Operational Optimization Formulation

This page documents the canonical FHOPS operational MILP formulation used by the Pyomo implementation in fhops.model.milp.operational.

The formulation is maintained from a shared source module so the SoftwareX manuscript and Sphinx docs stay synchronized.

FHOPS’ deterministic operational solver is formulated on a day-shift grid and maximizes weighted delivered production while penalizing leftovers, landing over-capacity slack, and machine movement costs. The equations below mirror the implemented Pyomo model in fhops.model.milp.operational.build_operational_model and the bundle normalization in fhops.model.milp.data.build_operational_bundle.

Problem statement. Given harvest blocks, machine roles, shift calendars, block windows, landing capacities, and harvest-system role prerequisites, choose machine-block assignments and per-shift production quantities to maximize weighted production subject to feasibility and sequencing constraints.

Sets and indices.

• \(m \in \mathcal{M}\): machines.

• \(b \in \mathcal{B}\): blocks.

• \(s=(d,\sigma) \in \mathcal{S}\): shift slots indexed by day \(d \in \mathcal{D}\) and shift label \(\sigma\).

• \(\mathcal{R}_b\): ordered machine roles required by the harvest system assigned to block \(b\).

• \(\mathcal{L}\): landings.

• \(\mathcal{P}^{\text{inv}} \subseteq \{(r,b): r \in \mathcal{R}_b\}\): role-block pairs with upstream prerequisites.

• \(\mathcal{P}^{\text{act}} \subseteq \mathcal{P}^{\text{inv}}\): role-block pairs that require positive head-start buffer activation.

• \(\mathcal{P}^{\text{load}} \subseteq \{(r,b): r \in \mathcal{R}_b\}\): loader role-block pairs.

Parameters.

• \(\bar{p}_{mb}\): production rate for machine \(m\) on block \(b\) (units per shift).

• \(W_b\): required total block production volume.

• \(A_{m,s} \in \{0,1\}\): machine availability for shift \(s\).

• \(\mathbf{1}^{\text{window}}_{b,d} \in \{0,1\}\): block window indicator (1 if day \(d\) is within block \(b\) window).

• \(\omega^{\text{prod}},\omega^{\text{mob}},\omega^{\text{trans}},\omega^{\text{land}}\): objective weights.

• \(\delta_{m,b',b}\): mobilization cost when machine \(m\) transitions from block \(b'\) to block \(b\).

• \(C_{\ell}\): daily assignment capacity for landing \(\ell\).

• \(\ell(b)\): landing associated with block \(b\).

• \(\mathcal{U}_{r,b}\): upstream roles that must feed role \(r\) on block \(b\).

• \(B_{r,b}\): required staged buffer volume before role \(r\) may activate on block \(b\).

• \(Q_{r,b}\): role production capacity upper bound per shift (used for activation linearization).

• \(q^{\text{batch}}_{r,b}\): loader batch size for loader role \(r\) on block \(b\).

• \(\mathcal{T}_b \subseteq \mathcal{R}_b\): terminal roles for block \(b\) (roles credited in block completion objective terms).

Decision variables.

• \(x_{m,b,s} \in \{0,1\}\): 1 if machine \(m\) is assigned to block \(b\) in shift \(s\).

• \(p_{m,b,s} \ge 0\): production by machine \(m\) on block \(b\) in shift \(s\).

• \(z_{r,b,s} \ge 0\): aggregated role-level production for role \(r\) on block \(b\) in shift \(s\).

• \(y_{m,b',b,s} \in \{0,1\}\): transition indicator for machine \(m\) from previous-shift block \(b'\) to current block \(b\) (defined for non-initial shifts).

• \(I^{\text{start}}_{r,b,s} \ge 0\): staged inventory available at start of shift \(s\) for role \(r\) on block \(b\).

• \(I_{r,b,s} \ge 0\): staged inventory at end of shift \(s\) for role \(r\) on block \(b\).

• \(g_{r,b,s} \in \{0,1\}\): role activation indicator for buffered downstream roles.

• \(n_{r,b,s} \in \mathbb{Z}_{\ge 0}\): loader batch count for loader role-block pair \((r,b)\).

• \(u_{r,b,s} \ge 0\): loader partial remainder volume.

• \(L_b \ge 0\): leftover unmet block volume slack.

• \(S_{\ell,d} \ge 0\): landing daily surplus slack.

Objective.

FHOPS maximizes weighted terminal production and subtracts penalty terms:

\[\begin{split}\begin{aligned} \max\; &\omega^{\text{prod}}\!\sum_{b\in\mathcal{B}}\sum_{r\in\mathcal{T}_b}\sum_{s\in\mathcal{S}} z_{r,b,s} - \omega^{\text{prod}}\!\sum_{b\in\mathcal{B}} L_b \\ &- \omega^{\text{land}}\!\sum_{\ell\in\mathcal{L}}\sum_{d\in\mathcal{D}} S_{\ell,d} \\ &- \omega^{\text{mob}}\!\sum_{m,b',b,s} \delta_{m,b',b}\, y_{m,b',b,s} - \omega^{\text{trans}}\!\sum_{m,b',b,s} y_{m,b',b,s}. \end{aligned}\end{split}\]

If a block has no terminal-role metadata, the implementation falls back to machine-level production sums for the production reward term.

Constraints.

