Performance benchmarking and sales forecasting using firm-level data for US retailers Vishal Gaur for presentation at “Rocket Science” en Retail 2008 Industrial Engineering, University of Chile June 19, 2008 (C) Vishal Gaur, Cornell University, 2008 1 Outline 1. Applications of public financial data – – Benchmarking of inventory turnover Forecasting sales using inventory and gross margin data 2. A distribution management problem at a supermarket chain (Albert Heijn, BV) 3. Experiment design in a retail chain (toy retailer in US) (C) Vishal Gaur, Cornell University, 2008 2 Importance of Inventory Management in Retailing • • • More than $300 billion of investment in inventory in the U.S. retailing industry in 2008 (~$500 billion including motor vehicles and spare parts). Inventory represents 36% of total assets and 53% of current assets of retailing firms. Inventory turnover – Routinely used for productivity comparisons by retailers, manufacturers, consultants and analysts. • Benefits of high inventory turnover – – – – Lower working capital requirement Lower inventory holding and obsolescence costs Greater ability to respond to market dynamics Fewer stock-outs (C) Vishal Gaur, Cornell University, 2008 3 Variation in Inventory Turnover • Within-firms variation – Range of inventory turnover of commonly known firms in 19852000: Best Buy Co. Inc. 2.8 – 8.5 Circuit City Stores, Inc. 4.0 – 5.6 The Gap, Inc. 3.6 – 6.3 Radio Shack Corp. 1.1 – 3.1 Wal-Mart Stores, Inc. 4.9 – 7.2 • Across-firms variation – Range of inventory turnover of supermarket chains during the year 2000: 4.7 to 19.5. (C) Vishal Gaur, Cornell University, 2008 4 Time-Series Plot of Annual Inventory Turnover of Four Consumer Electronics Retailers for 1987-2000 Annual Inventory Turnover 10 9 8 7 6 5 4 3 2 1 0 1986 1988 1990 1992 1994 1996 1998 2000 Time (years) Best Buy Co. Circuit City Stores, Inc. Radio Shack Corp CompUSA, Inc. (C) Vishal Gaur, Cornell University, 2008 5 Research Questions • Why does inventory turnover vary so much • • • • Correlation with gross margin, capital intensity and deviation of sales from forecast. Characterize the “earns versus turns” tradeoff. How to benchmark inventory productivity. Understand how firms make aggregatelevel inventory decisions. (C) Vishal Gaur, Cornell University, 2008 6 Description of Data • Data: – Obtained from S&P’s Compustat database – 311 firms across 10 retailing segments for years 1985-2000. – 3407 observations across firms and years; 11 annual observations per firm. • Preparation: – At least five consecutive years of observation for each firm • Causes of missing data: new entry, mergers, acquisitions, liquidations. – Missing data other than at the beginning or the end of the period • Bankruptcy and reorganization – Inventory valuation method • FIFO, LIFO, Average cost method, Retail method. (C) Vishal Gaur, Cornell University, 2008 7 Variables 1. Inventory Turnover IT = Cost of Goods Sold 2. Gross Margin GM = 3. Capital Intensity Avg Gross Fixed Assets CI = Avg Inventory + Avg Gross Fixed Assets 4. Sales Surprise SS = Average Inventory (C) Vishal Gaur, Cornell University, 2008 Sales - Cost of Goods Sold Sales Sales Realized Sales Forecast 8 Model Specification logITsit = Fi + c t + b1s logGMsit + b2s logCIsit + b3s logSS sit + ε sit where • s denotes segment index, i the firm index, and t the year index. • Fi : firm-specific fixed effects. Control for differences in the intercept between firms, such as between