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An Application of Newsboy Problem in Supply Chain Optimisation of Online Fashion E-Commerce

Chandramouli K, Gopinath A, Nachiappan Sundaram, T Ravindra Babu, Chaithanya Bandi

Myntra Designs Pvt. Ltd.

August 2020

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Contents

Motivation Business metrics and problem statement Our solution Algorithm Experiments and conclusion

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Motivation

Online fashion e-commerce and Myntra Fast and efficient delivery of an order Clusters and placement of Forward Deployment Centers (FDCs) Periodic movement, say, weekly

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Business metrics

Definition

Fulfilment Index (FI): The ratio of number of SKUs delivered from the FDC of the cluster to number of SKUs ordered from the cluster in a week. FI(cluster) = #Delivered SKUs in the week #Ordered SKUs in the week Higher values of FI signify efficient storage of high demand products and better customer service for the customers of the FDC cluster

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Business metrics ctd.

Definition

Utilization Index (UI): The ratio of number of SKUs predicted for sale by the model for a given FDC in a week to number of SKUs delivered from the FDC in the previous week. UI(cluster) = #predicted SKUs for the week #SKUs sold in the previous week High UI for large number of weeks signify that the FDC is getting congested.

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Problem statement

FI of 100% at 1 UI is an ideal situation where every week all the items moved to FDC are sold completely FI and UI are dependent metrics Our objective is to come up with a solution that maximizes FI and minimizes UI simultaneously.

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Solution

Define approximations for the FI and UI of a cluster at the level of SKU and model the reduced problem as an instance of newsboy problem.

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Solution ctd.

• Notation. Given a SKU, denote by

D : Demand random variable of the given SKU f : Probability density function of D with support in (0, ∞) F : Cumulative distribution function of D q : Quantity of a given SKU to be transported to the FDC s : Last week sales of SKU r : Relative importance between FI and UI ❊ : Expectation with respect to F

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Solution ctd.

Similar to FI/UI of a cluster we look into FI/UI of the given SKU. FI(SKU) :=#Deliveries of the SKU in the week # Orders of the SKU in the week = min{q, D} D UI(SKU) := #Predicted quantity of the SKU for the week #Quantity of the SKU sold in the previous week = q s

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Solution ctd.

To achieve the objective consider the following allocation function of q, a(q). a(q) := ❊ [FI(SKU) − r UI(SKU)] = ❊ min{q, D} D − r q s

• = F(q) + q

∞

q

1 x f (x)dx − r q s

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Solution ctd.

Theorem

q∗ given by ∞

q∗

1 x f (x)dx = r s is the global maximizer of allocation function a.

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Solution ctd.

Heuristics: The demand of a SKU in the case of fashion e-commerce is a discrete distribution. We assume that each SKU follows a Poisson distribution. ∞

q

1 x f (x)dx ≈

• k≥q

1 k e−λ λk k! ≈ 1 λ

• k≥q

e−λ λk+1 (k + 1)! = 1 − FPoisson(q) λ where λ is the parameter of the Poisson distribution.

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SKU allocation algorithm

Algorithm

Input: Historic weekly sales of each SKU Output: Quantity to allocate for each SKU

1: procedure ALLOCATE: 2:

for each SKU do

3:

Estimate the parameter of FPoisson for the SKU utilizing historic weekly sales of the SKU.

4:

Evaluate q∗

SKU = F −1 Poisson(1 − r λSKU sSKU ) for the SKU.

5:

return q∗

SKU for each SKU

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Comments

Observe that F −1

Poisson

• 1 − r

λSKU sSKU

• is well defined if and only if

0 <

• 1 − r

λSKU sSKU

• ⇐

⇒ r λSKU < sSKU. Hence there is a recommendation q∗

SKU for the SKU only under the

condition that the last week sales, sSKU, is larger than the relative importance, r, times the mean of sales λSKU of the SKU, a intuitive thing to do, and the quantity to recommend is obtained as a quantile

• f the demand distribution.

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Experiments

We compare our algorithm with a ARMAX time series based earlier production model at Myntra. Our solution only utilizes a subset of the features utilized by the time series based model.

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Experiments ctd.

Table: Comparison of cluster wise FI and UI metrics for both models

Region Regression Model Our Algorithm FI UI FI UI FDC 1 25% 1.24 36% 0.85 FDC 2 47% 1.98 67% 0.97 FDC 3 18% 0.81 31% 0.83 FDC 4 54% 2.06 62% 0.98 FDC 5 21% 1.08 36% 0.88 FDC 6 41% 2.10 52% 0.98 FDC 7 21% 0.70 35% 0.83 FDC 8 21% 0.84 34% 0.83 FDC 9 32% 0.83 45% 0.90 FDC 10 15% 0.75 28% 0.75 FDC 11 48% 1.99 69% 0.98 FDC 12 33% 1.81 37% 0.86 Application of Newsboy Problem August 2020 16 / 18

Conclusion

The boost in performance, unlike in time series based model where the objective is mean square error, is a result of direct optimization of an objective that depends on FI and UI. To conclude we have modeled our business problem as an instance of newsboy problem and efficiently solved it.

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Thanks

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