Integration of routing into a resource-constrained project scheduling problem

Abstract

The resource-constrained project scheduling problem (RCPSP), one of the most challenging combinatorial optimization scheduling problems, has been the focus of a great deal of research, resulting in numerous publications in the last decade. Previous publications focused on the RCPSP, including several extensions with different objectives to be minimized and constraints to be checked. The present work investigates the integration of the routing, i.e., the transport of the resources between activities into the RCPSP, and provides a new resolution scheme. The two subproblems are solved using an integrated approach that draws on both a disjunctive graph model and an explicit modeling of the routing. The resolution scheme takes advantage of an indirect representation of the solution to define both the schedule of activities and the routing of vehicles. The routing solution is modeled by a set of trips that define the loaded transport operations of vehicles that are induced by the flow in the graph. The numerical experiments prove that the models and the methods introduced in this paper are promising for solving the RCPSP with routing.

Appendix

1.1 A.1: MILP formulation for the RCPSPR

Data of the problem

$p_{i}$:

processing time of activity $i$

$b_{i}$:

number of resource required for activity $i$

$t_{ij}$:

transfer time from activity $i$ to activity $i$

$V$:

set of activities to schedule

$E$:

set of couple of activities linked by a precedence constraints

$T$:

set of vehicle

$T_{i}$:

vehicle $i$

$C_{i}$:

capacity of vehicle $i$

Variables

${\varphi }_{\text{ijk}}$:

number of resource $k$ transferred from $i$ to $j$

${\text{z}}_{\text{ij}}$:

binary variable that value 1 if activity $i$ is scheduled before activity $j$

${\text{S}}_{\text{i}}$:

starting time of activity $i$

$y_{ijut}$:

amount of resource transported from activity $i$ to $j$ by vehicle $u$ during the $t^{th}$ transport operation

$x_{ijut}$:

binary variable such $x_{ijut} = 1$ if vehicle $u$ make the transport of resource from activity $i$ to $j$ during the $t^{th}$ transport operation

$a_{ijtpqv}^{u}$:

binary variable with $a_{ijtpqv}^{u} = 1$ if transport operation number $t$ of vehicle $u$ from activity $i$ to $j$ is scheduled before the transport operation number $v$ of vehicle $u$ from activity $p$ to q.

The constraints numbered 1 to 9 in $P1$ are the classical RCPSP constraints (scheduling constraint), and constraints numbered 10 to 25 in $P2$ are the constraints tackling the transport.

$$ P1:\left\{ {\begin{array}{*{20}l} {} \hfill &\quad {Minimize\; S_{n + 1} } \hfill &\quad {(1)} \hfill \\ {\forall \left( {i ,j} \right) \in E} \hfill &\quad {z_{ij} = 1} \hfill &\quad {(2)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} } \hfill &\quad {S_{j} \ge S_{i} + p_{i} + M\left( {z_{ij} - 1} \right)} \hfill &\quad {(3)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} } \hfill &\quad {\varphi_{ij} \le min\left( {b_{i} ,b_{j} } \right)z_{ij} } \hfill &\quad {(4)} \hfill \\ {\forall j \in V} \hfill &\quad {\mathop \sum \limits_{i \in V} \varphi_{ij} = b_{j} } \hfill &\quad {(5)} \hfill \\ {\forall i \in V} \hfill &\quad {\mathop \sum \limits_{j \in V} \varphi_{ij} = b_{i} } \hfill &\quad {(6)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} } \hfill &\quad {z_{ij} \in \left\{ {0,1} \right\},\;z_{ii} = 0} \hfill &\quad {(7)} \hfill \\ {\forall i \in V} \hfill &\quad {S_{i} \in {\mathbb{N}}} \hfill &\quad {(8)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} } \hfill &\quad {\varphi_{ij} \in {\mathbb{N}},\text{ }\varphi_{ii} = 0} \hfill &\quad {(9)} \hfill \\ \end{array} } \right. $$