their managerial efficiency, location, accounting policies, marketing, etc. • ct : year-specific fixed effects. Control for differences in economic conditions over time. • b1s, b2s, b3s: segment-wise coefficients. b1s ≠ 0 for hypothesis 1, b2s > 0 for hypothesis 2, b3s > 0 for hypothesis 3. • εsit denotes the error term. (C) Vishal Gaur, Cornell University, 2008 9 Summary of Data Retail Industry Segment Apparel And Accessory Stores Catalog, Mail-Order Houses Department Stores Drug & Proprietary Stores Food Stores Hobby, Toy, And Game Shops Home Furniture & Equip Stores Jewelry Stores Radio,TV, Cons Electr Stores Variety Stores Aggregate statistics # annual Average Sales Inventory # of firms observations Turnover ($ million) 72 786 979.1 4.57 2.13 45 441 439.9 8.60 9.11 23 309 6058.6 3.87 1.45 23 256 2309.5 5.26 2.90 57 650 4573.6 10.78 4.58 10 98 1455.5 2.99 1.08 13 125 391.2 5.44 10.43 15 156 475.2 1.68 0.58 17 200 1585.0 4.10 1.54 36 386 6548.7 4.45 2.92 311 3407 2791.4 6.08 5.41 (C) Vishal Gaur, Cornell University, 2008 Gross Margin 0.37 0.08 0.39 0.17 0.34 0.08 0.28 0.07 0.26 0.06 0.35 0.07 0.40 0.07 0.42 0.13 0.31 0.11 0.29 0.09 0.33 0.11 Capital Intensity 0.59 0.14 0.50 0.18 0.63 0.10 0.48 0.12 0.75 0.08 0.46 0.14 0.55 0.16 0.36 0.11 0.44 0.09 0.51 0.15 0.57 0.17 10 Overall Fit Statistics • Model explains 66.7% of the within-firm variation and 97.2% of the total variation (within and across firms) in log(IT). • Intercept of the regression line varies across firms and across years. • The coefficients of gross margin, capital intensity and sales surprise are statistically significant. They differ by segment. (C) Vishal Gaur, Cornell University, 2008 11 Coefficients’ Estimates Segment-wise coefficients Apparel And Accessory Stores Catalog, Mail-Order Houses Department Stores Drug & Proprietary Stores Food Stores Hobby, Toy, And Game Shops Home Furniture & Equip Stores Jewelry Stores Radio,TV,Cons Electr Stores Variety Stores Pooled coefficients Gross Margin Capital Intensity Sales Surprise -0.153 -0.226 -0.310 -0.186 -0.351 -0.571 -0.017* -0.438 -0.500 -0.313 -0.285 0.977 -0.039* 0.861 0.361 1.085 -0.015* 0.562** 0.038* 0.268 0.106 0.252 0.053 0.225 0.189 0.143 0.179 0.215 0.174 0.279 0.140 0.176 0.143 • Coefficients marked * are not significant, coefficients marked ** have p<0.02, all other coefficients have p<0.001. (C) Vishal Gaur, Cornell University, 2008 12 Application to Benchmarking • Tradeoff curve – model specifies the tradeoff between IT, GM and CI, and corrects for the effect of sales surprise. ( GM ) 0.283 ( CI ) −0.252 ( SS ) −0.143 IT = Firm-specific constant × Time-specific constant • Adjusted Inventory Turnover (AIT) – equals the residual from the model and shows the distance of a firm from its tradeoff curve (benchmark). Residual = log AITsit = log ITsit + 0.285log GM sit − 0.252 log CI sit − 0.143log SS sit (C) Vishal Gaur, Cornell University, 2008 13 Example 1: Comparison of Four Consumer Electronics Retailers 10 9 Inventory Turns 8 7 6 5 4 3 2 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Gross Margin (%) Best Buy Co. Inc. (C) Vishal Gaur, Cornell University, 2008 Circuit City Stores Radio Shack CompUSA 14 Example 1: Values of Adjusted Inventory Turns for different gross margins for the four consumer electronics retailers 14 Best Buy Co. Inc. Circuit City Stores CompUSA Radio Shack 12 Inventory Turns 10 8 6 4 2 0 0 0.1 0.2 0.3 Gross Margin 0.4 0.5 0.6 Note: Figures are drawn using the average values