Machine assignment feasibility:

\[\sum_{b\in\mathcal{B}} x_{m,b,s} \le A_{m,s} \qquad \forall m\in\mathcal{M},\; s\in\mathcal{S}.\]

Role compatibility (machines can only work roles allowed by the block’s assigned harvest system):

\[x_{m,b,s}=0 \quad \text{if role}(m)\notin\mathcal{R}_b.\]

Production upper bound per assignment:

\[p_{m,b,s} \le \bar{p}_{mb}\,x_{m,b,s} \qquad \forall m,b,s.\]

Block window enforcement:

\[x_{m,b,s}=0 \quad \text{if } \mathbf{1}^{\text{window}}_{b,d}=0 \text{ for } s=(d,\sigma).\]

Role-production aggregation:

\[z_{r,b,s} = \sum_{m\in\mathcal{M}(r)} p_{m,b,s} \qquad \forall (r,b), s.\]

Transition linking (for non-initial shifts only):

\[y_{m,b',b,s} \le x_{m,b',\operatorname{prev}(s)}, \qquad y_{m,b',b,s} \le x_{m,b,s},\]

\[y_{m,b',b,s} \ge x_{m,b',\operatorname{prev}(s)} + x_{m,b,s} - 1.\]

Role inventory start and balance for prerequisite-driven downstream roles:

\[\begin{split}I^{\text{start}}_{r,b,s}= \begin{cases} 0, & s \text{ is first shift}\\ I_{r,b,\operatorname{prev}(s)}, & \text{otherwise} \end{cases} \qquad \forall (r,b)\in\mathcal{P}^{\text{inv}}, s,\end{split}\]

\[I_{r,b,s}=I^{\text{start}}_{r,b,s}+\sum_{u\in\mathcal{U}_{r,b}} z_{u,b,s}-z_{r,b,s} \qquad \forall (r,b)\in\mathcal{P}^{\text{inv}}, s,\]

\[z_{r,b,s} \le I^{\text{start}}_{r,b,s} \qquad \forall (r,b)\in\mathcal{P}^{\text{inv}}, s.\]

Head-start activation for buffered downstream roles:

\[z_{r,b,s} \le Q_{r,b}\,g_{r,b,s} \qquad \forall (r,b)\in\mathcal{P}^{\text{act}}, s,\]

\[I_{r,b,\operatorname{prev}(s)} \ge B_{r,b}\,g_{r,b,s} \qquad \forall (r,b)\in\mathcal{P}^{\text{act}}, s,\]

\[\sum_{m\in\mathcal{M}(r)} x_{m,b,s} \le |\mathcal{M}(r)|\, g_{r,b,s}, \qquad g_{r,b,s} \le \sum_{m\in\mathcal{M}(r)} x_{m,b,s} \qquad \forall (r,b)\in\mathcal{P}^{\text{act}}, s.\]

Loader batching:

\[z_{r,b,s}=q^{\text{batch}}_{r,b}\,n_{r,b,s}+u_{r,b,s} \qquad \forall (r,b)\in\mathcal{P}^{\text{load}}, s,\]

\[0 \le u_{r,b,s} \le q^{\text{batch}}_{r,b} \qquad \forall (r,b)\in\mathcal{P}^{\text{load}}, s.\]

Block completion balance with leftover slack:

\[\sum_{r\in\mathcal{T}_b}\sum_{s\in\mathcal{S}} z_{r,b,s} + L_b = W_b \qquad \forall b\in\mathcal{B}.\]

Landing daily assignment capacity with surplus slack:

\[\sum_{b:\,\ell(b)=\ell}\sum_{m\in\mathcal{M}}\sum_{\sigma:(d,\sigma)\in\mathcal{S}} x_{m,b,(d,\sigma)} \le C_{\ell} + S_{\ell,d} \qquad \forall \ell\in\mathcal{L},\; d\in\mathcal{D}.\]

Domain restrictions:

\[x, y, g \in \{0,1\},\quad n \in \mathbb{Z}_{\ge 0},\quad p,z,I^{\text{start}},I,u,L,S \ge 0.\]

Implementation mapping (equation blocks to code).

Equation/constraint block	Pyomo component / helper	Data provenance
Machine capacity and availability	model.machine_capacity (machine_capacity_rule)	availability_day / availability_shift from build_operational_bundle(...)
Role compatibility	model.role_compatibility (role_compatibility_rule)	machine_roles, block_system, and systems from build_operational_bundle(...)
Production-assignment coupling	model.production_cap (prod_cap_rule)	production_rates from build_operational_bundle(...)
Block windows	model.block_windows (window_rule)	windows from Scenario.window_for(...)
Role aggregation	model.role_prod_balance (role_prod_balance_rule)	machine_roles + harvest-system role metadata
Transition linkage	model.transition_prev, model.transition_curr, model.transition_link	mobilisation_params / mobilisation_distances
Inventory dynamics and guards	model.inventory_start_eq, model.inventory_balance, model.inventory_guard	harvest-system prerequisites and role buffers
Head-start activation	model.activation_prod, model.head_start, model.role_active_upper, model.role_active_lower	role_headstart_shifts in harvest-system configs
Loader batching	model.loader_batch, model.loader_partial_cap	loader_batch_volume_m3 in harvest-system configs
Block balance with leftovers	model.block_balance (block_balance_rule) + model.leftover	work_required from blocks
Landing capacity with slack	model.landing_capacity (landing_capacity_rule) + model.landing_surplus	landing_capacity and landing_for_block
Objective assembly	model.objective plus prod_weight, landing_weight, mobilisation_weight, transition_weight terms	ObjectiveWeights from the scenario contract
Data/parameter normalization	build_operational_bundle(...)	fhops.model.milp.data

This formulation is the canonical mathematical reference for FHOPS operational MILP documentation and thesis-level reporting.