Scheduling for multi-robot routing with blocking and enabling constraints

Abstract

This paper considers the problem of servicing a set of locations by a fleet of robots so as to minimize overall makespan. Although motivated by a specific real-world, multi-robot drilling and fastening application, the problem also arises in a range of other multi-robot domains where service start times are subject to precedence constraints and robots must be routed in space and time to avoid collisions. We formalize this general problem and analyze its complexity. We develop a heuristic local search procedure for solving it and analyze its performance on a set of synthetically generated problem instances, some of which capture the specific structure of the motivating drilling and fastening application, and others that generalize to other application settings. We provide a differential analysis of our local search procedure and a comparison to other approaches to demonstrate the efficacy of the proposed heuristic.

Appendices

Appendix

Example for a case where the number of minimal models for $CF_{s_{curr}}^{Ps}$ in Equation (14) is exponential in M

Assume the SSp instance contains no CCs, the number of robots M is even and the consistent positional selection Ps has no arc with its head to a node in $B_{s_{curr}}$. For notational convenience, let $v^i = s_{curr}[i]$ and let $w^i = next_{v^i}$. The ECs for each $w^i, \, i \in \left[ 1, M\right]$ is defined as follows:

(30)

Under those assumptions, we can safely assume that $CF_{s_{curr}}^{Ps}$ only contains clauses for the enabling constraints (Equation (11)) and of course the clause to select at least one node in $B_{s_{curr}}$ to appear in the successor. Also observe that each node in $B_{s_{curr}}$ corresponds to a maximal scc in $G_{Ps}\left[ B_{s_{curr}}\right]$. So instead of stating which maximal sccs occur in the minimal solution, we will instead mention which nodes of $B_{s_{curr}}$ occur in each minimal solution.

For $M = 8$ case, we illustrate the enablers provided in Equation (30) graphically in Fig. 17. For example, if $w^1$ is to appear in $s_{next}$ (minimal successor of $s_{curr}$), then from Equation (30) notice that at least one among $w^7, w^8$ must also appear $s_{next}$. We represented this requirement using arcs $(w^8, w^1), (w^7, w^1)$ in Fig. 17. Similarly the enabling requirements for all other nodes are also represented using arcs in Fig. 17.

We next enumerate the minimal models of $CF_{s_{curr}}^{Ps}$. We claim that nodes occurring on any chordless cycle in the graph shown in Fig. 17 corresponds to a minimal model. To see this, observe that the nodes occurring on any cycle of the graph shown in Fig. 17 corresponds to a feasible model for $CF_{s_{curr}}^{Ps}$, as every node in the cycle is enabled by its predecessor on the cycle. If the cycle contained a chord, then the solution is not minimal since we can form a shorter cycle and the nodes belonging to the shorter cycle also correspond to a feasible model of $CF_{s_{curr}}^{Ps}$. We can continue short cutting the cycle until we obtain a chordless cycle. There are at least $2^{\mathbf {\frac{M}{2}}}$ chordless cycles ( Lavrov (2018)) in the graph shown in Fig. 17 and corresponding to each of those cycles we can obtain a minimal successor.

Fig. 17

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Proof of proposition 2

Under the conditions mentioned in proposition 2 we first provide a simple procedure for enumerating the minimal models of $CF_{s_{curr}}^{Ps}$ and then argue that the number of such minimal models generated by the procedure does not exceed M. Briefly, the enumeration procedure omits the collision related clauses (see Equation (12)) in $CF_{s_{curr}}^{Ps}$ and outputs the minimal models feasible for the remaining clauses. The minimal models obtained from the previous step are then checked whether any collision related clause is violated. Our procedure essentially builds a directed graph to capture the dependencies between maximal sccs in $G_{Ps}\left[ B_{s_{curr}} \right]$ and then retrieves the minimal models from the dependency graph. The description below interlaces the procedure with the reasons for those steps being performed. The procedural steps are highlighted in bold italics.