$$ P2:\left\{ {\begin{array}{*{20}l} {\forall \left( {i ,j} \right) \in V^{2} } \hfill & {\mathop \sum \limits_{u \in T} \mathop \sum \limits_{{t \in X_{ij} }} y_{ijut} = \varphi_{ij} } \hfill & {(10)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T,\quad \forall t \in X_{ij} } \hfill & {y_{ijut} \le min\left( {b_{i} ,b_{j} ,C_{u} } \right)x_{ijut} } \hfill & {(11)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T,\quad \forall t \in X_{ij} } \hfill & {x_{ijut} \le y_{ijut} } \hfill & {(12)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T,\forall t \in X_{ij} } \hfill & {A_{ijut} \ge B_{ijut} + t_{ij} + \left( {x_{ijut} - 1} \right).H} \hfill & {(13)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T,\forall t \in X_{ij} } \hfill & {S_{j} \ge A_{ijut} + \left( {x_{ijut} - 1} \right).H} \hfill & {(14)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T,\forall t \in X_{ij} } \hfill & {B_{ijut} \ge S_{i} + p_{i} + \left( {x_{ijut} - 1} \right).H} \hfill & {(15)} \hfill \\ {\forall \left( {i ,j,p ,q} \right) \in V^{4} ,\forall u \in T,\forall \left( {t,v} \right) \in X_{ij}^{2} } \hfill & {a_{ijtpqv}^{u} + a_{pqvijt}^{u} \le x_{ijut} } \hfill & {(16)} \hfill \\ {\forall \left( {i ,j,p ,q} \right) \in V^{4} ,\forall u \in T,\forall \left( {t,v} \right) \in X_{ij}^{2} } \hfill & {a_{ijtpqv}^{u} + a_{pqvijt}^{u} \le x_{pquv} } \hfill & {(17)} \hfill \\ {\forall \left( {i ,j,p ,q} \right) \in V^{4} ,\forall u \in T,\forall \left( {t,v} \right) \in X_{ij}^{2} } \hfill & {a_{ijtpqv}^{u} + a_{pqvijt}^{u} \ge 1 - \left( {2 - x_{ijut} - x_{pquv} } \right).H} \hfill & {(18)} \hfill \\ {\forall \left( {i ,j,p ,q} \right) \in V^{4} ,\forall u \in T,\forall \left( {t,v} \right) \in X_{ij}^{2} } \hfill & {B_{pquv} \ge A_{ijut} + t_{jp} + \left( {a_{ijtpqv}^{u} - 1} \right).H} \hfill & {(19)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T, \forall t \in 1,\;\left[ {min\left( {b_{i} ,b_{j} } \right) - 1} \right]} \hfill & {y_{ijut + 1} \le y_{ijut} } \hfill & {(20)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T,\forall t \in X_{ij} } \hfill & {y_{ijut} \in {\mathbb{N}}} \hfill & {(21)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T,\forall t \in X_{ij} } \hfill & {x_{ijut} \in \left\{ {0,1} \right\}} \hfill & {(22)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T,\forall t \in X_{ij} } \hfill & {A_{ijut} \in {\mathbb{N}}} \hfill & {(23)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T,\forall t \in X_{ij} } \hfill & {B_{ijut} \in {\mathbb{N}}} \hfill & {(24)} \hfill \\ {\forall \left( {i ,j,p ,q} \right) \in V^{4} ,\forall u \in T,\forall \left( {t,v} \right) \in X_{ij}^{2} } \hfill & {a_{ijtpqv}^{u} \in \left\{ {0,1} \right\}} \hfill & {(25)} \hfill \\ \end{array} } \right. $$

1.2 A.2: CP formulation for the RCPSPR

The CP model was formulated from the MILP, and both CP data and CP decision variables have been introduced considering the same trend as the MILP, in order to favor readability first, and to facilitate second a better knowledge and understanding for researchers coming from the scheduling community and who are more experienced in MILP than CP. A CP model resembles an integer programming model in terms of syntax. It contains decision variables with their domains, a set of constraints, and possibly, an objective function. However, the CP modeling paradigm is much more expressive. In fact, the language is a superset of the integer linear programming modeling language. In addition to equality and inequality constraints between linear mathematical expressions, a CP model can contain nonlinear expressions, logical expressions, use decision variables as indices to other vectors of decision variables and include global constraints that capture a relationship between large sets of decision variables.

Disjunctive formulation of the CP

$$ P1:\left\{ {\begin{array}{*{20}l} {} \hfill &\quad {Minimize\; S_{n + 1} } \hfill &\quad {(1)} \hfill \\ {\forall \left( {i ,j} \right) \in E} \hfill &\quad {z_{ij} = 1} \hfill &\quad {(2)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} } \hfill &\quad {z_{ij} = 1 \Rightarrow S_{j} \ge S_{i} + p_{i} } \hfill &\quad {(3)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} } \hfill &\quad {\varphi_{ij} \le min\left( {b_{i} ,b_{j} } \right)z_{ij} } \hfill &\quad {(4)} \hfill \\ {\forall j \in V} \hfill &\quad {\mathop \sum \limits_{i \in V} \varphi_{ij} = b_{j} } \hfill &\quad {(5)} \hfill \\ {\forall i \in V} \hfill &\quad {\mathop \sum \limits_{j \in V} \varphi_{ij} = b_{i} } \hfill &\quad {(6)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} } \hfill &\quad {z_{ij} \in \left\{ {0,1} \right\},z_{ii} = 0} \hfill &\quad {(7)} \hfill \\ {\forall i \in V} \hfill &\quad {S_{i} \in {\mathbb{N}}} \hfill &\quad {(8)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} } \hfill &\quad {\varphi_{ij} \in {\mathbb{N}},\text{ }\varphi_{ii} = 0} \hfill &\quad {(9)} \hfill \\ \end{array} } \right. $$

