Table A1 The return costs model Notation...

Table A1

The return costs model

Notation	Description	Equation/value
$C_{ret}$	Total yearly cost of returns. The yearly total cost of returns, $C_{ret}$ ⁠, is the sum of product handling costs, $C_{hand}$ ⁠, order picking costs, $C_{pick}$ ⁠, tied-up capital, $C_{tied}$ ⁠, inventory holding costs, $C_{inv}$ ⁠, and transportation costs, $C_{tra}$ ⁠, and depends on the distribution of delivered products, $δ_{n} (q_{k}) :$ $C_{ret} = C_{hand} + C_{tied} + C_{inv} + C_{tra} + C_{pick}$	Equation (1)
$δ_{n} (q_{k})$	$n$ th distribution where $n \in N : n \in [1, 5]$ (⁠ $n$ is an integer between 1 and 5)
$C_{hand}$	Total yearly product handling cost of returns. The product handling cost of returns is the sum of receiving the returns when they arrive at the warehouse, $c_{rec}$ ⁠, and the subsequent quality inspection and restocking, $c_{res} :$ $C_{hand} = n_{del} r (c_{rec} + c_{res})$	Equation (2)
$C_{tied}$	Total yearly tied-up capital cost for returns. The tied-up capital depends on the number of returns that are in the customers' evaluation loops, $n_{r}$ ⁠, the lead time for which the items are in the loops, $t$ ⁠, and the retailer's purchase price of the item $p :$ $C_{tied} = n_{del} r t p / 365$	Equation (3)
$C_{inv}$	Total yearly inventory holding cost for returns when they are in customers' evaluation loops. The inventory holding cost is the tied-up capital multiplied by the interest rates: $C_{inv} = C_{tied} \cdot (1 + h)$	Equation (4)
$C_{tra}$	Total yearly transportation cost for returns. The transportation cost of returns depends on the distribution of delivered products, $δ_{n} (q_{k})$ and the fraction of items in order of how they are returned. Here, we calculated two cases. Case 1 concerns is where the customer decides to keep one of the ordered items and return the rest, and Case 2 is the worst-case scenario where the customer returns all ordered items. The transportation cost further depends on the weight of the parcel (the packed order), $c_{tra} (q_{k}) .$ Seeing how the two cases did not affect the calculation to any greater extent, the calculations are solely based on the first and most realistic case $C_{tra} = {\begin{matrix} \sum_{k = 1}^{11} \frac{r δ_{n} (q_{k})}{k} \cdot (c_{tra} (q_{k}) - c_{cus}), \| Case 1 : keep one item \\ \sum_{k = 1}^{11} r δ_{n} (q_{k}) \cdot (c_{tra} (q_{k}) - c_{cus}) \| Case 2 : return all \end{matrix}$	Equation (5)
$C_{pick}$	Total yearly order-picking cost of the returns. The order-picking cost of returns depends on the number of delivered products, $n_{d e l}$ ⁠, and how the delivered products are distributed, i.e. how many orders that contain $q_{k}$ items $C_{pick} = \sum_{k = 1}^{11} \frac{n_{del} (1 - μ_{s})}{q_{k}} \cdot q_{k - 1} \cdot (\frac{c_{pick}^{order}}{q_{k}} + c_{pick}^{item})$	Equation (6)
$n_{r}$	Total number of returns received by the warehouse each year	3,118
$n_{del}$	Total number of delivered/ordered products each year, $n_{del} = \sum_{k = 1}^{11} δ_{n} (q_{k})$	11,204
