Supply Uncertainty

Stockpyl contains code to solve the following types of single-echelon inventory optimization problems in the supply_uncertainty module:

  • Economic order quantity (EOQ)-based models

    • with disruptions

    • with additive and multiplicative yield uncertainty

  • Newsvendor-based models

    • with disruptions

    • with additive yield uncertainty

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

The EOQ Problem with Disruptions

The EOQ problem with disruptions (EOQD) assumes that the supplier can be disrupted randomly, following a 2-state Markov process. The eoq_with_disruptions() function solves the EOQD either exactly (using models by Parlar and Berkin (1991) and Berk and Arreola-Risa (1994) and optimizing numerically) or heuristically (using an approximation by Snyder (2014)).

>>> from stockpyl.supply_uncertainty import eoq_with_disruptions
>>> K = 8
>>> h = 0.225
>>> p = 5
>>> demand_rate = 1300
>>> disruption_rate = 1.5
>>> recovery_rate = 14
>>> Q, cost = eoq_with_disruptions(K, h, p, demand_rate, disruption_rate, recovery_rate)
>>> Q
772.8110739983106
>>> cost
173.95000257319708
>>> Q, cost = eoq_with_disruptions(K, h, p, demand_rate, disruption_rate, recovery_rate, approximate=True)
>>> Q
773.1432417118889
>>> cost
173.957229385175

The EOQ Problem with Yield Uncertainty

In EOQ problems with yield uncertainty, the supplier can always provide inventory, but the quantity delivered differs randomly from the quantity ordered. Stockpyl supports both additive yield uncertainty:

>>> from stockpyl.supply_uncertainty import eoq_with_additive_yield_uncertainty
>>> K = 18500
>>> h = 0.06
>>> demand_rate = 75000
>>> yield_mean = -15000
>>> yield_sd = 9000
>>> Q, cost = eoq_with_additive_yield_uncertainty(K, h, demand_rate, yield_mean, yield_sd)
>>> Q
230246.37046881882
>>> cost
12914.78222812913

and multiplicative yield uncertainty:

>>> from stockpyl.supply_uncertainty import eoq_with_multiplicative_yield_uncertainty
>>> from math import sqrt
>>> yield_mean = 0.8333
>>> yield_sd = sqrt(0.0198)
>>> Q, cost = eoq_with_multiplicative_yield_uncertainty(K, h, demand_rate, yield_mean, yield_sd)
>>> Q
254477.46130342316
>>> cost
13086.16169098594

The Newsvendor Problem with Disruptions

In the newsvendor (or base-stock) problem with disruptions, the demand is deterministic but the supply can be disrupted, following a 2-state Markov process. The newsvendor_with_disruptions() function returns both the optimal base-stock level and the optimal expected cost.

>>> from stockpyl.supply_uncertainty import newsvendor_with_disruptions
>>> h = 0.25
>>> p = 3
>>> d = 2000
>>> alpha = 0.04
>>> beta = 0.25
>>> S, cost = newsvendor_with_disruptions(h, p, d, alpha, beta)
>>> S
8000
>>> cost
2737.0689302470355

The Newsvendor Problem with Yield Uncertainty

In newsvendor problems with yield uncertainty, the supplier can always provide inventory, but the quantity delivered differs randomly from the quantity ordered. Stockpyl currently supports only additive yield uncertainty:

>>> from stockpyl.supply_uncertainty import newsvendor_with_additive_yield_uncertainty
>>> h = 15
>>> p = 75
>>> d = 1500000
>>> yield_mean = 250000
>>> yield_sd = 200000
>>> S, cost = newsvendor_with_additive_yield_uncertainty(h, p, d, yield_mean, yield_sd)
>>> S
1443484.3132203403
>>> cost
4497316.9310527835

If the yield mean and standard deviation are provided, as in the code above, the yield is assumed to have a normal distribution. Alternately, you can provide an arbitrary distribution as a scipy.stats.rv_continuous or scipy.stats.rv_discrete object:

>>> from scipy.stats import uniform
>>> newsvendor_with_additive_yield_uncertainty(h, p, d, yield_distribution=uniform(-500000, 1000000))
(1833333.3333333335, 6249999.997499999)