A two-stage robust approach for minimizing the weighted number of tardy jobs with objective uncertainty

Abstract

Minimizing the weighted number of tardy jobs on one machine is a classical and intensively studied scheduling problem. In this paper, we develop a two-stage robust approach, where exact weights are known after accepting the jobs to be performed, and before sequencing them on the machine. This assumption allows diverse recourse decisions to be taken in order to better adapt one’s mid-term plan. The contribution of this paper is twofold: First, we introduce a new scheduling problem and model it as a min-max-min optimization problem with mixed-integer recourse by extending existing models proposed for the deterministic case. Second, we take advantage of the special structure of the problem to propose two solution approaches based on results from the recent robust optimization literature: namely the finite adaptability (Bertsimas and Caramanis in IEEE Trans Autom Control 55(12):2751–2766, 2010) and a convexification-based approach (Arslan and Detienne in INFORMS J Comput 34(2):857–871, 2022). We also study the additional cost of the solutions if the sequence of jobs has to be determined before the uncertainty is revealed. Computational experiments are reported to analyze the effectiveness of our approaches.

A. Solution approaches for problem \((\widetilde{\mathcal {P})}\)

1.1 A.1 Finite adaptability

In this section, we give a K-adaptability formulation of the second problem. The main constraints are identical to those derived in Sect. 4.1. Only the modified constraints and variables are explained below:

$$\begin{aligned}&\text {minimize} \nonumber \\&\sum _{j\in {\mathcal {J}}} \left[ w_jU_j + (1-U_j)f_j + v_j \right] + \varGamma u - \sum _{q=1}^K\sum _{k\in {\widetilde{{\mathcal {J}}}}}f_k\psi _k^q \end{aligned}$$

(67)

$$\begin{aligned}&\text {subject to}\nonumber \\&(37)-(39)\nonumber \\&y_k^q\le x_k\quad \forall k\in {\widetilde{{\mathcal {J}}}},q=1,\dots ,K \end{aligned}$$

(68)

$$\begin{aligned}&z_k^q \le y_k^q\quad \forall k\in {\widetilde{{\mathcal {J}}}},q=1,\dots ,K \end{aligned}$$

(69)

$$\begin{aligned}&\sum _{k\in {\mathcal {G}}_j}x_k = 1 - U_j\quad \forall j\in {\mathcal {J}} \end{aligned}$$

(70)

$$\begin{aligned}&(42)-(48)\nonumber \\&t^q_k \ge 0\quad \forall k\in {\widetilde{{\mathcal {J}}}},q=1,\dots ,K \end{aligned}$$

(71)

$$\begin{aligned}&y_k^q\in \{0,1\}\quad \forall k\in {\widetilde{{\mathcal {J}}}},q=1,\dots ,K \end{aligned}$$

(72)

$$\begin{aligned}&z_k^q\in \{0,1\}\quad \forall k\in {\widetilde{{\mathcal {J}}}},q=1,\dots ,K \end{aligned}$$

(73)

$$\begin{aligned}&\psi _k^q\ge 0\quad \forall k\in {\widetilde{{\mathcal {J}}}},q=1,\dots ,K \end{aligned}$$

(74)

$$\begin{aligned}&\zeta _k^q\ge 0\quad \forall k\in {\widetilde{{\mathcal {J}}}},q=1,\dots ,K \end{aligned}$$

(75)

$$\begin{aligned}&U_j\in \{0,1\}\quad \forall j\in {\mathcal {J}}\end{aligned}$$

(76)

$$\begin{aligned}&u\ge 0 \end{aligned}$$

(77)

$$\begin{aligned}&v_j\ge 0\quad \forall j\in {\mathcal {J}}\end{aligned}$$

(78)

$$\begin{aligned}&\beta _q \ge 0\quad q=1,\dots ,K \end{aligned}$$

(79)

Here, the binary (first-stage) decision variable $x_k$ represents the selection of the kth job occurrence in the non-decreasing order of the deadlines, while $U_j$ denotes the variable indicating whether a job is executed tardy or not. Constraint (68) links the first- and second-stage decisions, constraint (69) ensures that a job is repaired only if it is scheduled, and constraint (70) ensures that exactly one job occurrence is selected for on-time jobs. The other variables and constraints have the same meaning as in KAdapt1. This model will be referred to as KAdapt2.

1.2 Convexification of the recourse set

In a very similar way as what has been done for problem $(\mathcal P)$, we can derive an exact formulation for this problem variant by using Proposition 1 on the set of eligible second-stage solutions $\widetilde{{\mathcal {Y}}}$. Then, by enumerating the extreme points of the convex hull of ${{\mathcal {Y}}}$, we can derive the following model:

$$\begin{aligned}&\text {minimize} \nonumber \\&\sum _{j\in {\mathcal {J}}}\left[ w_jU_j + f_j(1-U_j) \right] - \sum _{k\in {\widetilde{{\mathcal {J}}}}}\sum _{e\in E} f_k{\textbf{y}}_k^e\alpha _e + \varGamma u \end{aligned}$$

(80)

$$\begin{aligned}&+ \sum _{j\in {\mathcal {J}}}v_j \nonumber \\&\text {subject to}\nonumber \\&\sum _{e\in E}\alpha _e = 1 \end{aligned}$$

(81)

$$\begin{aligned}&\sum _{e\in E}{\textbf{y}}_k^e\alpha _e \le x_k \quad \forall k\in {\widetilde{{\mathcal {J}}}} \end{aligned}$$

(82)

$$\begin{aligned}&\sum _{k\in {\mathcal {G}}_j}x_k = 1 - U_j \quad \forall j\in {\mathcal {J}} \end{aligned}$$

(83)

$$\begin{aligned}&u + v_j \ge \sum _{k\in {\mathcal {G}}_j}\left[ {\bar{\delta }}_k\sum _{e\in E}\left( {\textbf{y}}_k^e - {\textbf{z}}_k^e \right) \alpha _e \right] \quad \forall j\in {\mathcal {J}} \end{aligned}$$

(84)

$$\begin{aligned}&x_k\in \{0,1\}\quad \forall k\in {\widetilde{{\mathcal {J}}}} \end{aligned}$$

(85)

$$\begin{aligned}&U_j\in \{0,1\}\quad \forall j\in {\mathcal {J}}\end{aligned}$$

(86)

$$\begin{aligned}&\alpha _e\ge 0\quad \forall e\in E \end{aligned}$$

(87)

$$\begin{aligned}&u\ge 0 \end{aligned}$$

(88)

$$\begin{aligned}&v_j\ge 0\quad \forall j\in {\mathcal {J}}\end{aligned}$$

(89)

Again, decision vector $\alpha$ represents the convex combination multipliers from the inner description of $\text {conv}({{\mathcal {Y}}})$, and u and v are the dual variables associated to the constraint $\xi \in \varXi$. Constraint (81) ensures that the second-stage variables must be a convex combinations of some extreme points. Constraint (82) links the first-stage variables with the second-stage variables, while constraint (83) ensures that exactly one job occurrence per job is selected in the first stage. Finally, constraint (84) corresponds to the dualized cost implied by the venue of a scenario $\xi \in \varXi$.

This model will be referred to as ColGen2.

We solve this large-scale MILP model using a simple adaptation of the branch-and-price algorithm presented in Sect. 4.2.2. Algorithm 1 is modified by checking, at line 10, the integrality of both $U^*$ and $x^*$, and lines 13 and 14 are adapted to branch either on a U- or an x-variable. Surprisingly, the pricing problem differs only by the value of the dual variable $\mu$ in input, which is now associated with constraint (82) instead of (58).