Came across a Mixed-Integer Linear Programming (MILP) focused paper that operations research enthusiasts will find interesting, specifically if your interest lies in manufacturing as well. https://lnkd.in/gs9S8sYQ

In Industry 4.0 smart factories, robotic systems increasingly take over material handling along with machining, making scheduling + robot allocation a critical and complex challenge.

Traditional flexible job shop scheduling (FJSP) formulations often simplify or ignore robot diversity, buffer constraints, blocking, or real layout constraints (e.g. not every robot can reach every machine)

The authors tackle the Multi-Robot Flexible Job Shop (MRFJS) problem, which essentially comprises of:

a. Jobs must go through sequences of operations across machines
b. Robots must transport parts between machines/buffers
c. Buffers have limited capacity; robots have varying capabilities and reachable zones
b. Blocking may occur (i.e. waiting because downstream resources are occupied)

So what is the proposed solution? The key components of the proposed methodology are:

1. Mathematical / MILP Model

They formulate the MRFJS as a Mixed-Integer Linear Programming (MILP) problem. The objective is to minimize makespan (i.e. the time when the last job finishes)

The model includes constraints for:

1. Assignment of each operation to exactly one machine/robot resource
2. Precedence among operations of each job
3. Timing constraints combining processing, robot travel, buffer waiting, and blocking
4. Robot routing, travel times (distance/speed), load/unload times

2. Genetic Algorithm (GA) using Alternative Graph & Semi-Enumerative Fitness

To cope with larger, real-scale instances, they design a GA (metaheuristic) tailored to this MRFJS problem

Key aqspects of the GA:

A. Alternative graph modeling: they extend the disjunctive / conjunctive graph framework to encode alternative choices (e.g. ordering between conflicting operations, robot assignments) and blocking dependencies. This allows the GA’s fitness evaluation to consider realistic constraints and avoid infeasible schedules.

B. Semi-enumerative fitness evaluation: instead of blindly evaluating all possible orderings, they use heuristics (like “Select Most Critical Pair”) to guide the choice of disjunctive edges in the graph that yield better feasible schedules. This reduces search complexity while maintaining solution quality.

3. Parameter tuning & experiments

They use a Taguchi design of experiments approach to select good GA parameters (population size, crossover probability, mutation probability)
They categorize test instances by scale: small (< 60 operations), medium (60–70), large (> 70).