Power of pre-processing: production scheduling with variable energy pricing and power-saving states

Abstract

This paper addresses a single machine scheduling problem with non-preemptive jobs to minimize the total electricity cost. Two latest trends in the area of the energy-aware scheduling are considered, namely the variable energy pricing and the power-saving states of a machine. Scheduling of the jobs and the machine states are considered jointly to achieve the highest possible savings. Although this problem has been previously addressed in the literature, the reported results of the state-of-the-art method show that the optimal solutions can be found only for instances with up to 35 jobs and 209 intervals within 3 hours of computation. We propose an elegant pre-processing technique called SPACES for computing the optimal switching of the machine states with respect to the energy costs. The optimal switchings are associated with the shortest paths in an interval-state graph that describes all possible transitions between the machine states in time. This idea allows us to implement efficient integer linear programming and constraint programming models of the problem while preserving the optimality. The efficiency of the models lies in the simplification of the optimal switching representation. The results of the experiments show that our approach outperforms the existing state-of-the-art exact method. On a set of benchmark instances with varying sizes and different state transition graphs, the proposed approach finds the optimal solutions even for the large instances with up to 190 jobs and 1277 intervals within an hour of computation.

Appendix A : Evaluated CP models

In this appendix, we describe the variations of the CP-SPACES model, which were implemented and evaluated in the preliminary experiment. The description is divided into three parts, namely the modeling of the jobs, the modeling of the spaces, and the linking constraints.

1.1 Modeling of the jobs:

All the models contain an interval variable z_j with a fixed length of p_j for each job J_j ∈. This interval variable represents the time interval in which the corresponding job is processed, i.e, the job starts at time StartOf(z_j) and completes at time EndOf(z_j). The models for the jobs differ primarily in how the cost of scheduling the jobs is formulated in the objective.

• OPTIONAL: For each job $J_{j} \in \mathcal {J}$ and interval $I_{i} \in \mathcal {I}$, create an optional interval variable z_{j, i} having fixed length p_j and fixed start meaning that the job J_j starts to be processed at the beginning of interval I_i. By enforcing Alternative(z_j,{z_{j, i} : ∀I ∈}) on every job J_j ∈, we constraint that only one such variable will be present in the schedule. Then, every pair of job J_j ∈ and interval I_i ∈ adds term $\text {PresenceOf}(z_{j, i}) \cdot c^{\text {(job)}}_{j, i}$ into the objective.

• LOGICAL: Every pair of job J_j ∈ and interval I_i ∈ adds term $(\text {StartOf}({z_{j}}) = i - 1) \cdot c_{j,i}^{(job)}$ into the objective, i.e., if J_j starts at the beginning of I_i, the contribution of J_j into objective is cj, i(job).

• ELEMENT: Each J_j ∈ adds term cj,StartOf(z_j)+ 1(job) into the objective. The indexing by a variable can be done using Element expression.

• OVERLAP: Every pair of job J_j ∈ and interval I_i ∈ adds term Overlap(z_j, I_i) ⋅ c_i ⋅ P[proc,proc] into the objective.

• STEPFN: For every unique processing time p ∈{p_j : J_j ∈}, create a step function F_p representing the cost of starting a job with processing time p_NoJob at the beginning of an interval. Each job J_j ∈ adds term $\text {Start(Eval}{\text {StartOf}{(z_{j}}), F_{{p_{j}}}} )$ into the objective.

1.2 Modeling of the spaces:

Similarly as with the jobs, the modeling techniques for the spaces differ in how they contribute into the objective.

• FIXED: For each pair of intervals $I_{i},I_{i^{\prime }} \in \mathcal {I}$ such that $i < i^{\prime }$, create an optional interval variable $x_{{i},{i^{\prime }}}$ having fixed start to i and having fixed end to $i^{\prime } - 1$. Each such pair of intervals adds term PresenceOf(x_i, i^′) ⋅ c^∗[i, i^′] into the objective.

• FREE: For each possible space length $\ell \in \{1, \dots , h\}$, we create $K(\ell ) = \left \lfloor \frac { h}{\ell } \right \rfloor$ optional interval variables x_ℓk that are indexed by $k \in \{ 1, 2, \dots , K(\ell ) \}$ and have fixed length ℓ. Each pair of length $\ell \in \{{1}\dots ,{h}\}$ and $k \in \{ 1, 2, \dots , K{(\ell )} \}$ adds term c^∗[StartOf(),_ℓk,EndOf()(x_ℓ, k) + 1] into the objective.

  The difference between FIXED and FREE is that in FREE the start times of the space variables are not fixed.

• NOVARS: In this case, the spaces are not modeled by variables at all. Instead, we create a sequence variable π over all jobs variables z_j. Each position ℓ ∈{1,…,n − 1} in π adds term c^∗(EndOf(π_ℓ),StartOf(π_ℓ+ 1) + 1) into the objective (for brevity of the description, the cost of the switching from the first off and to the last off is omitted).

Linking constraints: The formulations of the jobs and spaces must be linked together with a linking constraint ensuring that every time instant of the scheduling horizon is occupied by either a job or a space interval variable. Moreover, we use NoOverlap on the jobs and spaces variables so that they do not overlap each other. Note that since NOVARS does not use variables for modeling the spaces, the linking constraint is not necessary in this case.

• SUM: Here we simply constraint that the sum of the lengths of all present jobs and spaces variables equals to the length of the scheduling horizon.

• PULSE: For each job and space variable, we create a pulse function having height of 1. Then, all the pulse functions are summed together into a cumul expression, which is forced to be 1 in every time instant of the scheduling horizon.

• STARTOFNEXT: We create a sequence variable π over all jobs and spaces variables. Then we constraint that each variable on position π_ℓ ends at the start of variable π_ℓ+ 1.