of CI and setting SS = 1. (C) Vishal Gaur, Cornell University, 2008 15 Example 2: Comparison across years within a firm Ruddick Corp. IT is decreasing with time, but AIT is increasing with time. Gross Margin and Capital Intensity are increasing with time. 9.5 0.9 9 y = -0.0194x + 8.3937 R2 = 0.0592 0.8 0.7 8.5 0.6 8 0.5 7.5 0.4 0.3 7 0.2 0 1986 y = 0.0704x + 6.2215 R2 = 0.5705 6.5 0.1 6 1988 1990 1992 Gross Margin 1994 1996 1998 Capital Intensity 2000 1987 1989 1991 1995 1997 1999 Time (in years) Inventory Turnover (C) Vishal Gaur, Cornell University, 2008 1993 Adjusted Inventory Turnover 16 (C) Vishal Gaur, Cornell University, 2008 17 Average stock returns of public retailers & S&P 500 firms during 1978-2005 257 Retail firms 15.3% average return, 25.8% among survivors Number of Firms Firms in the S&P 500 index in 1978 14.1% average return, 21.7% among survivors 100 180 90 160 80 140 70 120 60 100 50 80 40 60 30 40 20 20 10 0 0 BNRT -30 -20 -10 35% retailers went bankrupt 0 10 20 30 40 50 60 70 80 BNRT -20 -10 0 10 20 30 40 50 60 70 80 90 100 Average Annual Stock Return (%) $1000 invested in Wal-mart in 78 worth $706,000 in Dec. 2005 $1000 in Circuit City in 78 worth $828,000 in Dec. 2005 $1000 in Gap in 78 worth $351,500 in Dec. 2005 (C) Vishal Gaur, Cornell University, 2008 18 Construction of portfolios of firms 1 Jan 1, 2002 1 Dec 31, 2002 2 Jun 30, 2003 3 Jun 30, 2004 Obtain annual financial statements for fiscal year ending between Jan 1 and Dec 31, 2002. For example, fiscal year 2001 may end on Jan 31, 2002. In each retail segment, rank firms by chosen metric and divide into 5 or 10 equal portfolios. 2 Invest $1 in each portfolio on June 30, 2003 equally divided among the firms in the portfolio. 3 Sell holdings on June 30, 2004 and form new portfolios using data available up to Dec 31, 2003. (C) Vishal Gaur, Cornell University, 2008 19 Variables used to form portfolios IT: inventory turnover [ = closing inventory/cost of goods sold] ΔIT: IT for current year – IT for previous year GM: gross margin [ = (sales – cost of goods sold)/sales] ΔGM: GM for current year – GM for previous year AIT: adjusted inventory turnover ΔAIT: AIT for current year – AIT for previous year Ranking is done within each retail segment for each year (C) Vishal Gaur, Cornell University, 2008 20 Comparison of total returns on portfolios formed in different ways Variables used to form portfolios Portfolio rankings IT ΔIT GM ΔGM AIT ΔAIT Low 0.81 0.70 1.21 1.02 0.90 0.77 2 0.84 1.05 0.88 1.14 0.86 0.99 3 1.15 1.04 1.14 1.26 1.03 1.11 4 1.10 1.28 1.04 0.99 1.23 1.18 High 1.54 1.36 1.11 0.98 1.41 1.38 • Table reports average monthly stock return (%) for 5 portfolios each formed on IT, ΔIT, GM, ΔGM, AIT and ΔAIT. • Number of observations in a portfolio range between 5253 and 6329. • Standard deviation of monthly stock returns = 20% (approx.) in each portfolio. (C) Vishal Gaur, Cornell University, 2008 21 Results of performance-attribution regressions Summary • Using AIT – Estimate of the intercept, α, increases as portfolio rank increases. – Low ranked portfolios have significantly negative intercept, showing belowaverage returns. – Five out of ten portfolios have statistically significant intercept (p=0.10) – Abnormal return on a zero investment portfolio (buy top 30% and short-sell bottom 30% firms at the beginning of each year) = 0.9 bp/month = 11.25% per year. (p<0.01) • Using IT – Estimate of the intercept, α, has a less evident trend as portfolio