A novel stochastic programming approach for scheduling of batch processes with decision dependent time of uncertainty realization

Abstract

Uncertainty modelling is key to obtain a realistically feasible solution for large-scale optimization problems. In this study, we consider two-stage stochastic programming to model discrete-time batch process operations with a type II endogenous (decision dependent) uncertainty, where time of uncertainty realizations are dependent on the model decisions. We propose an integer programming model to solve the problem, whose key feature is that it does not require auxiliary binary variables or explicit non-anticipativity constraints to ensure non-anticipativity. To the best of our knowledge this is the first model dealing with such type II uncertainties that has these characteristics, which makes it a much more computationally attractive model. We present a proof that non-anticipativity is enforced implicitly as well as computational results using a large-scale scientific services industrial plant. The computational results from the case study depicts significant benefits in using the proposed stochastic programming approach.

Appendices

Appendices

Nomenclature

Indices

i :

job

j :

task

k :

task in the path of a job i

$j^{'}$ :

imperfect task

t :

timepoint

$k^{'}$ :

task

$\varphi , r$ :

timepoints

Parameters/sets

I :

Jobs

J :

Tasks

$R_j$ :

Resources of task j

$c_j$ :

Capacity of task j

$\psi _s$ :

Probability of scenario s

$\phi (j)$ :

Completion time of task j

$A_i$ :

Number of units to be processed in job i

$P^{i}$ :

Path of a job i

$q_i$ :

Number of tasks in the path of a job i

$\varepsilon (j)$ :

predetermined time points for a task j

$N_G^+ (P_k^i )$ :

Set of tasks to which units are transferred from a task $P_k^i$

$N_G^{-} (P_k^{i} )$ :

Set of tasks from which units are transferred to a task $P_k^i$

$\rho _{ik^{'}k}$ :

Fixed value of task outcomes to obtain the first stage decisions

$\rho _{ik^{'}k}^{s}$ :

Actual value of task outcome realized in scenario s

Decision variables

$x_{ikt}$ :

Number of units waiting to be processed in the task $P_k^i$

$y_{ikt}$ :

Number of units from a job i to be processed in the task $P_k^i$

$z_{jt}$ :

Number of resources to be operated at timepoint t for a task j

$x_{ikt}^{s}$ :

Second stage decision for the number of units waiting to be processed in the task $P_k^i$ after the realization in scenario s

$y_{ikt}^{s}$ :

Second stage decision for number of units from a job i to be processed in the task $P_k^i$ after the realization in scenario s

$w_{ikt}^{s}$, $v_{ikt}^{s}$:

Final implementable decisions in scenario s

A Normalized process data for case study 2—scientific services facility

The normalized data for the 189 tasks in the scientific services facility are presented below in Table 5.

Table 5 Normalized process data used in experiments