Finding an optimal Nash equilibrium to the multi-agent project scheduling problem

Abstract

Large projects often involve a set of contractors, each in charge of a part of the project. In this paper, we assume that every contractor is self-interested and can control the duration of his/her activities, which can be shortened up to an incompressible limit, by gathering extra resources at a given cost. In this context, the resulting project makespan depends on all the contractors’ decisions. The customer of the project is interested in a short project makespan and offers a reward, proportional to the project makespan reduction, to be shared by the contractors. In practice, either the reward sharing policy results from an upfront agreement or payments are freely allocated by the customer. Each contractor is only interested in the maximization of his/her profit and behaves accordingly. This paper addresses the problem of finding a Nash equilibrium and a sharing policy that minimize the project makespan. The aim is to help the customer to determine the duration of the activities and the reward sharing policy such that no agent has an incentive to unilaterally deviate from this solution. We show that this problem is NP-hard and how it can be modeled and solved by mixed integer linear programming. Computational analysis on large instances proves the effectiveness of our approach. Based on an empirical investigation of the influence of reward sharing policies on the project makespan, the paper provides new insight into how a project’s customer should offer rewards to the contractors.

Appendices

Appendix 1: Problem complexity

Proof

The complexity of MAPSP can be shown using a reduction from the MAPSP assuming fixed w (MAPSP$^F$) which is known to be NP-complete in the strong sense [see Briand et al. (2012)]. While in MAPSP w is a part of its solution, in MAPSP$^F$ $w_u$ is fixed for each agent and it equals $w_u^F$.

An instance of MAPSP$^F$, given by a tuple $\left\langle {{\mathcal {G}}}^F, {{\mathcal {A}}}^F, {\underline{S}}^F, {\overline{S}}^F, C^F, \pi ^F, w^F \right\rangle$, polynomially reduces to an instance of MAPSP by constructing the project network ${{\mathcal {G}}}$ illustrated in Fig. 9. This network assumes the same set of agents, i.e., ${{\mathcal {A}}} = {{\mathcal {A}}}^F$, and exactly the same daily reward $\pi = \pi ^F$. The network has two subnetworks, i.e., ${{\mathcal {G}}}_{l,r}$ between nodes l and r and ${{\mathcal {G}}}_{r,n}$ between nodes r and n. Subnetwork ${{\mathcal {G}}}_{l,r}$ is ${{\mathcal {G}}}^F$, while the aim of the second subnetwork is to fix $w_u$ to $w_u^F$. The second subnetwork consists of m arcs starting from node r, each belonging to one agent $u \in {{\mathcal {A}}}$. These activities have duration $p_{i,j} \in \left\{ {\underline{p}}_{i,j}=0, {\overline{p}}_{i,j} = D({\overline{S}})^F + 1 \right\}$ where $D({\overline{S}})^F$ is the maximal project duration of problem MAPSP$^F$. Crashing cost of those activities is $c_{i,j} = w_u^F + \xi$ where $\xi$ is an arbitrarily small number. Network ${{\mathcal {G}}}_{r,n}$ may have only one decreasing cut w, which is profitable iff $w_u = w_u^F$ $\forall u \in {{\mathcal {A}}}$. Assuming $w = w^F$ then the makespan of ${{\mathcal {G}}}_{r,n}$ is 0 and the solution associated with ${{\mathcal {G}}}_{l,r}$ is the solution of MAPSP$^F$. On the other hand, if the makespan of ${{\mathcal {G}}}_{r,n}$ is $D({\overline{S}})^F + 1$ then the solution cannot be optimal. Therefore each optimal solution of the project network ${{\mathcal {G}}}$ in Fig. 9 has $w = w^F$. $\square$

