Industrial job-shop scheduling with random operations and different priorities

A classical job-shop scheduling problem with n jobs (orders) and m machines is considered. Each job-operation O_il (the l^{- th} operation of job i, l = 1,...,m, i = 1, 2,...,n) has a random time duration t_il with the average value t ̄_il and the variance V_il. Each job J_i has its due date D_i and its priority index ρ{variant}_i. Given p_i^*, the desired probability for job J_i to be accomplished on time, and p_i^**, the least permissible probability for the job to meet its due date on time, the problem is to determine starting time values S_il for each job-operation O_il. Those values are not calculated beforehand and are values conditioned on our decisions. Decision-making, i.e., determining values S_il is carried out at the moments when at least one of the machines is free for service and at least one job is ready to be processed on that machine. If at a certain moment t more than one job is ready to be processed, these jobs are compared pairwise. The winner of the first pair will be compared with the third job, etc., until only one job will be left. The latter has to be chosen for the machine. The competition is carried out by calculating the job's delivery performance, i.e., the probability for a certain job to meet its due date on time. Such a calculation is carried out by determining the probability to meet the deadline for the chain of random operations. Two different heuristics for choosing a job from the line will be imbedded in the problem. The first one is based on examining delivery performance values together with priority indices ρ{variant}_i. The second one deals with examining confidence possibilities p_i^* and p_i^** and does not take into account priority indices. A numerical example is presented. Both heuristics are examined via extensive simulation in order to evaluate their comparative efficiency for practical industrial problems.