An Application of Newsboy Problem in Supply Chain Optimisation of Online Fashion E-Commerce
Chandramouli Kamanchi, Gopinath Ashok Kumar, Nachiappan Sundaram, Ravindra Babu T, Chaithanya Bandi
AAn Application of Newsboy Problem in Supply Chain Optimisation of Online FashionE-Commerce
K Chandramouli ∗ [email protected] Science DivisionMyntra DesignsBangalore, Karnataka, India A Gopinath ∗ [email protected] Chain ManagementMyntra DesignsBangalore, Karnataka, India Nachiappan Sundaram ∗ [email protected] Science DivisionMyntra DesignsBangalore, Karnataka, India T Ravindra Babu [email protected] Science DivisionMyntra DesignsBangalore, Karnataka, India
Chaithanya Bandi [email protected] School of ManagementIllinois, USA
ABSTRACT
We describe a supply chain optimization model deployed in anonline fashion e-commerce company in India called Myntra. Ourmodel is simple, elegant and easy to put into service. The modelutilizes historic data and predicts the quantity of Stock KeepingUnits (SKUs) to hold so that the metrics “Fulfilment Index" and “Uti-lization Index" are optimized. We present the mathematics centralto our model as well as compare the performance of our model withbaseline regression based solutions.
KEYWORDS
Newsboy Problem, Fulfilment Index (FI), Utilization Index (UI),Stock Keeping Unit (SKU)
The single-item single-period newsboy problem has been exten-sively studied in the literature. It is still an active research problemwith several extensions to real-world scenarios. In its basic formu-lation the problem aims to obtain a replenishment strategy for aperishable item with a stochastic demand such that the expectedprofit is optimized. Many extensions and modifications that intro-duce additional complexity to this basic formulation are proposedin the literature.Some of the research effort on extensions of newsboy problemare focused on businesses that wish to maximize expected profitwith diverse range of assumptions on demand distribution andinventory models. For example [9] assumes additive as well as mul-tiplicative price-demand relationship in the newsboy problem tomaximize the expected profit. [3] presents a multi-period model forinventory control with price-dependent stochastic demand. In [10], ∗ Authors contributed equally to this work.Permission to make digital or hard copies of all or part of this work for personal orclassroom use is granted without fee provided that copies are not made or distributedfor profit or commercial advantage and that copies bear this notice and the full citationon the first page. Copyrights for components of this work owned by others than ACMmust be honored. Abstracting with credit is permitted. To copy otherwise, or republish,to post on servers or to redistribute to lists, requires prior specific permission and/or afee. Request permissions from [email protected].
San Diego ’20, 23 August 2020, San Diego, California - USA © 2020 Association for Computing Machinery.ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00https://doi.org/10.1145/nnnnnnn.nnnnnnn a variation of newsboy problem is considered where the objectiveis to maximize the minimum profit given only the mean and vari-ance of the demand distribution. Moreover it has been shown thatthe distribution that achieves this objective is close to a Poissondistribution.Other modifications to classical newsboy problem include multi-item considerations [5, 7] with capacity constraints for the items,multi-location models [1, 2] with dependence on season. Recentlydeep-learning based methods have also been employed [8, 11] tosolve the newsboy problem by mitigating the dependency on de-mand distribution.In the area of e-commerce [4, 6] invoke newsboy problem anddescribe models similar to our model. However our primary focus ison fashion e-commerce. Here, as a consequence of dynamic natureof fashion and limited total quantity of a typical style/item, we facechallenges like limited history of sales, frequent change of productson the platform etc. We approach these challenges with a simplechoice for demand distribution based on heuristics.In our work in fashion e-commerce we model the problem ofsystematic placement of inventory as an instance of newsboy prob-lem and obtain an efficient algorithm that optimizes our businessmetrics “Fulfilment Index" and “Utilization Index". Our key contri-butions are summarised as follows.
Our contributions: • We define and mathematically formulate our business objec-tives. • We model the problem of optimization of our objectives asan instance of newsboy problem and solve it analytically toobtain globally optimal solution. • Based on heuristic analysis of this optimal solution, we de-velop a SKU allocation algorithm. • We establish better performance of our algorithm comparedto regression based models to solve the SKU allocation prob-lem on real world data.
Most business organizations admit that the supply chain is thebackbone of their day to day operations. So it is critical for thesupply chain to be efficient and optimized. a r X i v : . [ s t a t . A P ] J u l an Diego ’20, 23 August 2020, San Diego, California - USA Chandramouli et al. At Myntra our supply chain outbound works as follows. Thecountry is divided into clusters based on the zipcode. In each clustera local warehouse/ Forward Deployment Center (FDC) hosts a selectgroup of products. An order of a product from the customer of acluster first reaches the FDC of the cluster and gets delivered if it isavailable at the FDC. If the product is unavailable at the FDC theorder is forwarded to the central warehouse/Big Box (BB).For better customer satisfaction through faster fulfillment oforders and efficient delivery it is essential that an ordered productbe available at the FDC of the cluster most of the time. One wayto accomplish this task is to hold a diverse range and quantity ofproducts at the FDC and replenish them every week. But due tospace and financial constraints it is infeasible to host a large numberof products at each FDC of the country. Clearly there is a trade-offbetween these two requirements. The following definitions quantifythese requirements.Definition 1.