Fig. 9

Reduction from MAPSP$^F$

Appendix 2: MILP model

$$\begin{aligned}&~~\min \left( (t_n - t_1) + \frac{\sum _{\forall (i,j) \in U} c_{i,j} \left( {\overline{p}}_{i,j}-p_{i,j} \right) }{ 1 + \sum _{\forall (i,j) \in U} c_{i,j} \left( {\overline{p}}_{i,j}-{\underline{p}}_{i,j} \right) } \right) \end{aligned}$$

(27)

$$\begin{aligned}&s.t. \nonumber \\&~~t_j - t_i - p_{i,j} - s_{i,j} = 0 \quad \forall (i,j) \in U\end{aligned}$$

(28)

$$\begin{aligned}&~~\displaystyle \sum _{A_u \in {{\mathcal {A}}}} w_u = 1 \end{aligned}$$

(29)

$$\begin{aligned}&~~\epsilon - z_{i,j} \le s_{i,j} \le {\overline{s}}_{i,j} \left( 1 - z_{i,j}\right) \quad \forall (i,j) \in U \end{aligned}$$

(30)

$$\begin{aligned}&~~z_{i,j} \le \displaystyle \sum _{\forall (k,i) \in U} z_{k,i} \quad \forall (i,j) \in U: i > 1 \end{aligned}$$

(31)

$$\begin{aligned}&~~z_{i,j} \le \displaystyle \sum _{\forall (j,l) \in U} z_{j,l} \quad \forall (i,j) \in U: j < n \end{aligned}$$

(32)

$$\begin{aligned}&~~x_{i,j} \le \left( {\overline{p}}_{i,j} - p_{i,j}\right) \le \left( {\overline{p}}_{i,j} - {\underline{p}}_{i,j}\right) x_{i,j} \quad \forall (i,j) \in U \end{aligned}$$

(33)

$$\begin{aligned}&~~y_{i,j} \le \left( p_{i,j} - {\underline{p}}_{i,j}\right) \le \left( {\overline{p}}_{i,j} - {\underline{p}}_{i,j}\right) y_{i,j} \quad \forall (i,j) \in U \nonumber \\\end{aligned}$$

(34)

$$\begin{aligned}&~~z_{i,j} \ge x_{i,j} \quad \forall (i,j) \in U \end{aligned}$$

(35)

$$\begin{aligned}&~~{\underline{c}}_{i,j}^u = c_{i,j} - \left( 1 - x_{i,j}\right) c_{i,j} \quad \nonumber \\&\quad \forall (i,j) \in U, \forall A_u \in \mathcal {A} : (i,j) \in \mathcal {T}_u \end{aligned}$$

(36)

$$\begin{aligned}&~~{\overline{c}}_{i,j}^u = c_{i,j} + \left( 1 - y_{i,j}\right) C - \left( 1 - z_{i,j}\right) c_{i,j} \quad \nonumber \\&\quad \forall (i,j) \in U, \forall A_u \in \mathcal {A} : (i,j) \in \mathcal {T}_u \end{aligned}$$

(37)

$$\begin{aligned}&~~{\underline{c}}_{i,j}^u = 0 \quad \forall (i,j) \in U, \forall A_u \in \mathcal {A} : (i,j) \notin \mathcal {T}_u \end{aligned}$$

(38)

$$\begin{aligned}&~~{\overline{c}}_{i,j}^u = C z_{i,j} \quad \forall (i,j) \in U, \forall A_u \in \mathcal {A} : (i,j) \notin \mathcal {T}_u \end{aligned}$$

(39)

$$\begin{aligned}&~~\alpha _{i,j}^u - \beta _{i,j}^u - \gamma _i^u + \gamma _j^u \le 0 \quad \forall (i,j) \in U, \forall A_u \in \mathcal {A} \end{aligned}$$

(40)

$$\begin{aligned}&~~\gamma _1^u = 1 \quad \forall A_u \in \mathcal {A} \end{aligned}$$

