A store-oriented approach to determine order packaging quantities in grocery retailing

Abstract

Instore operations in bricks-and-mortar grocery retailing account for the highest share of operational logistics costs within the internal retail supply chain. The order packaging quantity (OPQ) is regarded as one important driver of instore logistics efficiency. We define the OPQ as the number of consumer units that are bundled into one order and distribution unit for supplying the individual stores. Therefore, the OPQ corresponds to the smallest possible order size and determines the possible granularity of order sizes with an impact on instore operations costs. In this paper, we develop a cost-minimization model including instore handling and inventory carrying costs to determine OPQs. The model developed builds on inventory management theory and is based on discrete probability distributions of consumer demand. We apply the model in an industry case study with real retail data for 39 stock keeping units and 1,180 stores of a European retail company. By applying the minimal-cost OPQ for all stores, the costs considered can be reduced by 9.4 %. This paper can be considered as a first in-depth analysis of the dormant instore efficiency potential in connection with adjusted OPQs that seems to be largely untapped in retail research and practice.

Appendix

1.1 A1: Calculation of the undershoot and corresponding inventory position at the time of ordering

By assuming that the undershoot and the relevant demand in the review period $\left( D^{RP}\right)$ are stochastically independent (Tempelmeier 2011), we derive the discrete probability distribution of the undershoot with the help of the following approximate calculation with resulting probabilities that represent a non-increasing function of $u$ (Tempelmeier 2011):

$$\begin{aligned} P \lbrace U=u+1 \rbrace = \frac{ 1-P\lbrace D^{RP} \le u \rbrace }{E\lbrace D^{RP} \rbrace } \end{aligned}$$

(13)

with $u$ in the interval $[0,1,...,d^{RP, max}-1]$ resulting in probabilities of an undershoot when an order is released in the interval $[1,2,...,d^{RP, max}]$. As we assume lost sales when customer demand faces empty shelves, the undershoot cannot be larger than the order level $s$. However, the probabilities of larger undershoots should not be ignored. That is why we modify the resulting undershoot distributions by placing the mass, which is larger than $s$, right on $s$. We denote the resulting distribution as $U^{*}$ with $u^{*}$ in the interval $[1,2,...,s]$, which is now characterized by the following expression:

$$\begin{aligned} \sum _{u=1}^{s}{P\, \lbrace U^{*}=u^{*}\rbrace }=1 \end{aligned}$$

(14)

With this probability distribution of the undershoot we can easiliy calculate the probabilities of the inventory position $P(IP_O=ip_O)$ at the time of ordering:

$$\begin{aligned} IP_O = s-U^{*} \end{aligned}$$

(15)

$$\begin{aligned} P\, \lbrace IP_O = s-u^{*} \rbrace =P\,\lbrace U^{*}=u^{*} \rbrace ; \,\,\,u^{*}=[1,2,...,s] \end{aligned}$$

(16)

1.2 A2: Calculation of the distribution of the number of CUs that can be stacked directly onto the shelf

The distribution of the number of CUs that can be stacked directly onto the shelf $P\left\{ \#CU^{direct}=cu^d\right\}$ is calculated as follows:

$$\begin{aligned} P\left\{ \#CU^{direct}=cu^d\right\} = \sum _{\begin{array}{c} Min(Max(S-(ip_O+d^{LT});0);\\ q(ip_O))=cu^d \end{array}} P \left\{ Min \left( Max \left( S-\left( ip_O+d^{LT}\right) ;0\right) ;q(ip_O)\right) \right\} \end{aligned}$$

(17)

with $cu^d$ in the interval $[Min (Max(S-(s-1);0);OPQ), ..., Min(\lceil \frac{s}{OPQ}\rceil \cdot OPQ; S)]$. This distribution serves as a basis to calculate the expected number of CUs that can be put directly onto the shelf during initial shelf stocking ($E\left\{ \#CU^{direct}\right\}$).

For the cost calculations of OPQs that are greater than or equal to the order level $s$, calculation (17) –which continues to be valid– can be simplified as the order size $q$ is independent of the undershoot because in every possible case only one OPQ will be ordered $(q=OPQ)$. For these instances, we can simply convolve the probability distributions of $U$ and $D^{LT}$ and base all further calculations on the resulting distribution of physical inventory in stock instore $(PI_D)$ at the time when the order consisting of one OPQ arrives in the store (Tempelmeier 2011):

$$\begin{aligned} P\left\{ PI_D = s- pi_D\right\} = \sum _{u^{*}+d^{LT}=pi_D} P \, \left\{ U^{*}=u^{*}, D^{LT}=d^{LT} \right\} \end{aligned}$$

(18)

Again, the resulting probability distribution of $PI_D$ has to be modified analogous to the undershoot distribution by cutting the distribution for values greater than $s$ and placing the correspondent mass right on $s$. This is necessary, because the physical inventory cannot be further reduced. We denote the resulting distribution as $PI^{*}_D$ with $pi^{*}_D$ in the interval $[0,1,...,s-1]$, which is characterized by the following expression:

$$\begin{aligned} \sum _{pi_D=0}^{s-1}{P\, \lbrace PI^{*}_D=pi^{*}_D\rbrace }=1 \end{aligned}$$

(19)

In the case of $OPQ \ge s$, the probability distribution of the CUs that can be stacked directly after delivery can be calculated as follows:

$$\begin{aligned} & P\left\{ \#CU^{direct}=cu^d\right\} \\ & \quad =\sum _{\begin{array}{c}Min(Max(S-pi^{*}_D;0);\\ OPQ)=cu^d \end{array}} P \left\{ Min \left( Max (S-pi^{*}_D;0);OPQ\right) \right\} \,\,\ \text {for}\,\,OPQ \ge s \end{aligned}$$

(20)