Unreliable Job Selection and Sequencing Problem

• UJSSP is a stochastic combinatorial optimization problem that integrates job selection and sequencing under the risk of stochastic failures.
• It employs a Z-ordering strategy alongside MILP, dynamic programming, and stepwise exact algorithms to optimize expected net profit.
• Empirical results demonstrate its effectiveness for large-scale scheduling, with extensions applicable to distributed and real-time frameworks.

The Unreliable Job Selection and Sequencing Problem (UJSSP) is a stochastic combinatorial optimization problem that generalizes classical sequencing and scheduling models to incorporate explicit job- or resource-level unreliability. It encompasses single-machine stochastic breakdown models, deadline-constrained multi-worker scheduling, and robust job shop formulations under parametric uncertainty. The defining feature is that both which jobs are selected and the sequential order in which they are executed must be optimized in the presence of uncertain, potentially catastrophic failure events, with the objective of maximizing an expected utility such as net profit or aggregate completion rate.

1. Formal Problem Definition and Stochastic Failure Model

Let $J = \{1,2,\ldots,n\}$ be a set of jobs. Each job $j \in J$ has a nonnegative setup cost $c_j \geq 0$, reward $r_j \geq 0$, and a success probability $\pi_j \in [0,1]$. When job $j$ is selected for processing, a cost $c_j$ is incurred. The machine is subject to failure: if the machine fails before or during job $j$, no reward is obtained for $j$ and no subsequent jobs can be executed (permanent breakdown). For a selected subset $S \subseteq J$, the sequence (permutation) $\sigma$ determines the order in which jobs from $S$ are processed.

The expected reward for processing sequence $(S,\sigma)$ is: $$R(S, \sigma) = \sum_{k=1}^{|S|} r_{\sigma(k)} \prod_{i=1}^k \pi_{\sigma(i)}$$ Total cost is $c(S) = \sum_{j \in S} c_j$. The expected net profit is $z(S, \sigma) = R(S, \sigma) - c(S)$. Once the subset $S$ is fixed, the optimal sequencing $\sigma$ ranks jobs by descending $Z_j = \frac{\pi_j r_j}{1-\pi_j}$. The UJSSP is then formulated as

$\max_{S \subseteq J} \left\{ R(S) - c(S) \right\},$

where $R(S) = R(S,\sigma_Z)$, the expected reward under the optimal $Z$-order (Agnetis et al., 21 Nov 2025).

2. Computational Complexity and Special-Case Solvability

The general UJSSP is strongly NP-hard. NP-hardness is established by reduction from the Product Partition Problem (PPP) via a parameterization mapping integers $a_j\geq 2$ to $\pi_j=1/a_j$, $r_j=\sqrt W (a_j-1)$, $c_j = \ln a_j$ where $W = \prod_j a_j$, such that achieving a specified net profit is equivalent to solving PPP. Hence, no polynomial-time solution is likely to exist for general parameter choices (Agnetis et al., 21 Nov 2025).

However, two important special cases admit efficient solutions:

• Identical costs $(c_j \equiv c)$: The problem reduces to a monotone submodular maximization with cardinality constraint, solved optimally by the classical greedy algorithm based on marginal increases in net profit [Stadje 1995; Chade & Smith 2006].
• Identical success probabilities $(\pi_j\equiv \pi)$: The optimal subset consists of the jobs with the highest $r_j$ values, sequenced arbitrarily, again solvable greedily as a polymatroid base maximization [Olszewski & Vohra 2016].

Thus, while UJSSP is hard in general, parameter-homogeneous settings are tractable.

3. Optimization Techniques and Exact Algorithms

Three classes of exact algorithms have been developed:

1. Compact Mixed-Integer Linear Programming (MILP) Formulation (Z-ordering):
   • Binary selection $x_j \in \{0,1\}$, continuous variables $P_j \geq 0$ for cumulative survival up to job $j$, sequencing in descending $Z_j$ order. The MILP optimizes

   $\max \sum_{j=1}^n (r_j P_j - c_j x_j)$

   under linear constraints propagating survival probabilities.

2. Dynamic Programming (DP) Approach:

   • When costs are integer and problem size is moderate, define $g(b,i)$ as the maximal expected reward from jobs $i,\ldots,n$ with budget $b$. Standard forward recursion computes $g(b,i+1)$ with or without selecting $i$, yielding the optimal net profit by maximization over feasible budgets. Time complexity is $O(n \cdot \sum_j c_j)$ (pseudopolynomial for costs) (Agnetis et al., 21 Nov 2025).
3. Stepwise Exact Methods (Forward/Backward Scan, Editor's term):
   • These novel algorithms maintain upper envelopes of affine marginal functions $F(S;r)$ as partial subsets $S$ are constructed. At each iteration, dominated partial solutions are pruned using parametrized linear criteria, retaining only those that can be optimal for some suffix reward $r$ (forward) or prefix product $p$ (backward). Lines are managed by dynamic convex hull/upper-envelope algorithms [Preparata & Shamos 1979]. In practical computation, candidate set sizes grow subexponentially, enabling solution of real-sized instances with $n$ up to $10^4$ jobs in seconds to minutes.