rank increases. – Two out of ten portfolios have statistically significant intercept (p=0.10) – Abnormal return on a zero investment portfolio is not statistically significant. • All regressions yield significant F-statistics (p<0.01) with R2 ranging between 36.5% and 61.2%. (C) Vishal Gaur, Cornell University, 2008 22 Incorporating Price and Inventory Endogeneity in Firm-Level Sales Forecasting Saravanan Kesavan Vishal Gaur Ananth Raman 23 (C) Vishal Gaur, Cornell 23 Sales, Inventory, and Margin for a Retailer are Interrelated • Customer purchases depend on product availability, variety, and prices. Sales • Sales are input into retailer’s decisions on inventory and margin. • Inventory and Margin influence each other. Inventory Margin The variables Sales, Inventory, and Margin are measured for a given firm for a given year. Margin (or relative markup) is used as a measure for price. (C) Vishal Gaur, Cornell University, 2008 24 24 Implications of Interrelationships among Sales, Inventory, and Margin Data are the joint outcome of all six interrelationships!! z z Simple correlation among sales, inventory, and margin insufficient to determine causation Change to one variable affects the other two variables ¾ z Not possible to evaluate observed practices easily Time-series methods for forecasting sales ignoring inventory and margin are inadequate ¾ Past sales managed using inventory and margin (C) Vishal Gaur, Cornell University, 2008 25 25 Measurement of Endogenous Variables Let i = firm index, t = year index, q = quarterly index [Cost of goods sold]it Average sales per store, CSit = [Number of stores]it 4 Average inventory per store, ISit = Margin, ∑ Ending inventory itq q=1 4*[Number of stores]it MUit = Revenueit [Cost of goods sold]it Notes: 1. Sales measured at cost and not at price since revenue encompasses margin. 2. Normalization of data by the number of stores. 3. Inventory and COGS adjusted for LIFO reserves. (C) Vishal Gaur, Cornell University, 2008 26 26 Predetermined Variables SGA per store, SGA it = [Selling, general & administrative expenses]it [Number of stores]it 4 Proportion of new inventory, PIit = ∑ [ Accounts payable] itq q=1 4 ∑ [Ending inventory ] itq q=1 Store growth, Git = [Number of stores]it [Number of stores]i,t-1 4 ∑ [Net property, plant, & equipment] itq Capital investment per store,CAPSit = q=1 [ 4*Number of stores]it Others: Index of consumer sentiment (ICS); Lagged sales per store; Lagged margin Notes: 1. Net property, plant, & equipment is adjusted for operating leases. 2. Proportion of new inventory differs from days of inventory. (C) Vishal Gaur, Cornell University, 2008 27 27 Predetermined variables Lagged SGA per store, sgait-1 Lagged index of consumer sentiment, icsi,t-1 Sales per store, csit Store growth, git Lagged proportion of new inventory, pii,t-1 Inventory per store, isit Lagged sales per store, csi,t-1 Lagged cap. investment per store, pii,t-1 Gross margin, muit Lagged gross margin, gmi,t-1 (C) Vishal Gaur, Cornell University, 2008 28 28 Model Specification: Structural Equations cs: Sales per store is: Inventory per store z Aggregate Sales Equation Inventory Elasticity mu: Margin Price Elasticity ⎧Index of ⎫ ⎧Proportion ⎫ ⎪ ⎪ csit = Fi + α11isit + α12 muit + α13 {SGA per store}it −1 + α14 ⎨ ⎬ + α15 {Store Growth}it + α16 ⎨Consumer ⎬ + ε it ⎩of new inventory ⎭i ,t −1 ⎪Sentiment ⎪ ⎩ ⎭t −1 z Aggregate Inventory Equation Stocking Propensity to Sales Stocking Propensity to Margin ⎧ Proportion of ⎫ ⎧Capital Inv.