Price Markdowns and Linear Programming
Fashion retailers are often limited by the fashion designers/houses on the timing and range of prices they can charge for the brand name merchandise. To gain more control over their pricing functions, many of these retailers have developed their own private brands (merchandise with fully developed, supported, and advertised brand profiles) and/or private labels (fillin merchandise with no specific brand profiles), which are sold exclusively through their network of stores. Macy’s, for example, targets customers with various needs through its suite of private brands and labels including Alfani (men/women), Charter Club (women), Club Room (men), and Greendog (children) (for more details, see Macy’s Inc, 2013). Similarly, Saks Fifth Avenue’s Men’s Collection and Crown and Bloomingdale’s The Men’s Store attempt to appeal to sophisticated yet priceconscious shoppers in need of quality menswear (Palmieri, 2011; Racked Staff, 2011). The trend has also been embraced by online fashion retailers who could not let such an opportunity slip away. The giant ShopBop.com, for example, has introduced its own brand, Bop Basics, as an alternative for its customers to the more expensive designer collections (Business Insider, 2013). BlueFly.com has tried to achieve similar goals by introducing and promoting its private label brands including Harrison, Hayden, and Cullen (Bloomberg Businessweek, 2013).
From an operations perspective, private brands/labels allow style and seasonal goods retailers to be more responsive to the markets they serve. In particular, since no binding agreements with the designer/fashion houses are in place to specify tight pricing terms and markdown conditions, fashion retailers can use price as an effective means to drive profitability. In the absence of any contractual obligations, two actions are often employed to immediately impact the retailer’s bottom line. First, retailers with private brands/labels are free to set the initial markup as low or as high as they would like as there is nothing in place to enforce it to be within a certain range. Second, retailers can take immediate actions and consider a price markdown the moment sales drop and an item starts to underperform.
Motivated by the specific issues put forth by the management of private brands/labels, we focus on some of the recent efforts of brickandmortar and online retailers that attempt to streamline their fashionrelated markdown practices. In these instances, retailers intend to exploit the pricing flexibility that comes with the selling of private brands/labels to recommend price markdown strategies that would maximize margins. We illustrate this approach using an example from a major online fashion retailer that offers collections consisting of a mix of designer and private brand/label items. Some conceptual ideas of our approach are present in the work of Caro & Gallien (2012).
For one of its private label items, the retailer starts the new season with 500 units in stock. Because the supply of the item comes from overseas, the retailer cannot restock the item during the selling season. A typical season at this retailer lasts about 16 weeks. The item’s full selling price is €60, which is anticipated to be offered for at least one week. If markdowns are needed, the retailer prefers fixed discrete price discounts that can be easily communicated to its customers. For this reason, price markdowns of 25% and 50% off, corresponding to selling prices of €45 and €30, respectively, are considered. All markdowns are permanent and irreversible. While still in the preseason, the retailer wants to understand what its optimal markdown strategies should be based on various probable fullprice weekly sales rates. Among these strategies, selling the item at the full price throughout the season is preferred. If this is not profitable because of lowerthan originally expected sales, the retailer wants to explore alternative strategies that account for seasonality, inventory depletion effects, and special online events such as the timing of email campaigns. Furthermore, once the selling season starts, the retailer wants to have dynamic control over its pricing function to be able to revise or implement price markdowns that reflect updated market conditions.
In a business environment such as this one, product demand is always difficult to predict. In many instances, seasonal products show sales patterns that do not repeat from one season to the next. To complicate things further, within the organization itself, opinions are typically divergent on how products will likely perform in the marketplace. Given the uncertainty that surrounds the demand processes, you may ask how fashion retailers can operationalize their markdown initiatives. Often, although individual product histories cannot be recycled to get relevant product intelligence, histories of groups of similar products can be analyzed to learn the likely demand response of a typical group member. For the specific item introduced earlier, the results of such an undertaking are provided in Table 7.2. To estimate the product group demand models, the product group the item belongs to was identified, all items in this group were selected, and multiplicative models of the types discussed in Chapter 3 (i.e., model types B1 and B2) were explored. The product group identification and the withingroup product selection are inexpensive tasks that are typically driven by the product hierarchies in use at the retailer. Finding the preferred model specification(s) is a more involved task that builds on existing theory and requires extensive testing and tweaking.
The demand models depicted in Table 7.2 are exponential models of the B2 type. We prefer this functional form over its B1 counterpart because it performs marginally better in regard to the quality of the model fit. Since seasonality within the product group appears weak, we do not consider it explicitly. We also prefer to provide only an excerpt from the full output since the product sales baselines are irrelevant to the subsequent markdown optimization process. In our search for the preferred model specification, we build on previous retail studies and find that product group sales are time dependent and explained by markdown values and special online events. In spite of our findings, we choose to show results for two competing models to subsequently illustrate the impact the presence of the special online events has on the expected profitability of recommended markdown policies. Focusing on the parameter estimates of the full model, it is obvious that price markdowns impact sales nonlinearly. This is an intuitive result that confirms to the retailer’s expectations. In addition, within the product group, we observe that sales tend to decline toward the end of the items’ selling season. The retailer speculates that this behavior is mainly a reflection of the assortment being broken, that is, the onhand inventory not providing a complete selection of colors and sizes. Although more sophisticated approaches can be employed, we model sales’ time dependency and, indirectly, the impact of the inventory level and mix using three timerelated variables. As shown in Table 7.2, the corresponding parameter estimates are all statistically significant and quite large in magnitude. For example, all else being equal, the last weeks of the selling season are expected to experience about a tenth of the regular sales (i.e., P3 multiplier equals 0.13). Last but not least, we note that the special online events such as the email campaigns tend to positively impact sales on average by a factor of 1.61.
Although specific to an average group product, the insights gained from the figures in Table 7.2 can be used to initialize the computation of the optimal markdown policies. In the absence of any sales data in the preseason, the retailer could explore the likely product performance using hypothetical weekly sales rates. In season, however, it can decide on the best course of action in regard to the pricing of the item based on actual sales rates and continuously updated productspecific demand multipliers. The differentiation of the latter
Table 7.2 Product Group Demand Analysis
Reduced Model 
Full Model 