Results on artificial and PPP-derived datasets show the stepwise algorithms handle large instances efficiently; MILP is limited by solver scalability, and DP is practical up to a few thousand jobs unless costs are very large (Agnetis et al., 21 Nov 2025).

4. Relationship to Distributed and Real-Time Scheduling Variants

UJSSP theory unifies several lines of research:

• Distributed stochastic real-time job scheduling with unreliable workers (Hsu et al., 2019): In this extension, each of $N$ applications produces random jobs requiring subtasks on $M$ unreliable workers, each with type-specific completion probabilities $P_{i,j}$. There is a single-frame deadline, and a feasible schedule must select a non-interfering set of jobs with completed tasks on distinct (available) workers. The stochastic completion rate region $\mathcal{R}_{\text{max}}$ is optimized using a Lyapunov drift-based virtual queue construction and NP-hard set-packing for the “max-weight” policy. A $\sqrt{M}$-approximate greedy policy is proved to achieve a fraction $1/\sqrt{M}$ of $\mathcal{R}_{\text{max}}$ at $O(N\log N)$ complexity per frame.
• Uncertainty-robust Job Shop Scheduling (Infantes et al., 4 Mar 2024): Here, each operation's processing time is a random variable, and robust schedules are learned via GNN+PPO (Proximal Policy Optimization) architectures, minimizing expected makespan. The “Wheatley” GNN/DRL policy natively handles machine/process failure and dynamic job arrival by augmentation of the graph model and Markov Decision Process, providing a flexible framework for generalizations of UJSSP with correlated network failures or dynamically arriving task sets.

UJSSP thus constitutes a core combinatorial structure for stochastic scheduling under both selection and execution risk across a spectrum of application domains.

5. Empirical Results and Large-Instance Performance

Systematic computational experiments on diverse instances demonstrate that:

• The stepwise exact algorithms solve random UJSSP instances with $n\leq 10\,000$, probabilities $\pi_j \sim$ Unif(0.01,0.99), and randomly assigned costs/rewards in under a second for most parameter schemes; the hardest product-partition-derived instances with non-integer costs and tightly grouped $\pi_j$ are tractable up to $n=25$–$50$.
• MILP solvers reliably find optima for $n\leq 40$; performance degrades sharply for larger or more degenerate cases (large root LP gaps, slow branch pruning).
• DP methods are effective up to $n\approx 2500$ for moderate integer costs, but are subject to coarse-graining limitations.
• For distributed and deadline settings (Hsu et al., 2019), the $\sqrt{M}$-approximate greedy policy tracks the (intractable) optimal scheduling region closely in all tested cases ($N$ up to 30), with $O(1/N)$ scaling in symmetric per-application rates.

These computational findings establish the practicality of UJSSP formulations and exact solution techniques for large-scale real-world selections and sequence optimization under stochastic failure.

6. Connections, Variants, and Open Questions

UJSSP generalizes and subsumes classical test sequencing, optimal college admission, and single-machine breakdown models. By mapping combinatorial submodular profit functions to special parameter regimes (e.g., uniform $c_j$ or $\pi_j$), UJSSP admits efficient greedy optimization, whereas the full generality with arbitrary continuous parameters is NP-hard (Agnetis et al., 21 Nov 2025).

Extensions to time-window constraints, strong/weak correlated failures, multi-machine selection, and streaming arrivals are possible by embedding the UJSSP core in richer stochastic scheduling frameworks (Hsu et al., 2019, Infantes et al., 4 Mar 2024). Notably, the computational complexity with strictly fractional costs remains unresolved, as does the existence (or tightness) of approximation algorithms for the general parameter set.

A promising direction is the integration of deep RL/GNN optimization policies with the envelope-pruning approach for hybrid data-driven and combinatorial exact methods (Infantes et al., 4 Mar 2024). The UJSSP also serves as a template for evaluating and benchmarking algorithms in stochastic decision-making under cascading failure risk scenarios.

---

Key References:

• "The Unreliable Job Selection and Sequencing Problem" (Agnetis et al., 21 Nov 2025)
• "Scheduling Stochastic Real-Time Jobs in Unreliable Workers" (Hsu et al., 2019)
• "Learning to Solve Job Shop Scheduling under Uncertainty" (Infantes et al., 4 Mar 2024)