⎫ isit = J i + α 21csit + α 22 muit + α 23 {Sales per store}i ,t −1 + α 24 ⎨ ⎬ + α 25 {Store Growth}it + α 26 ⎨ ⎬ + ηit ⎩ new inventory ⎭i ,t −1 ⎩ per store ⎭i ,t −1 z Gross Margin Equation Markup Propensity to Sales Markup Propensity to Inventory muit = H i + α 31csit + α 32isit + α 33 {Store Growth}it + α 34 {Margin}it −1 + υit (C) Vishal Gaur, Cornell University, 2008 29 29 Application to Forecasting z Reduced Form Model ¾ ¾ Forecasts simultaneously for sales per store, inventory per store, and margin We evaluate forecasts of sales revenue per store and total sales revenue (C) Vishal Gaur, Cornell University, 2008 30 Comparison with financial analysts z z z Consensus analysts’ forecast of sales obtained from I/B/E/S Use forecasts generated 11-12 months before the fiscal year-end date For our model: ¾ z Expected number of new store openings taken from previous year’s 10-K statement Exclude observations in which number of stores changed by more than 30% ¾ M&A, bankruptcies. (C) Vishal Gaur, Cornell University, 2008 31 Results of comparison with financial analysts IBES 12 months Reduced Form Model ARIMA Model (1,1,1) Double Exponential Smoothing Model Year “n” MAPE MAPE MAPE MAPE 2004 77 4.97% 4.77% 5.24% 6.04% 2005 79 4.63% 4.15% 5.31% 5.56% 2006 64 5.18% 4.03% 4.86% 5.63% (C) Vishal Gaur, Cornell University, 2008 32 Is forecast error correlated with the principles of our model? Positive analyst bias for current year Over-inventoried Inventory/store residual in the previous year Negative analyst bias for current year Under-inventoried Underpriced Overpriced Gross margin residual in the previous year Regression results show that both types of bias are present in analysts’ forecasts and are statistically significant. (C) Vishal Gaur, Cornell University, 2008 33 Uses of our model z To financial analysts / outside experts ¾ z To firms ¾ ¾ z For forecasting future sales of firms in aggregate planning In performance analysis Enable better communication between the firm and its investors (C) Vishal Gaur, Cornell University, 2008 34 (C) Vishal Gaur, Cornell University, 2008 35 An Optimization Algorithm for the Inventory Routing Problem and Its Implementation at a Supermarket Chain Vishal Gaur, Marshall L. Fisher (C) Vishal Gaur, Cornell University, 2008 36 Albert Heijn Store and DC Locations 650 Albert Heijn stores + 350 other stores 4 Regional DCs 1 National DC (C) Vishal Gaur, Cornell University, 2008 37 Problem Definition • Givens – A set of stores served by a single distribution center (DC) – A one week planning horizon; time is divided into discrete epochs of one hour each – Demand forecast for each hour at each store – Travel times and costs between all locations • Objective: to determine a periodic weekly delivery schedule specifying the times when each store should be replenished and the vehicle routes that service these requirements at minimum cost. (C) Vishal Gaur, Cornell University, 2008 38 Albert Heijn Supply Chain and Problem Definition Suppliers National DC Slow moving items Suppliers Regional DCs 4 DCs Fast moving items Stores 650 Albert Heijn stores + 350 other stores Short-term Problem (frequency: at least once a day) Solve VRPTW using actual order volumes and planned delivery times. Long-term Problem (frequency: once in 3 months) Determine delivery times and vehicle routes using forecasted volume. Previously solved by a hierarchical approach. (C) Vishal Gaur, Cornell University, 2008 39 