Coefficient 
Standard Error 
t value 
p value 
Coefficient 
Standard Error 
t value 
p value 

(irrelevant output removed due to space constraints) Markdown 
1.83 
0.2 
7.5 
0.00 
1.56 
0.2 
6.6 
0.00 
Multiplier 0% offSj 
1.00 
1.00 

Multiplier 25% off 5, 
1.58 
1.48 

Multiplier 50% off 5_{3} 
2.49 
2.18 

Periods in between 85% and 90% of the selling season (PI) 
1.10 
0.2 
5.2 
0.00 
0.90 
0.2 
4.4 
0.00 
PI multiplier 
0.33 
0.41 

Periods in between 90% and 95% of the selling season (P2) 
1.77 
0.2 
8.4 
0.00 
1.53 
0.2 
7.4 
0.00 
P2 multiplier 
0.17 
0.22 

Periods above 95% of the selling season (P3) 
2.28 
0.2 
12.1 
0.00 
2.04 
0.2 
11.0 
0.00 
P3 multiplier E,_{3} 
0.10 
0.13 

Special Events (SE) Indicator 
 
 
 
 
0.48 
0.1 
4.9 
0.00 
SE multiplier q 
 
1.61 
Full Model: Multiplier 25% off 1.48=exp(0.25 X 1.56); PI multiplier 0.41=exp(0.90); SE multiplier 1.61=exp(0.48) Reduced Model: Multiple Rsquared: 0.58, Adjusted Rsquared: 0.56 Full Model: Multiple Rsquared: 0.62, Adjusted Rsquared: 0.60
happens throughout the season when relevant information becomes available. For example, all products in a group may start the season with a special online event demand multiplier of 1.61 but could end the season with such multipliers in the 1.252.50 range, based on each product’s independent performance. Updating the multipliers typically requires the use of various weighted moving averages of which exponential moving average is the most frequently used. Since illustrating the dynamic character of markdown optimization is beyond the scope of this discussion, we show next how the retailer can structure its preseason markdown initiatives to prepare for more accurate inseason pricing decisions. The same underlying markdown mechanism, however, applies to both of these cases.
As part of how it runs its ebusiness, the retailer sends out customized newsletters intended to promote new collections, raise awareness for specific brands or item groups, or inform customers of imminent sales opportunities. While the effectiveness of these initiatives largely depends on the content of the actual message, the retailer plans to run recurrent email campaigns directly targeting our item’s group a week after products are introduced to the market and every four weeks thereafter (i.e., weeks 2, 6, 10, and 14). These campaigns are of the same type as those we used to estimate the grouplevel demand models shown in Table 7.2. Based on discussions among several buyers at the retailer, a consensus has been reached on the market expectations for this important item. In the absence of any auxiliary activities, there are high hopes that the product will sell at full price at a weekly rate of 25 units. In this context, the initial inventory of 500 units is perceived as sufficient to serve the market requirements with a sufficient amount of leftover to create some endoftheseason estore excitement through permanent markdowns. To investigate possible preseason strategies for inseason markdowns, the retailer can use the group level demand multipliers computed previously to adjust the expected baseline sales of 25 units to account for product lifecycle events such as markdowns, time dependency, and special online initiatives. Because after purchasing the items the purchase price becomes a sunk cost, the retailer wishes to maximize revenues from the inventory it starts the season with such that several market constraints are satisfied. In formal terms, the retailer needs to solve the following revenue maximization problem:
where X_{ti} are 0/1 decision variables that specify whether or not the discrete price р_{г} is to be offered in week t, p_{t} is one of the possible prices in the discrete price set S ={€60.0, €45.0, €30.0}, D is the baseline sales of 25 units per week, S_{t} is the demand multiplier corresponding to price р_{г} (see Table 7.2 for values for S), ? is 1 or ?_{k} based on the position of the current week t within the selling season (see Table 7.2 for time brackets and values for ?_{k}), g_{t} is 1 or abased on whether or not a special online event is scheduled for week t (see Table 7.2 for the value for g), X_{s} is the inventory left over at the end of the season that needs to be salvaged and is the unit salvage value of €10.
Constraints C1C7 bound the optimal solution and enforce inventory limitations and other operations practices in use at the retailer. Constraint C1 limits the amount of inventory the retailer can sell to the initial value of 500 units. Constraints C2 enforce the use of a single price point in each of the 16 weeks of the selling horizon. Constraint C3 makes sure that the item is offered at full price for at least one week. Constraints C4C6 implement the common retail practice that stipulates that price markdowns are irreversible. Lastly, constraint C7 imposes sign and value restrictions on all decision variables. Figure 7.1 shows the demand values (D • S) ? ?_{t} ? g,, which enter both the objective function and the C1 constraint. These values are specific to the full model of Table 7.2. To repeat the task for the reduced model, we simply update the S and ? multipliers appropriately and set g to 1. The corresponding demand profiles should be smoother than those depicted in Figure 7.1.
For the reduced and full demand models of Table 7.2, the solutions of the price markdown optimization problem are provided in Table 7.3. These results suggest that the
Figure 7.1 PriceDependent Demand Profiles.
Table 7.3 Preseason Optimal Markdown Policies
Expected Revenue (€) 
Markdown Policy 

Reduced Model 
23,403.1 
Sell at €60.0 for 3 weeks. Switch to €45.0 in week 4. Sell at €45.0 until the end of the season. 
Full Model 
25,156.1 
Sell at €60.0 for 6 weeks. Switch to €45.0 in week 7. Sell at €45.0 until the end of the season. 
demand model specifications that describe the item’s market performance lead better to more profitable markdown strategies. In this example, by modeling the impact of the special online events explicitly, the retailer is advised to markdown its full price to €45 in week 7, which follows the anticipated email campaign of week 6. In the absence of this intelligence, the retailer is advised to reduce the price to €45 in week 4 or soon after the first email campaign of week 2.
Just because the retailer possesses this type of information before the season starts does not mean that the retailer should stick to this plan once actual demand for the item starts to become available. In particular, once the season starts, the retailer should confirm the hypothetical baseline sales used in the preseason markdown optimization exercise. The actual sales, once available, could be used then to rerun the optimization procedure and adjust the depth and the timing of the suggested markdowns. In addition, all product group demand multipliers can be revised in season to reflect the item’s actual performance. By dynamically resolving the markdown optimization with updated information, revenues can be maximized and endoftheseason spoilage can be minimized.