$c_{rec}$	Receiving handling fee per item for incoming returns [EUR]	2.25
$c_{res}$	Handling fee per item for quality inspecting and restocking the returns [EUR]	4.50
$q_{k}$	Quantity of items in an order, where $q_{k} \in N : q_{k} \in [1,11]$ (⁠ $q_{k}$ is an integer between 1 and 11)	[1,11]
$μ_{s}$	Fraction of single-item (⁠ $q_{k} = 1$ ⁠) customer orders	0.67
$r$	Return rate, assumes any of the values of $r_{1}$ ⁠, $r_{2}$ ⁠, or $r_{3}$	–
$r_{1}$	Overall return rate used in Situation 1	0.28
$r_{2}$	Return rate for fit-related returns used in Situation 2. $r_{2} = 0.55 r_{1}$	0.15
$r_{3}$	Return rate derived from pilot test used in Situation 3. $r_{3} = (1 - 0.8) r_{2}$	0.03
$c_{pick}^{order}$	Cost per order [EUR]	1.4
$c_{pick}^{item}$	Cost per item [EUR] (includes cost per order line, €0.56)	0.94
$p$	The retailer's average purchase price of an item [EUR]	30
$t$	Lead time for which the items are in customers' evaluation loops, i.e. the time between the order having left the warehouse until it is back at the warehouse [days]	15
$h$	Inventory holding interest rate	0.25
$c_{tra} (q_{k})$	Transportation fee to pick-up points for parcels containing $q_{k}$ items [EUR] $c_{tra} (q_{k}) = {\begin{matrix} 12.3, \| 1 \leq q_{k} \leq 2 \\ 14.2, \| 3 \leq q_{k} \leq 4 \\ 16.2, \| 5 \leq q_{k} \leq 6 \\ 18.7, \| 7 \leq q_{k} \leq 8 \\ 21.6, \| 9 \leq q_{k} \leq 10 \end{matrix}$	–
$c_{cus}$	Customer's cost to return, paid to the retailer [EUR]	4.9
$δ_{n} (q_{k})$	$n$ th distribution where $n \in N : n \in [1, 5]$ (⁠ $n$ is an integer between 1 and 5)	–
$δ_{1} (q_{k})$	Customer order distribution based on $μ_{s}$ for single-item orders (⁠ $q_{k} = 1$ ⁠), and $1 - μ_{s}$ for multi-orders evenly distributed on orders for $1 < q_{k} \leq 10$ $δ_{1} (q_{k}) = {\begin{matrix} n_{del} μ_{s}, \| q_{k} = 1 \\ n_{del} (1 - μ_{s}) / 10, \| 1 < q_{k} \leq 10 \end{matrix}$	–
$δ_{2} (q_{k})$	Customer order distribution based on $μ_{s}$ for single-item orders and based $1 - μ_{s}$ for multi-orders (⁠ $2 \leq q_{k} \leq 10$ ⁠) exponentially declining by $2$ for each $k$ in $q_{k}$ $δ_{2} (q_{k}) = {\begin{matrix} n_{d e l} μ_{s}, \| q_{k} = 1 \\ n_{d e l} (1 - μ_{s}) / 2, \| q_{k} = 2 \\ δ_{2} (q_{k - 1}) / 2, \| 3 \leq q_{k} \leq 10 \end{matrix}$	–
$δ_{3} (q_{k})$	Customer order distribution based on $μ_{s}$ for single-item orders and based $1 - μ_{s}$ for multi-orders (⁠ $2 \leq q_{k} \leq 10$ ⁠) exponentially declining by $3$ for each $k$ in $q_{k}$ $δ_{3} (q_{k}) = {\begin{matrix} n_{del} μ_{s}, \| q_{k} = 1 \\ n_{del} (1 - μ_{s}) / 3, \| q_{k} = 2 \\ δ_{3} (q_{k - 1}) / 3, \| 3 \leq q_{k} \leq 10 \end{matrix}$	–
$δ_{4} (q_{k})$	Customer order distribution based on $μ_{s}$ for single-item orders and based $1 - μ_{s}$ for multi-orders (⁠ $2 \leq q_{k} \leq 10$ ⁠) exponentially declining by $4$ for each $k$ in $q_{k}$ $δ_{4} (q_{k}) = {\begin{matrix} n_{del} μ_{s}, \| q_{k} = 1 \\ n_{del} (1 - μ_{s}) / 4, \| q_{k} = 2 \\ δ_{4} (q_{k - 1}) / 4, \| 3 \leq q_{k} \leq 10 \end{matrix}$	–