Characteristics of Store Demand Variation in demand during a week Demand Rate (units/hour) 3.5 Variation in demand during the year Total Weekly Sales Volume 140 3.0 120 2.5 100 2.0 80 1.5 60 1.0 40 0.5 Mon Tue Wed Thu Fri 20 Sat 0 0 0 12 24 36 48 Time (in hours) 60 72 0 10 20 30 40 50 60 Time (in weeks) Demand varies considerably by hour but the same pattern repeats every week. Coefficient of variation of demand is about 5%. (C) Vishal Gaur, Cornell University, 2008 40 Use of generalized minimum weight matching on non-bipartite graphs when each cluster is restricted to at most two stores • Matching: A subset of edges in a graph with the property that every vertex has at most one edge in the subset incident to it. • Types of matching problems: – Maximum weight matching – Minimum weight perfect matching • Generalized minimum weight matching problem: minimize the sum of the costs of the edges in the matching plus the sum of the costs of the unmatched vertices. • We convert our problem into a maximum weight matching problem. Algorithms for this problem include Edmonds O(n4), Gabow O(n3), Micali and Vazirani O(m√n). (C) Vishal Gaur, Cornell University, 2008 41 Sequential application of the generalized minimum weight matching algorithm gives us clusters with progressively larger number of stores. First iteration Second iteration 3 2 5 3 4 1 1 2 5 7 7 6 6 (C) Vishal Gaur, Cornell University, 2008 4 42 Flowchart of the Randomized Algorithm For stores i, j, and iteration l, Tijl = number of times i and j have combined together up to iteration l. Pijl = 1/(1 + Tijl) = probability of splitting a route at link (i, j) in iteration l. Randomly split some clusters Minimum Weighted Matching Update probabilities of combination of store pairs Construct new routes and delivery schedule Termination Criteria (C) Vishal Gaur, Cornell University, 2008 43 Additional Constraints for the Implementation at Albert Heijn, BV • Time window constraints • Store-vehicle restrictions • Loading-unloading time delays and cost • Traffic delays • Delivery time variability • Vehicle waiting time restrictions and cost • Time between departure and last stop (C) Vishal Gaur, Cornell University, 2008 44 Assignment of routes to a heterogeneous fleet of trucks • Deterministic scheduling of jobs with arrival times and due times on parallel machines. • NP-Hard. Heuristics do not even guarantee a feasible solution. • Three phase heuristic – Lagrangian relaxation with fixed penalty cost assigned to each truck type – Progressive reduction of problem size – Face validity: pair-wise exchange of routes between trucks (C) Vishal Gaur, Cornell University, 2008 45 Implementation • Began in June 1999. • Phased implementation to facilitate testing and get buy-in from the company – Solve VRPTW with existing delivery times and at most two stores per route – Solve VRPTW with existing delivery times and unlimited number of stores per route – Solve IRP with at most two stores in a cluster and restricted delivery times – Solve IRP with unrestricted cluster size and delivery times. • Main activities: pilot implementation and testing, training of route planners, design of graphical user interface. (C) Vishal Gaur, Cornell University, 2008 46 Summary of Computational Results Each problem has 207 stores across 6 days, vehicle capacity 26 units, and daily store demand ranging between 3 units and 68 units. Problem Scenarios 1 Cost ($) A Albert Heijn's solution 242,751.39 B VRPTW with at most 2 deliveries