$$ P2:\left\{ {\begin{array}{*{20}l} {\forall \left( {i ,j} \right) \in V^{2} } \hfill & {\mathop \sum \limits_{u \in T} \mathop \sum \limits_{{t \in X_{ij} }} y_{ijut} = \varphi_{ij} } \hfill & {(10)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T,\forall t \in X_{ij} } \hfill & {y_{ijut} \le min\left( {b_{i} ,b_{j} ,C_{u} } \right)x_{ijut} } \hfill & {(11)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T,\forall t \in X_{ij} } \hfill & {y_{ijut} = 0 \Rightarrow x_{ijut} = 0 \wedge A_{ijut} = 0 \wedge B_{ijut} = 0 } \hfill & {(12)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T,\forall t \in X_{ij} } \hfill & {x_{ijut} = 1 \Rightarrow A_{ijut} \ge B_{ijut} + t_{ij} } \hfill & {(13)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T,\forall t \in X_{ij} } \hfill & {x_{ijut} = 1 \Rightarrow S_{j} \ge A_{ijut} } \hfill & {(14)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T,\forall t \in X_{ij} } \hfill & {x_{ijut} = 1 \Rightarrow B_{ijut} \ge S_{i} + p_{i} } \hfill & {(15)} \hfill \\ {\forall \left( {i ,j,p ,q} \right) \in V^{4} ,\forall u \in T,\forall \left( {t,v} \right) \in X_{ij}^{2} } \hfill & {a_{ijtpqv}^{u} + a_{pqvijt}^{u} \le x_{ijut} } \hfill & {(16)} \hfill \\ {\forall \left( {i ,j,p ,q} \right) \in V^{4} ,\forall u \in T,\forall \left( {t,v} \right) \in X_{ij}^{2} } \hfill & {a_{ijtpqv}^{u} + a_{pqvijt}^{u} \le x_{pquv} } \hfill & {(17)} \hfill \\ {\forall \left( {i ,j,p ,q} \right) \in V^{4} ,\forall u \in T,\forall \left( {t,v} \right) \in X_{ij}^{2} } \hfill & {\left( {x_{ijut} = 1 \wedge x_{pquv} = 1} \right) \Rightarrow a_{ijtpqv}^{u} + a_{pqvijt}^{u} \ge 1} \hfill & {(18)} \hfill \\ {\forall \left( {i ,j,p ,q} \right) \in V^{4} ,\forall u \in T,\forall \left( {t,v} \right) \in X_{ij}^{2} } \hfill & {a_{ijtpqv}^{u} = 1 \Rightarrow B_{pquv} \ge A_{ijut} + t_{jp} } \hfill & {(19)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T, \forall t \in 1,\;\left[ {min\left( {b_{i} ,b_{j} } \right) - 1} \right]} \hfill & {y_{ijut + 1} \le y_{ijut} } \hfill & {(20)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T,\forall t \in X_{ij} } \hfill & {y_{ijut} \in \left\{ {0,B} \right\}} \hfill & {(21)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T,\forall t \in X_{ij} } \hfill & {x_{ijut} \in \left\{ {0,1} \right\}} \hfill & {(22)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T,\forall t \in X_{ij} } \hfill & {A_{ijut} \in {\mathbb{N}}} \hfill & {(23)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} , \forall u \in T,\forall t \in X_{ij} } \hfill & {B_{ijut} \in {\mathbb{N}}} \hfill & {(24)} \hfill \\ {\forall \left( {i ,j,p ,q} \right) \in V^{4} ,\forall u \in T,\forall \left( {t,v} \right) \in X_{ij}^{2} } \hfill & {a_{ijtpqv}^{u} \in \left\{ {0,1} \right\}} \hfill & {(25)} \hfill \\ \end{array} } \right. $$

Cumulative formulation of the CP for the scheduling part

In the cumulative formulation, the activities are represented by interval variables. An interval variable represents an interval of time during which an activity is performed:

interval act[i in 0…n+1] size p_i

The cumulative constraint to define $f$ enforces that at each point in time, the cumulated resource demands of the set of activities that require the resource at this time, does not exceed a given limit.

