Optimal job-shop scheduling with random operations and cost objectives

We consider a job-shop manufacturing cell of n jobs (orders), J_i, 1≤i≤n, and m machines M_k, 1≤k≤m. Each job-operation O_iℓ (the ℓth operation of job i) has a random time duration t_iℓ with the average value t̄_iℓ and the variance V_iℓ. Each job J_i has its due date D_i and the penalty cost C*_i for not delivering the job on time (to be paid once to the customer). An additional penalty C**_i has to be paid for each time unit of delay, i.e., when waiting for the job's delivery after the due date. If job J_i is accomplished before D_i it has to be stored until the due date with the expenses C***_i per time unit. The problem is to determine optimal earliest start times S_i of jobs J_i, 1≤i≤n, in order to minimize the average value of total penalty and storage expenses. Three basic principles are incorporated in the model: (1) At each time moment when several jobs are ready to be served on one and the same machine, a competition among them is introduced. It is based on the newly developed heuristic decision-making rule with cost objectives. (2) A simulation model of manufacturing the job-shop and comprising decision-making for each competitive situation, is developed. (3) Optimization is carried out by applying to the simulation model the coordinate descent search method. The variables to be optimized are the earliest start times S_i. A numerical example of a simulation run is presented to clarify the decision-making rule. The optimization model is verified via extensive simulation.