Fulfilment Index (FI):It is defined as the ratio between number of SKUs delivered from theFDC of the cluster and number of SKUs ordered from the cluster in aweek. FI ( cluster ) = Delivered SKUs in the week
Ordered SKUs in the week
Definition 2.
Utilization Index (UI):It is defined as the ratio between number of SKUs predicted for sale bythe model for a given FDC in a week and number of SKUs deliveredfrom the given FDC in the previous week.
U I ( cluster ) = predicted SKUs for the week SKUs sold in the previous week
Now from the definition of FI it is clear that the metric measuresthe efficiency of our storage. Higher values of FI signify efficientstorage of high demand products and better customer service forthe customers of the FDC cluster. On the other hand UI measuresthe space efficiency or congestion at the FDC. Moreover the abilityto control this metric, accomplished by incorporating a tunableparameter r in the solution, accounts for fluctuations in manpowerallotment for the SKU movement. High UI for large number ofweeks signify that the FDC is getting congested. In summary a FIof 100% at 1 UI is an ideal situation where every week all the itemsmoved to FDC are sold completely.Also observe that FI and UI are dependent metrics. For examplehigher values of UI account for large and diverse range of productsat FDC thereby ensuring higher values of FI but with significantstorage cost as well as eviction cost of unsold items. Our objectiveis to come up with a solution that maximizes FI and minimizes UIsimultaneously. We achieve this objective by modeling our problemas an instance of newsboy problem. First we define approximations for the FI of a cluster and UI of acluster at the level of SKU and model the reduced problem as aninstance of newsboy problem. To begin we setup some notation. Given a SKU, denote by D : Demand random variable of the given SKU f : Probability density function of D with support in ( , ∞) F : Cumulative distribution function of Dq : Quantity of a given SKU to be transported to the FDC s : Last week sales of SKU r : Relative importance between FI and UI E : Expectation with respect to F Similar to FI/UI of a cluster we look into FI/UI of the given SKU. FI ( SKU ) (cid:66) = min { q , D } DU I ( SKU ) (cid:66) = qs To achieve the objective consider the following allocation functionof q , a ( q ) . a ( q ) (cid:66) E [ FI ( SKU ) − r U I ( SKU )] = E (cid:20) min { q , D } D − r qs (cid:21) = E (cid:20) min { q , D } D { D ≤ q } + min { q , D } D { D > q } − r qs (cid:21) = F ( q ) + q E (cid:20) D { D > q } (cid:21) − r qs = F ( q ) + q ∫ ∞ q x f ( x ) dx − r qs Theorem 1. q ∗ given by ∫ ∞ q ∗ x f ( x ) dx = rs is the global maxi-mizer of allocation function a . Proof. Differentiating a ( q ) we get a ′ ( q ) = f ( q ) + ∫ ∞ q x f ( x ) dx − q f ( q ) q − rs = ∫ ∞ q x f ( x ) dx − rs Moreover on twice differentiating a ( q ) we get a ′′ ( q ) = − f ( q ) q ≤ . So a ′ ( q ∗ ) = a ′′ ( q ∗) ≤ . Hence a ( q ) is convex and q ∗ is aglobal maximizer of allocation function a . Now observe that for q < q ∫ ∞ q x f ( x ) dx > ∫ ∞ q x f ( x ) dx ⇐⇒ ∫ q q x f ( x ) dx > . pplication of Newsboy Problem San Diego ’20, 23 August 2020, San Diego, California - USA Therefore a ′ ( q ) is strictly decreasing function of q . Hence a ′ ( q ) = q ∗ is the unique global maximizer of theallocation function. □ Motivated by [10], we employ the following heuristics as thedemand of a SKU in the case of fashion e-commerce is a discrete dis-tribution. We assume that each SKU follows a Poisson distribution.Therefore ∫ ∞ q x f ( x ) dx ≈ (cid:213) k ≥ q k e − λ λ k k ! ≈ λ (cid:213) k ≥ q e − λ λ k + ( k + ) ! = − F Poisson ( q ) λ where λ is the parameter of the Poisson distribution. With thisheuristics we obtain the solution q ∗ = F − ( − λrs ) Algorithm 1
SKU allocation algorithm
Input:
Historic weekly sales of each SKU
Output:
Quantity to allocate for each SKU procedure ALLOCATE: for each SKU do Estimate the parameter of F Poisson for the SKU utilizinghistoric weekly sales of the SKU. Evaluate q ∗ SKU = F − ( − r (cid:98) λ SKU s SKU ) for the SKU. return q ∗ SKU for each SKUOur algorithm obtains a cluster level demand distribution F Poisson for each SKU by Maximum Likelihood Estimation (MLE) of theparameter of F Poisson for the SKU. It is easy to see that the estimator (cid:98) λ SKU given by MLE is the mean of the historical weekly sale samplesof the SKU at the cluster. Observe that F − (cid:18) − r (cid:98) λ SKU s SKU (cid:19) is welldefined if and only if 0 < (cid:18) − r (cid:98) λ SKU s SKU (cid:19) ⇐⇒ r (cid:98) λ SKU < s SKU .Hence there is a recommendation q ∗ SKU for the SKU only under thecondition that the last week sales, s SKU , is larger than the relativeimportance, r , times the mean of sales (cid:98) λ SKU of the SKU, a intuitivething to do, and the quantity to recommend is obtained as a quantileof the demand distribution.