per route 221,732.78 C D Savings (%) Cost ($) 3 Savings (%) 247,501.22 Cost ($) Savings (%) 243,146.80 8.66% 227,687.77 8.01% 223,889.22 7.92% VRPTW with unlimited 215,414.55 number of deliveries per route 11.26% 220,400.90 10.95% 217,094.64 10.71% IRP with at most 2 stores per 208,323.03 cluster 14.18% 214,054.01 13.51% 209,983.17 13.64% Savings with respect to B E 2 IRP with unlimited number of stores per cluster 6.05% 200,463.69 Savings with respect to C (C) Vishal Gaur, Cornell University, 2008 17.42% 6.94% 5.99% 205,939.19 16.79% 6.56% 6.21% 202,440.14 16.74% 6.75% 47 Personal Experience of Jeroen Hirdes: A Route Planner’s Perspective • Working with the latest developments • Better understanding of route scheduling algorithms • Benefits of storing data to create long term history • Better judgement of store behaviour (C) Vishal Gaur, Cornell University, 2008 48 Example VRSA Routes Intertour routes (C) Vishal Gaur, Cornell University, 2008 49 Summary • Lessons from the implementation - Decision support role: enabling Albert Heijn to evaluate the cost impact of tactical decisions such as delivery frequency constraint, time window constraint, etc. - Performance analysis: determine the cost of order variability at stores, and deviations from scheduled delivery times. • Further issues addressed by the application – Assignment of routes to trucks, – Load balancing at the warehouse, – Determining which DC to assign to a store, – Management of deliveries from suppliers to stores. (C) Vishal Gaur, Cornell University, 2008 50 In-store experiments to determine the impact of price on sales Objectives • Select a subset of stores in a retail chain for conducting an experiment – Short lifecycle products • Estimate price elasticity of the demand for the retail chain (C) Vishal Gaur, Cornell University, 2008 51 Sample Data – Total Store Sales Sales ($) Store 101 102 104 105 106 107 108 109 110 111 201 202 203 Opening Year 91 92 93 93 94 94 96 95 95 96 94 94 94 Month to date This Year 243360.89 200584.79 161107.45 167564.02 166967.08 156529.29 159969.72 138550.52 199359.06 204735.67 281504.57 127881.34 159432.69 (C) Vishal Gaur, Cornell University, 2008 Month to date Last Year 225825.43 161526.19 135974.13 125992.23 134297.03 110326.95 125368.47 108505.13 140331.28 168748.16 271636.39 88210.31 144225.67 Year to date This Year 774970.63 680822.05 494684.61 492248.49 542107.98 523500.39 481693.31 426937.44 595843.71 633289.33 904793.24 403403.03 537945.19 Year to date Last Year 776310.7 616936.85 497567.43 462623.69 506260.25 472678.06 480267.04 433574.55 538571.22 598483.91 913364.1 339888.69 522730.67 52 Sample Input Data – Fraction of Sales from Each Product Category Store 101 102 104 105 106 107 108 109 110 111 201 202 203 20 5.2 6.5 5.9 5.0 16.1 5.2 19.0 5.0 11.0 14.9 6.2 5.8 6.8 (C) Vishal Gaur, Cornell University, 2008 Product Categories 10 15 22 6.0 11.2 6.0 7.4 12.4 7.7 5.2 8.2 5.9 4.0 8.7 6.2 3.3 9.2 8.1 3.9 10.6 7.6 3.0 9.2 6.3 5.1 6.7 6.4 3.2 11.1 6.7 4.4 9.3 5.9 5.3 13.3 9.3 4.3 9.8 7.9 5.3 11.4 7.9 40 9.1 9.2 7.0 9.1 6.1 6.3 7.4 8.3 8.2 10.0 10.3 9.7 9.0 53 Experiment Design Store Selection Problem – Block Design We select 3 stores from each cluster such that • They are relatively isolated from other stores in the chain • They are similar in age and size Prices High Clusters Medium Low 1 2 3 4 5 6 (C) Vishal Gaur, Cornell University, 2008 54 How to group stores using historical sales mix: distance metric Distance estimates