Newsvendor ModelCalculator

    Calculate the optimal order quantity and view the profit curve in real time

    Parameters

    Demand Distribution

    units
    0–500
    units
    1–200

    Cost Structure

    $
    1–200
    $
    1–200
    $
    0–200

    Optimal Order (Q*)

    100units

    Expected Profit

    $1,601.118

    Critical Ratio (CR)

    50.0%

    Cu=20 / Co=20

    Underage Cost (Cu)

    $20

    Overage Cost: $20

    Expected Profit Curve

    Demand Distribution & Optimal Order

    Newsvendor Model determines the optimal order quantity for a single-period inventory problem. Q* is found by applying the critical ratio CR = Cu/(Cu+Co) to the inverse normal CDF. Cu (underage cost) = Price − Cost, Co (overage cost) = Cost − Salvage.

    Newsvendor Model: Theory, Insights, and Practice

    The newsvendor (newsboy) model is a classic single-period inventory decision under uncertain demand. It balances lost sales (underage) against leftover inventory (overage) and yields an optimal order quantity that maximizes expected profit. [1][2]

    Core Assumptions

    One selling season, uncertain demand with a known distribution, unit selling price and unit cost, and a salvage or disposal value for leftovers. These assumptions make the expected-profit function concave and the solution tractable. [2][3]

    Critical Fractile Rule

    Let Cu = Price − Cost (underage cost) and Co = Cost − Salvage (overage cost). The optimal order Q* satisfies F(Q*) = Cu/(Cu+Co), where F is the demand CDF. Intuitively, you order until the probability of selling one more unit equals the ratio of marginal gain to total marginal cost. [1][2]

    Research Extensions

    Research generalizes the model to pricing and assortment decisions, risk-averse preferences, supply-chain contracts, and multi-product or multi-period settings. These extensions preserve the basic trade-off while changing objectives or constraints. [3][4]

    How to Use It

    Estimate demand distribution from historical data or forecasts, compute Cu and Co, then solve for Q* using the inverse CDF or simulation. Stress-test with sensitivity analysis and align with service-level targets. [2][5]

    Practical Examples

    • Daily fresh-food ordering for a bakery or grocer.
    • Seasonal fashion buys where leftovers are discounted.
    • Printed materials or promotional items with end-of-campaign salvage value.
    • Staffing or cloud capacity for a one-day event or launch.

    References

    • [1] Arrow, Harris, Marschak (1951) — Optimal inventory policy (Econometrica).
    • [2] Inventory Analytics (open book) — Newsvendor model and critical fractile.
    • [3] Khouja (1999); Qin, Wang, and Wei (2011) — Reviews of the newsvendor problem and extensions.
    • [4] Handbook of Newsvendor Problems (Springer, 2012) — Models, extensions, applications.
    • [5] Wikipedia — Newsvendor model overview.