Stack: Python, NumPy, Pandas, NetworkX, PySCIPOpt

Repository: inv-manage-scip

This project implements inventory models from Logic of Logistics, as a way for me to implement operations research and optimization to inventory management.

Constant-demand policies

With constant demand $D$, ordering cost $K$, holding cost $h$, instantaneous replenishment, and no shortages, the average cost of an order quantity $Q$ is

\[C(Q)=\frac{KD}{Q}+\frac{hQ}{2}.\]

The minimizer is the classical economic order quantity

\[Q^*=\sqrt{\frac{2KD}{h}},\]

with replenishment interval $T^=Q^/D$. ELSM_ConstDemand implements the formulas, simulation, grid search, and cost comparisons.

Finite horizons and practical policies

For a finite horizon $t$ with $m$ orders, equal cycle lengths minimize holding cost:

\[C(m)=Km+\frac{hDt^2}{2m}.\]

The implementation checks neighboring integer values of

\[m^*=t\sqrt{\frac{hD}{2K}},\]

and returns the selected order times, quantities, cycle lengths, and cost breakdown.

For standardized replenishment intervals, the project also evaluates policies of the form

\[T=T_B2^k,\]

where $T_B$ is a base period and $k$ is an integer. It compares adjacent powers of two around the unrestricted optimum for

\[f(T)=\frac{K}{T}+gT, \qquad g=\frac{hD}{2},\]

and selects the lower-cost policy. This restriction has a cost guarantee of approximately 6% over the unrestricted optimum.

Wagner-Whitin optimization

For variable demand $d_t$, setup cost $K_t$, purchase cost $c_t$, and holding cost $h_t$, the finite-horizon model uses

\[I_t=I_{t-1}+x_t-d_t,\]

where $x_t$ is the order quantity and $I_t$ is ending inventory. The PySCIPOpt implementation minimizes

\[\begin{aligned} \text{Minimize } & \sum_{t=1}^{T}\left(K_ty_t+c_tx_t+h_tI_t\right)\\ \text{subject to } & I_t=I_{t-1}+x_t-d_t,\\ & I_t\geq 0,\quad x_t\geq 0,\\ & x_t\leq \left(\sum_{s=t}^{T}d_s\right)y_t,\\ & y_t\in\{0,1\}. \end{aligned}\]

subject to nonnegative inventory, order quantities, and binary setup variables. WagnerWhitinSCIP exposes the resulting inventory and setup decisions for inspection.

Network formulation and T scaling

The zero-inventory-ordering property gives an acyclic network with $T+1$ nodes. An arc $(i,j)$ represents ordering in period $i$ to cover demand through period $j-1$, with setup and holding weight

\[\ell_{ij}=K_i+h_i\sum_{k=i}^{j-1}(k-i)d_k.\]

The shortest path from node $0$ to node $T$ identifies order periods. WagnerWhitinNetworkSolver reconstructs order quantities, setup decisions, and inventory levels from that path. Prefix sums make each of the $O(T^2)$ possible arcs constant-time to generate, so network construction is $O(T^2)$.

Validation and implementation

The test suite checks the constant-demand optima, simulation behavior, power-of-two selection, invalid inputs, and the zero-inventory-ordering property of the SCIP model. The notebook also compares SCIP and network objective values on the benchmark instances.

Implementation

The project is structured as a small reusable Python package rather than a collection of standalone notebook calculations:

Reproduce the environment with uv sync and run the suite with uv run pytest. Current models assume deterministic demand, instantaneous replenishment, and no shortages; extending the network model to time-varying purchase costs, lead times, and service constraints is future work.