A parallel greedy approach enhanced by genetic algorithm for the stochastic rig routing problem

Abstract

Scheduling drilling activities for oil and gas exploration involves solving a problem of optimal routing of a fleet of vehicles that represent drilling rigs. Given a set of sites in some geographic area and a certain number of wells to drill in each site, the problem asks to find routes for all the rigs, minimizing the total travel time and respecting the time windows constraints. It is allowed that the same site can be visited by many rigs until all the required wells are drilled. An essential part of the considered problem is the uncertain drilling time in each site due to geological characteristics that cannot be fully predicted. A mixed integer programming model and a parallel greedy algorithm proposed in an earlier study can be used for solving very small-sized instances. In this paper, a graphics processing unit (GPU) accelerated genetic algorithm is developed for using in the greedy algorithm as a subroutine. This approach was implemented and tested on a high-performance computing cluster and the experiments have shown good ability to solve large-scale problems.

Appendix: MIP model of the stochastic routing problem

In this section, the complete MIP model of the stochastic routing problem from [6] is reproduced and discussed. Recall that it is assumed that the same vehicle may re-visit the same site to perform the drilling work in several portions. To model this, each site is formally substituted by its $N^{vis}$ copies (all the notation are the same as in Sect. 2). Let $\mathcal{I}$ be a set of all the site-copies and $\mathcal{I}_i$ be a set of copies of the original site $i\in I$. To avoid confusions, indices i and j represent the original sites, while $i'$ and $j'$ are used for the copies. Since the open routing problem is considered, a dummy site, denoted by f, is introduced. All routes start and finish at this site. The distances between f and the other sites are equal to zero.

Introduce the following variables.

$x_{ui'j'}=1$ if vehicle u visits site-copy $i'$ and travels to site-copy $j'$, and $x_{ui'j'}=0$ otherwise.

$w_{sc}=1$ if scenario sc is taken into account when evaluating feasibility of the routes, and $w_{sc}=0$ otherwise.

$y_{ui'} \in Z$ is the number of wells of site-copy $i'$ drilled by vehicle u.

$t^s_{u,i',sc} \ge 0$ is the starting time of works for vehicle u on site-copy $i'$.

The MIP model is formulated to minimize the total travel time:

$$\begin{aligned} \sum _{u\in U}\sum _{i'\in I'_{f}} \sum _{j'\in I'_{f}} s_{ui'j'} x_{ui'j'}, \end{aligned}$$

(A1)

subject to

$$\begin{aligned} \sum _{j'\in I'_f} x_{ui'j'}= & {} \sum _{j'\in I'_f} x_{uj'i'}, \ u \in U, \ i'\in I'_u\setminus \{f_s\}, \end{aligned}$$

(A2)

$$\begin{aligned} \sum _{u\in U_{j'}}\sum _{i'\in I'_u} x_{ui'j'}\le & {} 1, \ j' \in I', \end{aligned}$$

(A3)

$$\begin{aligned} \sum _{i'\in I_i'}\sum _{u\in U_i} y_{ui'}= & {} n_i, \ i \in I, \end{aligned}$$

(A4)

$$\begin{aligned} \sum _{j'\in I'_u} x_{uj'i'} \le y_{ui'}\le & {} n_i \sum _{j'\in I'_u} x_{uj'i'}, \ i' \in I', \ u\in U_{i'}, \end{aligned}$$

(A5)

$$\begin{aligned} t^s_{u,i',sc} + y_{ui'} d_{u,i,sc} + s_{ui'j'}\le & {} t^s_{u,j',sc} + b_{\max }(2- x_{ui'j'}-w_{sc}),\nonumber \\{} & {} i'\ne j' \in I',\ i:\ i'\in I_i', \ u\in U_{i'}\cap U_{j'}, \ sc\in SC, \end{aligned}$$

(A6)

$$s_{u,f_s,j'}\le t^s_{uj'sc} + b_{\max }(2- x_{u,f,j'} -w_{sc}), j' \in I', \ u\in U_{j'}, \ sc\in SC,$$

(A7)

$$t^s_{u,i',sc}\ge \sum _{j'\in I'_u} a_i x_{uj'i'}-b_{\max }(1-w_{sc} ), \ i\in I, i'\in I_{i}', \ u \in U_i, \ sc\in SC,$$

(A8)

$$t^s_{u,i',sc} + y_{ui'} d_{i,sc}\le \sum _{j'\in I'_u} b_i x_{uj'i'}+b_{\max }(1-w_{sc} ), i\in I, i'\in I_{i}', \ u \in U_i, \ sc\in SC.$$

(A9)

$$\begin{aligned} \sum _{i'\in I'_u}x_{u,f,i'}= & {} \sum _{i'\in I'_u}x_{u,i',f} = 1, \ u \in U, \end{aligned}$$

(A10)

$$\begin{aligned} \sum _{sc\in SC} w_{sc} p_{sc}\ge & {} \alpha . \end{aligned}$$

(A11)

Constraints (A2) guarantee that each site-copy has the same number of incoming and outgoing arcs. Inequalities (A3) indicate that each vehicle visits each site-copy at most once. Conditions (A4)–(A5) ensure that the required number of wells are drilled at each site, and each well is drilled by one vehicle at some visit. Constraints (A6)–(A7) set the starting times of the works on sites for vehicles. Inequalities (A8)–(A9) ensure feasibility of routes with respect to time windows. Conditions (A10) indicate that each vehicle starts and completes its route in the dummy site f. In (A11) it is stated that the total probability of feasible scenarios is at least $\alpha$.

The deterministic one-scenario model can be obtained if one variable $w_{sc}$ is fixed to 1 and the others are fixed to 0. Some constraints can be simplified then, e.g. (A8) and (A9) should be applied only for this single scenario. If no returns to the same site are considered, i.e. there is only one copy of each site ($\mathcal{I}_i = \{i\}$ for each i), the model is similar to the one of [10]. The same way the multi-scenario deterministic model, which is used in the parallel greedy algorithm, can be obtained if variables $w_{sc}$ are fixed to one for the given set of scenarios and are fixed to zero for the others.