost spanning tree (MCST) games are permutationally convex (PC). Recently, Rosenthal (1987) gave an extension of MCST games to minimum cost spanning forest (MCSF) games and showed these games have nonempty cores. In this note we show any MCSF game is a PC game.

Maximizing set function formulation of two scheduling problems

In this note we consider two problems: (1) Schedulingn jobs non-preemptively on a single machine to minimize total weighted earliness and tardiness (WET). (2) Schedulingn jobs nonpreemptively on two parallel identical processors to minimize weighted mean flow time (WMFT). A new approach for these problems is presented. The approach is based on a problem of maximizing a submodular set function. Heuristic algorithm for the problems also is presented.

Optimal assignment of NOP due-dates and sequencing in a single machine shop

We consider the problem of optimal assignment of NOP due-dates ton jobs and sequencing them on a single machine to minimize a penalty function depending on the values of assigned constant waiting allowance and maximum job tardiness. It is shown that the earliest due date (EDD) order is an optimal sequence. For finding optimal constant waiting allowance, we reduce the problem to a multiple objective piecewise linear programming with single variable. An efficient algorithm for optimal solution of the problem is given.