obtained for a sample of 6 stores Distance between two stores = ( fi1 − f j1 ) ( 2 + fi2 − f j2 ) ( 2 + fi3 − fj3 ) 2 where fi1 = fraction of sales of store i from department 1, fi2 = fraction of sales of store i from department 2, etc. 101 102 103 104 102 0.24 103 0.25 0.26 104 0.41 0.41 0.55 105 0.14 0.16 0.13 0.45 106 0.48 0.34 0.48 0.29 105 0.43 Principle for grouping stores: Two stores are said to be similar if the distance between them is small, otherwise they are said to be dissimilar. We wish to group stores into clusters such that stores within a cluster are similar to each other and stores in different clusters are dissimilar from each other. (C) Vishal Gaur, Cornell University, 2008 55 How to group stores using previous sales mix: clustering • Objective function: – Minimize sum of all distances within clusters • Decision variables – Whether a store is the ‘seed’ for a cluster – Whether store i is clustered with store j, for all pairs (i,j) • This picture is a schematic of clusters. All interconnections are not shown to avoid cluttering the graph. (C) Vishal Gaur, Cornell University, 2008 Constraints – Each store must be in at least one cluster – If store i is clustered with store j, then store j must be a seed – Minimum number of clusters – Minimum size of a cluster 56 List of stores selected for the experiment Cluster Store 1 102 103 204 105 108 402 205 302 303 325 401 526 107 326 527 110 503 504 2 3 4 5 6 Year To Opening Date Sales Year ($ '000) 92 680.8 93 610.5 94 514.1 94 492.2 96 481.7 94 477.9 95 466.2 94 432.6 95 434.1 96 561.8 94 644.1 96 556.1 94 523.5 96 438.1 96 441.8 95 595.8 96 547 96 553.3 (C) Vishal Gaur, Cornell University, 2008 % Sales from each category 1 2 3 4 5 11.3 5.4 8.1 8.5 7.9 9.9 11.6 6.8 10.1 10.1 8.9 6 7.5 10.5 10.5 11 6.1 7.9 8.4 8.9 9.7 5 6.4 11.4 12.3 11.3 8.9 7.9 8.4 7.5 57 Summary Statistics of Products Tested Prices ($) Purchase Cost ($) Low Medium High A: Family Game Center 19.99 24.99 29.99 11.00 B: Phonics Traveler 24.99 29.99 34.99 18.00 C: Headset Walkie Talkie 14.99 19.99 24.99 11.00 Regular Prices (C) Vishal Gaur, Cornell University, 2008 58 Assignment of Products to Stores Products are assigned to stores using a randomized block design to avoid systematic interactions between products. Cluster 1 Cluster 2 Store 1 Store 2 Store 3 Store 4 Store 5 Store 6 Product A 29.99 24.99 19.99 24.99 19.99 29.99 Product B 24.99 29.99 34.99 29.99 24.99 34.99 Product C 14.99 19.99 24.99 19.99 14.99 24.99 Control Stores Precautions taken during the experiment • Price labels were changed to reflect new prices • Store managers were not informed about the experiment to control for execution differences. (C) Vishal Gaur, Cornell University, 2008 59 Sample Output Data from the Zany Brainy Study Product C Week ending 10/3 Beginning # of Sales Inventory Transactions Week ending 11/7 Beginning # of Sales Inventory Transactions Cluster 1 Store 1 3 0 2794 53 7 362 Store 2 Store 3 12 30 1 0 2565 2365 35 31 9 3 330 324 Store 4 8 3 2037 38 6 252 Store 5 Store 6 7 6 1 0 2020 1883 37 36 6 2 238 279 Store 7 11 1 1685 35 5 248 Store 8 Store 9 6 11 0 1 1681 1612 36 34 6 3 220 218 Cluster 2 Cluster 3 Total Test Period = 6 weeks (C) Vishal Gaur, Cornell University, 2008 60 Results of the Zany Brainy experiment Sales Data Total Unit Sales for the 3 test products at each of the 3 Price Levels 80 70 60 50 40 30 20 10 0 Low Product A (C) Vishal Gaur, Cornell University, 2008 Medium Price Product B High Product C 61