Coordination of Inventory Distribution and Price Markdowns for Clearance Sales at Zara

Abstract

Zara holds a clearance period for several weeks after each of its two annual selling seasons. Due to restrictions in shipping capacity, allocation decisions for the remaining warehouse inventory start 4–6 weeks prior to the clearance period. Our work addresses the problem of dynamically coordinating inventory and pricing decisions for unsold merchandise during the last month of the regular season and then clearance sales. The inventory allocation prior to markdowns is particularly challenging because it is a large-scale optimization problem and countries “compete” for scarce inventory. Moreover, there are many business rules that must be satisfied. Until recently, the decision process used by Zara for end-of-season inventory allocation and clearance pricing was essentially manual and based on managerial judgment. We propose a model-based approach that builds on a deterministic approximation. The deterministic problem is still too large so it is further broken down into an aggregate master plan and a store-level model per-country with feedback recourse between the two levels. After a working prototype of the new tool was completed, we performed a controlled field experiment during the 2012 summer clearance to estimate the model’s impact. The controlled experiment showed that the model increased revenue by 2.5%, which is equivalent to $24M in additional revenue. Given that unsold inventory is sunk at the time of clearance sales, the additional revenue translates directly into profits. The implementation of the tool coincided with the launch of Zara’s online portal. We discuss how the model-based process was adjusted to accommodate this new channel.

Appendices

Appendix 1: Base Model: Single-Item Continuous Formulation

To gain insights, we formulate a single-item base model in which inventory decisions are treated as continuous variables. First, consider a single-period problem with multiple countries that are sourced from the same central depot. Let F _m(p _m) be the demand in country m at price p _m. Here we assume that demand is deterministic and given by $\displaystyle F_m(p_m) = C_m \left (\frac {p_m}{p_m^T} \right )^{-\beta _m}$ where $p_m^T>0$ is the regular-season price for the item, β _m is the constant price elasticity, and C _m > 0 is a country specific constant that is proportional to the market size in country m. We assume that β _m > 1. The case with β _m < 1 is not interesting for our purposes because the revenue increases with price, which means that the retailer has no incentive to introduce markdowns and would rather keep the regular-season price $p_m^T$.¹

Let q _m be the inventory allocated to country m and let $J_m(q_m) :=\max _{p_m \ge 0} p_m \min \big \{F_m(p_m),q_m \big \}$ be the maximum revenue obtained by optimizing the price p _m. Note that when β _m > 1 the (unconstrained) revenue p _m F _m(p _m) is convex so standard results such as Proposition 1 in Bitran and Caldentey (2003) do not apply. However, the (constrained) revenue $p_m \min \big \{F_m(p_m),q_m \big \}$ is a unimodal function in p _m. In fact, the revenue increases until the price is such that supply exactly matches demand and then it decreases. In other words, the revenue has a unique maximizer that satisfies F _m(p _m) = q _m. Hence, the optimal price is $\displaystyle p_m^*(q_m) = p_m^T \left (\frac {C_m}{q_m}\right )^{\frac {1}{\beta _m}}$. Substituting the optimal price in the revenue function we obtain $\displaystyle J_m(q_m) = p_m^T C_m^{\frac {1}{\beta _m}} q_m^{1-\frac {1}{\beta _m}}$, which is concave in q _m.

Let I ⁰ be the total inventory available at the central depot. In the absence of additional business requirements or constraints, the inventory allocation problem faced by the retailer can be formulated as follows:

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} &\displaystyle \max &\displaystyle \sum_{m \in \mathcal{M}} J_m(q_m) \\ &\displaystyle s.t. &\displaystyle \sum_{m \in \mathcal{M}} q_m \le I^0 \\ &\displaystyle &\displaystyle q_m \ge 0 \quad \forall \, m \in \mathcal{M}. \end{array} \end{aligned} $$

(43)

Since $\displaystyle \frac {\partial J_m}{\partial q_m} = \Big (1-\frac {1}{\beta _m}\Big ) p_m^T C_m^{\frac {1}{\beta _m}} q_m^{-\frac {1}{\beta _m}} > 0, \ \forall \, m \in \mathcal {M}$, it follows that the constraint $\sum _{m \in \mathcal {M}} q_m \le I^0$ must be binding. Let ν be its Lagrangian multiplier or shadow price. From the Karush-Kuhn-Tucker conditions (Bertsekas 1999) it follows that the optimal quantities are given by

$$\displaystyle \begin{aligned} q_m^* = C_m \left(\Big(1-\frac{1}{\beta_m}\Big) \frac{p_m^T}{\nu}\right)^{\beta_m} \quad \forall \, m \in \mathcal{M}. \end{aligned} $$

(44)

Equation (44) shows that q _m is increasing in C _m and $p_m^T$. Therefore, all other things being equal, it is optimal to allocate more inventory to countries with larger market size and higher regular-season price. If there is ample inventory I ⁰ at the depot such that $\nu \le p_m^T$, then q _m is also increasing in β _m, so ceteris paribus, it is optimal to allocate more inventory to countries where demand is more elastic. Note that q _m > 0 for all m meaning that all countries get a positive allocation. Of course, this last observation hinges on fractional inventory being allowed.