• 1. We build the dependency graph $G_{dep}$ as follows. Corresponding to each maximal scc of $G_{Ps}\left[ B_{s_{curr}} \right]$, we introduce a vertex in $G_{dep}$. If e is a maximal scc of $G_{Ps}\left[ B_{s_{curr}} \right]$, then slightly abusing notation we will denote the vertex corresponding to scc e in $G_{dep}$ by $v_e$.

• 2. Under the condition mentioned in the proposition that $|K(u)| \le 1$, every clause that was generated in $CF_{s_{curr}}^{Ps}$ from Equations (7), (11) and (13) are either of the form $(y_c \rightarrow y_d)$ or $(y_c \rightarrow 0)$. Corresponding to each clause of the form $(y_c \rightarrow y_d)$ in $CF_{s_{curr}}^{Ps}$ we introduce an arc $(v_d, v_c)$ in $G_{dep}$, to represent the dependency of scc c on scc d.

• 3. After introducing those arcs in the previous step, we then remove some of the vertices and arcs from $G_{dep}$ as follows. For each clause of the form $(y_c \rightarrow 0)$ in $CF_{s_{curr}}^{Ps}$, we remove vertex $v_c$ and all the other vertices that are reachable from $v_c$ by traversing arcs in $G_{dep}$ from tail to head. While nodes comprising scc c cannot appear in a successor of $s_{curr}$ owing to the clause $(y_c \rightarrow 0)$, sccs reachable from c can also be ignored owing to their dependency on c by transitivity. Consequently we may remove $v_c$ and vertices dependent on $v_c$ from further consideration in the enumeration process.

• 4. We next describe a procedure to enumerate the minimal models of $CF_{s_{curr}}^{Ps}$ using the maximal sccs of $G_{dep}$. Compute the maximal sccs of $G_{dep}$. Observe that each maximal scc in $G_{dep}$ comprises of maximal sccs of $G_{Ps}\left[ B_{s_{curr}} \right]$. Further, if $W_1, W_2$ are maximal sccs of $G_{dep}$, such that, there exists a path from a vertex in $W_1$ to a vertex in $W_2$, then vertices in $W_2$ depend on $W_1$. It implies that if a member of $W_2$ is present in a feasible solution to $CF_{s_{curr}}^{Ps}$ then all members of $W_1$ must also necessarily be present in that solution, but not necessary for the other way around. Hence to enumerate minimal models of $CF_{s_{curr}}^{Ps}$, it is sufficient to consider only those maximal sccs of $G_{dep}$ that do not have an incoming arc from any other maximal scc of $G_{dep}$¹, since occurrence of these sccs in a feasible solution is not dependent on the occurrence other sccs.

• 5. Let us denote the maximal sccs of $G_{dep}$ that do not have an incoming arc from any other maximal sccs of $G_{dep}$ by $SCC_{dep}$. If there exists a pair of maximal sccs of $G_{Ps}\left[ B_{s_{curr}} \right]$, say c and d, such that $CF_{s_{curr}}^{Ps}$ contains a collision related clause $(y_{c} \rightarrow (1-y_d))$ while vertices $v_c$ and $v_d$ both belong to the same maximal scc of $G_{dep}$, then the STP to $s_{curr}$ cannot possibly be extended to a complete STP.

• 6. If the condition in the previous step does not occur, then from each member (maximal scc) of $SCC_{dep}$ we generate a minimal model of $CF_{s_{curr}}^{Ps}$. If W is a member of $SCC_{dep}$, then a minimal model of $CF_{s_{curr}}^{Ps}$ is derived by setting the $y_{\cdot }$ variables corresponding to just those maximal sccs of $G_{Ps}\left[ B_{s_{curr}} \right]$ that are present in W to 1. Since the number of maximal sccs of $G_{dep}$ is bounded by M, it follows that the number of minimal models of $CF_{s_{curr}}^{Ps}$ is $\le M$.