To evaluate and compare our algorithm we have extensively utilizedthe real world data available at Myntra.com. First we have chosen atarget week for evaluation. Then for our algorithm 1 based on news-boy problem we prepared the dataset as described below. Given aregion/FDC/cluster (see Table 1) and a SKU (see Figure 1) on our platform we obtained the weekly sales of the SKU in the region forthe 9 weeks that precede the target week. As discussed earlier inSection 4 we obtain the parameter of the Poisson demand distribu-tion of the SKU in the cluster by maximum likelihood estimation. Inour case the parameter estimate is the sample mean of the 9 weeklydemand samples of the SKU. We then utilize the sample mean andweekly demand sample for the week that immediately precedes thetarget week, denoted (cid:98) λ SKU and s SKU respectively, and evaluatestep 4 of the algorithm with the choice r = . r allows us to trade FI for UI and vice-versa. It enables usto handle fluctuations in the implementation of the solution. Thedependency of FI and UI on r is summarised in Table 1 for a set of12 FDCs.We compared our algorithm with a regression based model thatis an earlier production model and predicts the demand of a SKUat a cluster/FDC given features of the SKU. The regression modelis an ARMAX time series model that utilizes cluster level as wellas platform level features constructed from 14 week history ofthe SKU that precedes the target week and predicts the weeklydemand of the target week. Table 3 describes dominant features ofthe regression model.We summarise our observations from the comparison of themethods here. As shown in Table 2, in almost all clusters chosen wesee that both FI and UI see significant improvement over baselinemethod. We observe that the weekly sales data the precede thetarget week is also utilized (see Table 3) in the benchmark algorithm.However the boost in performance, unlike in benchmark where theobjective is mean square error, is a result of direct optimization ofan objective that depends on FI and UI. Figure 1: Typical SKUs in our dataset
We have modeled our business problem as an instance of newsboyproblem and solved it. Based on heuristic analysis of this solutionwe proposed a SKU allocation algorithm. We have compared andshown that our algorithm has superior business metrics comparedto regression based benchmark.
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Table 1: Cluster wise FI and UI metrics with different r -values r = . r = . r = . r = . r = . Region FI UI FI UI FI UI FI UI FI UI
FDC_1 44% 3.07 44% 2.92 36% 0.85 35% 0.67 32% 0.57FDC_2 70% 2.86 69% 2.64 67% 0.97 63% 0.80 58% 0.64FDC_3 38% 3.33 38% 3.22 31% 0.83 29% 0.67 28% 0.58FDC_4 68% 2.53 67% 2.30 62% 0.98 58% 0.82 53% 0.64FDC_5 44% 3.37 44% 3.18 36% 0.88 34% 0.70 31% 0.58FDC_6 61% 3.28 60% 3.04 52% 0.98 50% 0.82 46% 0.66FDC_7 42% 3.11 42% 2.97 35% 0.83 33% 0.67 31% 0.57FDC_8 40% 3.10 40% 2.97 34% 0.83 32% 0.66 31% 0.57FDC_9 54% 2.59 53% 2.38 45% 0.90 42% 0.71 38% 0.58FDC_10 34% 3.14 35% 3.03 28% 0.75 27% 0.60 25% 0.52FDC_11 71% 2.86 70% 2.63 69% 0.98 65% 0.81 61% 0.65FDC_12 46% 2.84 46% 2.67 37% 0.86 35% 0.69 32% 0.58
Table 2: Comparison of cluster wise FI and UI metrics for both modelsRegion Regression Model Our AlgorithmFI 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
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% 0.86Table 3: Dominant features of regression based modelFeatures Description atc_count Add to cart count of the SKUmrp_range Article retail price classificationinventory_refill_days Average days to refill the inventory from stock outt_1:12_quantity Weekly sales in the 12 weeks that precede the target weekthree_w_max_units Last three weeks maximum inventory in unitsthree_w_min_units Last three weeks minimum inventory in unitssku_selling_rt Ratio between cluster SKU sales and cluster style salessku_style_rt Ratio between overall SKU sales and overall style salescatalog_live_week_no_diff Age of the productwl_count Wish list count [6] Shouyu Ma, Zied Jemai, Evren Sahin, and Yves Dallery. 2017. The news-vendorproblem with drop-shipping and resalable returns.
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