Now consider a multi-period version of the single-item problem described above. Let $w \in \mathcal {W}=\{w: 1 \le w < W\}$ denote a period and let $I_m^w$ be the inventory in country m at the beginning of period w. An important feature in a multi-period setting is that the retailer can choose to “save” inventory for a future period. To capture this decision, we introduce the variable $\lambda _m^w$ that represents the amount of inventory withdrawn from $I_m^w$ and allocated to period w in country m. Since there is no incentive to allocate inventory that will not sell, it follows that $\lambda _m^w$ will be equal to the sales in period w, which is the interpretation we give to that variable in Sect. 4.²

With the additional variables, the pricing problem in country m can be formulated as the following dynamic program

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} J_m^w(I_m^w) = &\displaystyle \max &\displaystyle p_m^w \min\big\{F_m^w(p_m^w), \lambda_m^w\big\} + J_m^{w+1}(I_m^{w+1}) \\ &\displaystyle &\displaystyle I_m^{w+1} = I_m^w - \lambda_m^w \\ &\displaystyle &\displaystyle p_m^w, \lambda_m^w, I_m^{w+1} \ge 0, \end{array} \end{aligned} $$

(45)

where $F_m^w(p_m^w)$ is the (deterministic) demand in country m for the price $p_m^w$ in period w ≥ 1. Then, allocating the inventory at the depot across countries corresponds to solving

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle \max &\displaystyle \sum_{m \in \mathcal{M}} J_m^1(q_m) \\ &\displaystyle s.t. &\displaystyle \sum_{m \in \mathcal{M}} q_m \le I^0 \\ &\displaystyle &\displaystyle q_m \ge 0 \quad \forall \, m \in \mathcal{M}. \end{array} \end{aligned} $$

(46)

Given that the problem is deterministic, the sequential (closed-loop) optimization has an equivalent simultaneous (open-loop) formulation that is given by:

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} &\displaystyle \max &\displaystyle \sum_{m \in \mathcal{M}} \sum_{w \in \mathcal{W}} p_m^w \min\big\{F_m^w(p_m^w), \lambda_m^w\big\} \\ &\displaystyle s.t. &\displaystyle I_m^1 = q_m \quad \forall \, m \in \mathcal{M} \\ &\displaystyle &\displaystyle I_m^{w+1} = I_m^w - \lambda_m^w \quad \forall \, (m,w) \in \mathcal{M}\mathcal{W} \\ &\displaystyle &\displaystyle \sum_{m \in \mathcal{M}} q_m \le I^0 \\ &\displaystyle &\displaystyle p_m^w, \lambda_m^w, I_m^{w} \ge 0 \quad \forall \, (m,w) \in \mathcal{M}\mathcal{W}. \end{array} \end{aligned} $$

(47)

Note that the inventory variables $I_m^w$ in the formulation above can be omitted and the non-negative constraint $I_m^w \ge 0, \, \forall \, (m,w) \in \mathcal {M}\mathcal {W},$ can be replaced by $q_m \le \sum _{w \in \mathcal {W}} \lambda _m^w, \forall \, m \in \mathcal {M}$. Moreover, with no loss of optimality one can assume that $q_m = \sum _{w \in \mathcal {W}} \lambda _m^w, \forall \, m \in \mathcal {M}$, so the optimization problem (47) can be reformulated as

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} &\displaystyle \max &\displaystyle \sum_{m \in \mathcal{M}} \sum_{w \in \mathcal{W}} \widehat{J}_m^w(\lambda_m^w) \\ &\displaystyle s.t. &\displaystyle \sum_{m \in \mathcal{M}} \sum_{w \in \mathcal{W}} \lambda_m^w \le I^0 \\ &\displaystyle &\displaystyle \lambda_m^w \ge 0 \quad \forall \, (m,w) \in \mathcal{M}\mathcal{W}, \end{array} \end{aligned} $$

(48)

where $\widehat {J}_m^w(\lambda _m^w) = \max _{p_m^w \ge 0} \ p_m^w \min \big \{F_m^w(p_m^w), \lambda _m^w\big \}$. The optimization problem (48) has the same structure as the single-period problem (43). In particular, suppose that for country m there exists a parameter 0 < κ _m < 1 such that $\displaystyle F_m^w(p_m^w) = \kappa _m^{w-1} F_m^1(p_m^w) = \kappa _m^{w-1} C_m \left (\frac {p_m}{p_m^T} \right )^{-\beta _m}$ for w ≥ 1.³ Similar to Caro and Gallien (2012), the parameter κ _m represents a discount factor that captures how prices age, regardless of the inventory level. Then, from Eq. (44) the quantity allocated to country m is given by

$$\displaystyle \begin{aligned} q_m^* = \sum_{w \in \mathcal{W}} \lambda_m^w = C_m \frac{1-\kappa_m^{W-1}}{1-\kappa_m} \left(\Big(1-\frac{1}{\beta_m}\Big) \frac{p_m^T}{\nu}\right)^{\beta_m} \quad \forall \, m \in \mathcal{M}. \end{aligned} $$

(49)

Therefore, the insights from the single-period problem carry over to the multi-period case. Namely, the allocation $q_m^*$ to a given country m increases with the market size C _m, the regular-season price $p_m^T$, and the elasticity β _m (when $\nu \le p_m^T$). Of course, $q_m^*$ is also increasing in the parameter κ _m.

Appendix 2: System Snapshots