cumulFunction f=sum (i in 1…n) pulse (act[i], b_i)

$$ P1^{\prime } :\left\{ {\begin{array}{*{20}l} {} \hfill &\quad {Minimize \;end\;Of\left( {act\left[ {n + 1} \right]} \right);} \hfill &\quad {(1)} \hfill \\ {} \hfill &\quad {f \le B;} \hfill &\quad {(2)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} } \hfill &\quad {z_{ij} = 1 \Rightarrow endBeforeStart(act\left[ j \right], act\left[ i \right]); } \hfill &\quad {(3)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} } \hfill &\quad {\varphi_{ij} \le min\left( {b_{i} ,b_{j} } \right)z_{ij} } \hfill &\quad {(4)} \hfill \\ {\forall j \in V} \hfill &\quad {\mathop \sum \limits_{i \in V} \varphi_{ij} = b_{j} } \hfill &\quad {(5)} \hfill \\ {\forall i \in V} \hfill &\quad {\mathop \sum \limits_{j \in V} \varphi_{ij} = b_{i} } \hfill &\quad {(6)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} } \hfill &\quad {z_{ij} \in \left\{ {0,1} \right\},z_{ii} = 0} \hfill &\quad {(7)} \hfill \\ {\forall i \in V} \hfill &\quad {S_{i} \in {\mathbb{N}}} \hfill &\quad {(8)} \hfill \\ {\forall \left( {i ,j} \right) \in V^{2} } \hfill &\quad {\varphi_{ij} \in {\mathbb{N}},\text{ }\varphi_{ii} = 0} \hfill &\quad {(9)} \hfill \\ \end{array} } \right. $$

The constraints to tackle the routing remain the same as the CP disjunctive formulation with the model $P2$.

1.3 A.3: Additional numerical experiments

Extra modeled should be investigated to get around MILP drawbacks that come from the large number of variables used and from the big-M constraints required for disjunctions of both scheduling and routing.

Cumulative constraint-based models have been proved to be efficient in RCPSP resolution (Kreter et al. 2016), and in several RCPSP variants providing results that compete with MILP formulation. Because, the transport operations are fully defined by exchange of resources between activities, the RCPSPR that requires the proper coordination of both activities and routing constraints, presents characteristics that do not favor cumulative model and by consequence resolution (Table 10).

Table 10 CP resolution, disjunctive model vs. cumulative model

To obtain a fair comparative study, the two models (disjunctive MILP formulation and CP disjunctive formulation) do not encompass any extra constraint that could strengthen the model and the two models have been solved using the CPLEX 12.7 package running on the same computer. The default parameters are used in the resolution of both MILP and CP: The results are introduced in Table 11 where * is used for optimal solution and $t.l.$ for computational time exceeding 48 h.

Table 11 GRASP $\times$ ELS resolution vs. MILP and CP

The results prove the efficiency of constraint programming formulations first (that permit to obtain numerous optimal solutions) and that the CP optimizer of CPLEX competes with the MILP solver. The coordination between scheduling and routing operations should receive attention to strengthen both models. For example, the difference between starting time of $i$ and starting time of $j$, for two scheduling operations with a non-null flow ($\varphi_{ij} \ge 0$), could be used to create new transportation constraints by a careful evaluation of the number of loaded and unloaded transport operations required, and by consequence, of the delay between $S_{j}$ and $S_{i}$. The linear constraint $S_{j} \ge S_{i} + p_{i} + M\left( {z_{ij} - 1} \right)$ could be updated with $S_{j} \ge S_{i} + p_{i} + \delta + M\left( {z_{ij} - 1} \right)$ where $\delta$ is the extra delay (Fig. 17).

Fig. 17

Illustration of one additional constraint strengthening the model

Table 12 reports the best solutions obtained by CPLEX MILP and CPLEX CP within the time that the GRASP $\times$ ELS found its best upper bound. CPLEX MILP failed to provide a solution (except for instance LMQV_U5 and LMQV_C6), whereas CPLEX CP provides 7 solutions, with poor quality regarding the solutions of the GRASP $\times$ ELS. For LMQV_U1, CPLEX CP provides a solution of 91 that is strongly larger than 74 which is the best solution found by GRASP $\times$ ELS in 58 s. The CPLEX CP provides a first solution in shorten computational time than CPLEX MILP. Such results push us into considering that constraint programming is a promising method that could be used in investigating the RCPSPR.

Table 12 GRASP $\times$ ELS resolution vs. MILP and CP within the same time