I NTRODUCCIÓN

We will now turn to the more realistic case where the retailer only observes sales, i.e. censored demand observations if a product stocks out. This is common in most retail environments. In addition to recording total sales quantities, POS scanner systems record the time of the last sale and hourly sales quantities. We will use this information to estimate the total demand if one product stocks out.

Demand Estimation for Out-of-Stock Situations For products and days with censored demand observations Ci, i.e. stockouts, we need to calculate unobservable lost sales first. We estimate unobservable lost sales and demand substitution behavior without any distributional assumptions based on Lau and Lau (1996) and Sachs and Minner (2014). As the amount sold until a product stocks out is an unbiased estimator of the demand, we can use this information to estimate unobservable lost sales of the out-of-stock product.

Furthermore, sales of potential substitutes are inflated. As we can also observe original demand of this product up to the stockout of the other product, we can apply this information to determine an estimate of the additional demand and deflate sales of the substitute.

To determine the amount of original demand that occurred before product i stocked out (Lau and Lau1996; Sachs and Minner2014), we consider the hourly sales observations hⁱ_z;t. We denote the hourly time intervals by z. The store closes at Z. We consider that the ratio of cumulative demands until the end of the day H_Z;tⁱ

and the sum of all demand occurring during the day D_i;t is one. This holds for the

Our aim is to obtain a multiplier K_zⁱthat can be used to inflate unobservable lost sales and deflate sales with substitution. Recursively, we can calculate the ratios of mean cumulative sales and demand for all days with full availability based on mean hourly sales Nhⁱ_zby

In order to estimate demand, we replace the hourly time intervals z with the stockout time ki;t. If product i stocks out at time ki;t on day t , we have to correct for unobservable lost sales which is accomplished by

F_kⁱ

Replacing ki;t by kj;t in (5.15) allows to calculate the deflated demand of the substitute i if product j is out-of-stock.

Figure5.2illustrates demand estimation for the two-product case where product j is sold out. Demand for product j is filled until the order-up-to level (horizontal line) is reached. The time of the stockout is recorded as kj;t. Any demand occurring after kj;t cannot be satisfied from product j ’s inventory and is unobserved. The shaded area of product j is an estimate for the unobserved demand and depends on kj;t. The total demand for product j is estimated as F_k^j

j;t. Cumulative sales of product i before j stocks out corresponds to primary demand for i . After kj;t, demand for i is inflated by substitution (shaded area). A fraction of the unsatisfied demand for product j is shifted to product i . F_kⁱ

j;t is an estimate for the primary demand of product i deducting substitution.

66 5 Data-Driven Order Policies with Censored Demand and Substitution in Retailing

Fig. 5.2 Demand estimation

Product i

,

Demand ,

Product j

Stockout Observations of One Product Constraints (5.3) and (5.4) are not affected by censoring and remain unchanged. Constraints (5.5) and (5.6) have to be adapted since demand cannot be observed. They are therefore replaced by an estimate of demand [see (5.15)] in constraints (5.16) and (5.17).

si;t  F_kⁱ

j;t 8t 2 T (5.16)

s_j;t C s_ji;t  F_k^j

j;t 8t 2 T (5.17)

Additionally, observations have to be separated into censored Cj and full Fj

demand observations. Product i is always fully available, i.e. Fi. The amount of demand to be substituted may not exceed the past substitution quantity. Since the past substitution quantity cannot be directly observed, it is estimated from the total sales of product i and the estimate of primary demand of i F_kⁱ

j;tin (5.18) . The total sales D_i;t^c are inflated by the substitution from j to i so that by taking the difference we obtain an estimate of the substitution amount.

sji;t  D_i;t^c  F_kⁱ

j;t 8t 2 F_i\ C_j (5.18)

Since there are no out-of-stock observations for product i , no observations on the customers’ willingness to substitute from i to j can be observed (sij;t D 0). This holds also for product j on all days with full availability of both products (Fi\ F_j).

sji;t D 0 8t 2 F_i\ F_j (5.19)

Further, the following constraints for all decision variables apply:

s_i;t; s_j;t; s_ij;t; s_{j i;t}; F_kⁱ

t; F_k^j

t; B_i;t; B_j;t  0 (5.20)

ai; bii; bji; aj; bjj; bij2 <: (5.21)

Stockout Observations of Both Products We conclude this chapter by outlining the model for censored demand and stockouts of both products. Constraint (5.4) has to be adapted to account for the substituted demand from i to j :

s_j;tC s_ij;t  B_j;t 8t 2 T (5.22)

If both products stock out, the sequence of stockouts is important. Only sales observations before the first stockout occurs reflect primary demand. Therefore, the time of the first stockout kt is obtained as the minimum of the stockout time of product i and j :

kt D min.k_i;tI k_j;t/: (5.23)

In addition to constraints (5.3) and (5.16) to (5.19), the cases when product i or product i and j stock out have to be covered. If product i stocks out and j is fully available, constraints (5.24) ensures that substitution does not exceed its estimate.

In this case, no substitution from j to i can be observed (5.25).

sij;t  D_j;t^c  F_k^j

i;t 8t 2 C_i\ F_j (5.24)

s_ji;t D 0 8t 2 C_i\ F_j (5.25)

If both products stock out (t 2 C_i\ C_j), the demand and substitution estimates are calculated based on the first stockout time k_t in constraints (5.26) and (5.27).

s_ji;t  D_i;t^c  F_kⁱ

t 8t 2 C_i\ C_j (5.26)

sij;t  D_j;t^c  F_k^j

t 8t 2 C_i\ C_j (5.27)

Note that information on substitution behavior is limited by the historical inventory level and lowering Bi;tdoes not result in larger substitution quantities.

The complete LP model for all possible cases with censored demand can be summarized as follows:

68 5 Data-Driven Order Policies with Censored Demand and Substitution in Retailing

No stockouts:

s_ij;t D 0 8t 2 F_i \ F_jI i D 1; 2I j D 1; 2I i ¤ j (5.31) Both products are out-of-stock:

sji;t  D^c_i;t F_kⁱ

t 8t 2 C_i\ C_jI i D 1; 2I j D 1; 2I i ¤ j (5.32) All observations:

si;tC s_ji;t  B_i;t 8t 2 T I i D 1; 2I j D 1; 2I i ¤ j (5.33) si;tC s_ij;t  F_kⁱ

j;t 8t 2 T I i D 1; 2I j D 1; 2I i ¤ j (5.34)

s_i;t; s_j;t; s_ij;t; s_ji;t; F_kⁱ

t; F_k^j

t; B_i;t; B_j;t  0 (5.35)

ai; bii; bji; aj; bjj; bij2 < (5.36)