ss Module

Overview

The ss module contains code for solving the \((s,S)\) optimization problem.

Note

The notation and references (equations, sections, examples, etc.) used below refer to Snyder and Shen, Fundamentals of Supply Chain Theory (FoSCT), 2nd edition (2019).

See Also

For an overview of single-echelon inventory optimization in Stockpyl, see the tutorial page for single-echelon inventory optimization.

API Reference

s_s_cost_discrete(reorder_point, order_up_to_level, holding_cost, stockout_cost, fixed_cost, use_poisson, demand_mean=None, demand_hi=None, demand_pmf=None)[source]

Calculate the exact cost of the given solution for an \((s,S)\) policy with given parameters under a discrete (Poisson or custom) demand distribution.

Uses method introduced in Zheng and Federgruen (1991).

Parameters

  • reorder_point (float) – Reorder point. [\(s\)]

  • order_up_to_level (float) – Order-up-to level. [\(S\)]

  • holding_cost (float) – Holding cost per item per period. [\(h\)]

  • stockout_cost (float) – Stockout cost per item per period. [\(p\)]

  • fixed_cost (float) – Fixed cost per order. [\(K\)]

  • use_poisson (bool) – Set to True to use Poisson distribution, False to use custom discrete distribution. If True, then mean must be provided; if False, then hi and demand_pdf must be provied.

  • demand_mean (float, optional) – Mean demand per period. Required if use_poisson is True, ignored otherwise. [\(\mu\)]

  • demand_hi (int, optional) – Upper limit of support of demand per period (lower limit is assumed to be 0). Required if use_poisson is False, ignored otherwise.

  • demand_pmf (list, optional) – List of pmf values for demand values 0, …, hi. Required if use_poisson is False, ignored otherwise.

Returns

cost – Expected cost per period. [\(g(s,S)\)]

Return type

float

Raises

  • ValueError – If reorder_point or order_up_to_level is not an integer.

  • ValueError – If holding_cost, stockout_cost, or fixed_cost <= 0.

  • ValueError – If demand_mean < 0, or if demand_hi is not a non-negative integer.

  • ValueError – If demand_pmf is not a list of length demand_hi + 1.

Equations Used (equation (5.7)):

\[g(s,S) = \frac{K + \sum_{d=0}^{S-s-1} m(d)g(S-d)}{M(S-s)},\]

where \(g(\cdot)\) is the newsvendor cost function and \(M(\cdot)\) and \(m(\cdot)\) are as described in equations (4.71)–(4.75).

References

Y.-S. Zheng and A. Federgruen, Finding Optimal \((s,S)\) Policies is About as Simple as Evaluating a Single Policy, Operations Research 39(4), 654-665 (1991).

Example (Example 4.7):

>>> s_s_cost_discrete(4, 10, 1, 4, 5, True, 6)
8.034111561471642
s_s_discrete_exact(holding_cost, stockout_cost, fixed_cost, use_poisson, demand_mean=None, demand_hi=None, demand_pmf=None)[source]

Determine optimal \(s\) and \(S\) for an \((s,S)\) policy under a discrete (Poisson or custom) demand distribution.

Uses method introduced in Zheng and Federgruen (1991).

Parameters

  • holding_cost (float) – Holding cost per item per period. [\(h\)]

  • stockout_cost (float) – Stockout cost per item per period. [\(p\)]

  • fixed_cost (float) – Fixed cost per order. [\(K\)]

  • use_poisson (bool) – Set to True to use Poisson distribution, False to use custom discrete distribution. If True, then demand_mean must be provided; if False, then demand_hi and demand_pdf must be provied.

  • demand_mean (float, optional) – Mean demand per period. Required if use_poisson is True, ignored otherwise. [\(\mu\)]

  • demand_hi (int, optional) – Upper limit of support of demand per period (lower limit is assumed to be 0). Required if use_poisson is False, ignored otherwise.

  • demand_pmf (list, optional) – List of pmf values for demand values 0, …, demand_hi. Required if use_poisson is False, ignored otherwise.

Returns

  • reorder_point (float) – Reorder point. [\(s\)]

  • order_up_to_level (float) – Order-up-to level. [\(S\)]

  • cost (float) – Expected cost per period. [\(g(s,S)\)]

Raises

  • ValueError – If holding_cost, stockout_cost, or fixed_cost <= 0.

  • ValueError – If demand_mean < 0, or if demand_hi is not a non-negative integer.

  • ValueError – If demand_pmf is not a list of length demand_hi + 1.

Algorithm Used: Exact algorithm for periodic-review \((s,S)\) policies with discrete demand distribution (Algorithm 4.2)

References

Y.-S. Zheng and A. Federgruen, Finding Optimal \((s,S)\) Policies is About as Simple as Evaluating a Single Policy, Operations Research 39(4), 654-665 (1991).

Example (Example 4.7):

>>> s_s_discrete_exact(1, 4, 5, True, 6)
(4.0, 10.0, 8.034111561471642)
s_s_power_approximation(holding_cost, stockout_cost, fixed_cost, demand_mean, demand_sd)[source]

Determine heuristic \(s\) and \(S\) for an \((s,S)\) policy under a normal demand distribution.

Uses the power approximation by Ehrhardt and Mosier (1984).

Parameters

  • holding_cost (float) – Holding cost per item per period. [\(h\)]

  • stockout_cost (float) – Stockout cost per item per period. [\(p\)]

  • fixed_cost (float) – Fixed cost per order. [\(K\)]

  • demand_mean (float) – Mean demand per period. [\(\mu\)]

  • demand_sd (float) – Standard deviation of demand per period. [\(\sigma\)]

Returns

  • reorder_point (float) – Reorder point. [\(s\)]

  • order_up_to_level (float) – Order-up-to level. [\(S\)]

Raises

  • ValueError – If holding_cost, stockout_cost, or fixed_cost <= 0.

  • ValueError – If demand_mean or demand_sd < 0.

References

R. Ehrhardt and C. Mosier, A Revision of the Power Approximation for Computing \((s,S)\) Policies, Management Science 30, 618-622 (1984).

Equations Used (equations (4.77)-(4.80)):

\[ \begin{align}\begin{aligned}Q = 1.30 \mu^{0.494} \left(\frac{K}{h}\right)^{0.506} \left(1 + \frac{\sigma_L^2}{\mu^2}\right)^{ 0.116}\\z = \sqrt{\frac{Q}{\sigma_L} \frac{h}{p}}\\s = 0.973\mu_L + \sigma_L\left(\frac{0.183}{z} + 1.063 - 2.192z\right) \label{eq:sS_power_approx3}\\S = s + Q\end{aligned}\end{align} \]

where \(\mu_L = \mu L\) and \(\sigma^2_L = \sigma^2L\) are the mean and standard deviation of the lead-time demand.

Example (Example 4.7):

>>> s_s_power_approximation(0.18, 0.70, 2.5, 50, 8)
(40.19461695647407, 74.29017010980579)