Lotfi Tadj

Critical minerals such as rare earth elements and battery metals combine geographically concentrated supply, long capacity lead times and growing strategic demand, yet the policy levers that address the resulting risk (primary capacity, recycling, stockpiling, supplier diversification, substitution research and environmental mitigation) are usually analysed separately. We formulate a continuous-time optimal control model with eight state variables and six policy controls in which a planner minimises discounted cost while imports and unmet demand clear the market, and we solve it by direct multiple shooting with CasADi and IPOPT using an illustrative parameter set of the magnitude of a mid-sized mineral market. The model was implemented through a case study where different scenarios along with sensitivity analysis were conducted. The reported results show that: First, diversification of suppliers, rather than reduction of import volumes, is the economically optimal response to import-concentration risk. Second, a recycled-content requirement is costly only above a certain recycled materiel share (11%) of demand. Finally, meaningful precautionary stockpiling against an uncertain disruption requires the simultaneous presence of a binding constraint on emergency supply-response speed and sufficient lead time before the earliest plausible disruption date; policy designed around either condition alone is unlikely to induce the gradual, precautionary behavior it is presumably intended to encourage.

Critical minerals such as rare earth elements and battery metals combine geographically concentrated supply, long capacity lead times and growing strategic demand, yet the policy levers that address the resulting risk (primary capacity, recycling, stockpiling, supplier diversification, substitution research and environmental mitigation) are usually analysed separately. We formulate a continuous-time optimal control model with eight state variables and six policy controls in which a planner minimises discounted cost while imports and unmet demand clear the market, and we solve it by direct multiple shooting with CasADi and IPOPT using an illustrative parameter set of the magnitude of a mid-sized mineral market. The model was implemented through a case study where different scenarios along with sensitivity analysis were conducted. The reported results show that: First, diversification of suppliers, rather than reduction of import volumes, is the economically optimal response to import-concentration risk. Second, a recycled-content requirement is costly only above a certain recycled materiel share (11%) of demand. Finally, meaningful precautionary stockpiling against an uncertain disruption requires the simultaneous presence of a binding constraint on emergency supply-response speed and sufficient lead time before the earliest plausible disruption date; policy designed around either condition alone is unlikely to induce the gradual, precautionary behavior it is presumably intended to encourage.

IoT-enabled model predictive control (MPC) lowers cost and spoilage in perishable-goods production but depends on sensor data that an attacker can falsify, which network intrusion detection cannot verify. We study sensor attacks on a published IoT-MPC plant model and a defense that needs no additional model: a CUSUM test on the innovations of the Kalman filter already in the loop, isolation of flagged sensors, and re-estimation from the attack onset. With accurate sensors and an exact model, a falsified energy meter raises the tracking cost by 71% and replay raises spoilage by 27%; the defense detects them in 89% and 100% of runs at a nominal 1% false-alarm rate and limits the cost increases to 9% and 3%. In closed loop, static detectors miss moderate and slow attacks that predictive detectors find. Detection weakens with sensor noise and model error; a fully informed attacker raises the cost by 93% undetected, but model errors expose it, and a redundant meter on an independent path limits it to about 15%. On the HAI testbed, an identified-model residual detector finds 32 of 34 attacks at 0.24 false alarms per day, against 27 for a windowed autoencoder and 21 for a static range check.

Under the generalized-cμ rule with quadratic delay costs, which is an accumulating-priority rule with relative slope κ=c2μ2/(c1μ1), a server facing two saturated classes alternates between runs of class 1 and runs of class 2. On an integer lattice we show that a class-1 run is a ruin problem for a random walk with moving boundary and a class-2 run is a first-excess problem for a positive-increment walk; the deficit of the first is the starting level of the second, and the overshoot of the second is the starting level of the first. For rational interarrival transforms (geometric, Erlang, hyperexponential) the Wiener–Hopf factorisation gives both laws in closed form from the roots of a Lundberg polynomial. Wald’s identities and the stationary law of the run-start chain then give the long-run switching frequency. The construction covers random service times and class-dependent setup times, which enter as a modified first step of each run. The frequencies agree with an exact lattice computation to three or four digits in all cases tested, for several slopes and arrival shares, and a simulation of the finite queue confirms that the saturated chain is exact when the load including setups is at least one. We also give closed-form undershoot weights, which complete the Gerber–Shiu factorisation of the moving-boundary ruin theory, and for more than two classes we prove a share law (with a correction for starved classes) while no closed form for the switching frequency is available.

The generalized-cμ (Gcμ) rule is asymptotically optimal for a single multiclass queue with convex delay costs, no setups and traffic intensity near one. We ask what it is worth when applied only at the bottleneck of a job shop with family setup times. Without setups and with equal service rates, its heavy-traffic cost ratio to FIFO has the closed form 1/[(∑_iπ_i/a_i)(∑_iπ_ia_i)], equal to 0.66 for our shop and confirmed by simulation. In a two-class model the rule gains more over FIFO as setups grow, because it switches families less often; we derive the switching frequency exactly for a saturated chain. Against a setup-aware, weights-aware heuristic, in a simulation of a five-machine job shop from the literature, the rule’s advantage disappears at setup-to-processing ratios of about 0.1. A hysteresis variant extends the window to about 0.25, and in some variants 0.5, with gains of 3–12%; it beats a tuned limited-batching rule by 4–8%, but weighted shop flow time is 1–2% worse. Most benchmark instances and the one real-plant instance we found have ratios of 1 to 5, where the index rules lose 4–10%. The rule suits bottlenecks with small setups.

Tourism is a major driver of the global economy, accounting for approximately 10% of global GDP; however, the literature highlights that unsustainable tourism growth can lead to economic volatility and overdependence, making sustainable approaches essential for ensuring long-term stability and resilience. This paper presents the development of a nonlinear model predictive control (MPC) framework that incorporates key state variables, such as the number of tourists and average spending per tourist, along with control variables including investment in facilities and services and human capital development, in addition to the formulation of the cost function and system dynamics. Numerical experiments demonstrate that the proposed MPC framework effectively manages the complex interactions inherent in tourism systems, achieving sustained growth in tourist arrivals, spending, and employment without instability, stabilizing experience quality despite congestion effects, maintaining a controlled decline in affordability within feasible bounds, and producing smooth, interpretable policy trajectories consistent with institutional constraints. To evaluate the robustness of the framework and identify the most critical governing parameters, a sensitivity analysis was conducted using the Morris screening method.