(41)

$$\begin{aligned}&~~\gamma _n^u = 0~\forall A_u \in \mathcal {A} \end{aligned}$$

(42)

$$\begin{aligned}&~~\displaystyle \sum _{\forall (k,i) \in U} f_{k,i}^u = \displaystyle \sum _{\forall (i,k) \in U} f_{i,k}^u \quad \forall i \in X {\setminus } \left\{ 1,n\right\} , \forall A_u \in \mathcal {A} \end{aligned}$$

(43)

$$\begin{aligned}&~~{\underline{c}}_{i,j}^u \le f_{i,j}^u \le {\overline{c}}_{i,j}^u \quad \forall (i,j) \in U, \forall A_u \in \mathcal {A} \end{aligned}$$

(44)

$$\begin{aligned}&~~\phi _{i,j}^u \le \alpha _{i,j}^u c_{i,j} \quad \forall (i,j) \in U, \forall A_u \in \mathcal {A} \end{aligned}$$

(45)

$$\begin{aligned}&~~\phi _{i,j}^u \le {\underline{c}}_{i,j}^u \quad \forall (i,j) \in U, \forall A_u \in \mathcal {A} \end{aligned}$$

(46)

$$\begin{aligned}&~~\phi _{i,j}^u \ge {\underline{c}}_{i,j}^u - \left( 1-\alpha _{i,j}^u\right) c_{i,j} \quad \forall (i,j) \in U, \forall A_u \in \mathcal {A} \end{aligned}$$

(47)

$$\begin{aligned}&~~\varphi _{i,j}^u \le \beta _{i,j}^u 3C \quad \forall (i,j) \in U, \forall A_u \in \mathcal {A} \end{aligned}$$

(48)

$$\begin{aligned}&~~\varphi _{i,j}^u \le {\overline{c}}_{i,j}^u \quad \forall (i,j) \in U, \forall A_u \in \mathcal {A} \end{aligned}$$

(49)

$$\begin{aligned}&~~\varphi _{i,j}^u \ge {\overline{c}}_{i,j}^u - \left( 1-\beta _{i,j}^u\right) 3C \quad \forall (i,j) \in U, \forall A_u \in \mathcal {A} \end{aligned}$$

(50)

$$\begin{aligned}&~~\displaystyle \sum _{\forall (1,k) \in U} f_{1,k}^u \le \displaystyle \sum _{(i,j) \in U} \phi _{i,j}^u - \displaystyle \sum _{\forall (i,j) \in U} \varphi _{i,j}^u \quad \forall A_u \in \mathcal {A} \end{aligned}$$

(51)

$$\begin{aligned}&\displaystyle \sum _{\forall (1,k) \in U} f_{1,k}^u < \pi w_u \quad \forall A_u \in \mathcal {A} \end{aligned}$$

(52)

where

$$\begin{aligned}&~~t_i, D \in {\mathbb {R}}_{\ge 0}, ~~s_i \in {\mathbb {R}}_{\ge 0}, ~~p_{i,j} \in {\mathbb {Z}}_{\ge 0}, ~~{\underline{p}}_{i,j} \le p_{i,j} \le {\overline{p}}_{i,j}, \\&~~0 \le w_u \le 1, ~~ x_{i,j}, y_{i,j}, z_{i,j} \in \left\{ 0,1\right\} , \\&~~\alpha _{i,j}^u, \beta _{i,j}^u, \gamma _i^u \in {\mathbb {Z}}_{\ge 0}, ~~f_{i,j}^u \in {\mathbb {R}}_{\ge 0}, \\&~~{\underline{c}}_{i,j}^u, {\overline{c}}_{i,j}^u \in {\mathbb {R}}_{\ge 0}, ~~\phi _{i,j}^u, \varphi _{i,j}^u \in {\mathbb {R}}_{\ge 0}, ~~\epsilon = 1/n, \\&~~C = \sum _{(i,j) \in U} c_{i,j}, {\overline{s}}_{i,j} = D({\overline{S}}). \end{aligned}$$