$δ_{5} (q_{k})$	Customer order distribution based on $μ_{s}$ for single-item orders and based $1 - μ_{s}$ for multi-orders (⁠ $2 \leq q_{k} \leq 10$ ⁠) exponentially declining by $5$ for each $k$ in $q_{k}$ $δ_{5} (q_{k}) = {\begin{matrix} n_{del} μ_{s}, \| q_{k} = 1 \\ n_{del} (1 - μ_{s}) / 5, \| q_{k} = 2 \\ δ_{5} (q_{k - 1}) / 5, \| 3 \leq q_{k} \leq 10 \end{matrix}$	–

Notation	Description	Equation/value
$C_{ret}$	Total yearly cost of returns. The yearly total cost of returns, $C_{ret}$ , is the sum of product handling costs, $C_{hand}$ , order picking costs, $C_{pick}$ , tied-up capital, $C_{tied}$ , inventory holding costs, $C_{inv}$ , and transportation costs, $C_{tra}$ , and depends on the distribution of delivered products, $δ_{n} (q_{k}) :$ $C_{ret} = C_{hand} + C_{tied} + C_{inv} + C_{tra} + C_{pick}$	Equation (1)
$δ_{n} (q_{k})$	$n$ th distribution where $n \in N : n \in [1, 5]$ ( $n$ is an integer between 1 and 5)
$C_{hand}$	Total yearly product handling cost of returns. The product handling cost of returns is the sum of receiving the returns when they arrive at the warehouse, $c_{rec}$ , and the subsequent quality inspection and restocking, $c_{res} :$ $C_{hand} = n_{del} r (c_{rec} + c_{res})$	Equation (2)
$C_{tied}$	Total yearly tied-up capital cost for returns. The tied-up capital depends on the number of returns that are in the customers' evaluation loops, $n_{r}$ , the lead time for which the items are in the loops, $t$ , and the retailer's purchase price of the item $p :$ $C_{tied} = n_{del} r t p / 365$	Equation (3)
$C_{inv}$	Total yearly inventory holding cost for returns when they are in customers' evaluation loops. The inventory holding cost is the tied-up capital multiplied by the interest rates: $C_{inv} = C_{tied} \cdot (1 + h)$	Equation (4)
$C_{tra}$	Total yearly transportation cost for returns. The transportation cost of returns depends on the distribution of delivered products, $δ_{n} (q_{k})$ and the fraction of items in order of how they are returned. Here, we calculated two cases. Case 1 concerns is where the customer decides to keep one of the ordered items and return the rest, and Case 2 is the worst-case scenario where the customer returns all ordered items. The transportation cost further depends on the weight of the parcel (the packed order), $c_{tra} (q_{k}) .$ Seeing how the two cases did not affect the calculation to any greater extent, the calculations are solely based on the first and most realistic case $C_{tra} = {\begin{matrix} \sum_{k = 1}^{11} \frac{r δ_{n} (q_{k})}{k} \cdot (c_{tra} (q_{k}) - c_{cus}), \| Case 1 : keep one item \\ \sum_{k = 1}^{11} r δ_{n} (q_{k}) \cdot (c_{tra} (q_{k}) - c_{cus}) \| Case 2 : return all \end{matrix}$	Equation (5)
$C_{pick}$	Total yearly order-picking cost of the returns. The order-picking cost of returns depends on the number of delivered products, $n_{d e l}$ , and how the delivered products are distributed, i.e. how many orders that contain $q_{k}$ items $C_{pick} = \sum_{k = 1}^{11} \frac{n_{del} (1 - μ_{s})}{q_{k}} \cdot q_{k - 1} \cdot (\frac{c_{pick}^{order}}{q_{k}} + c_{pick}^{item})$	Equation (6)
$n_{r}$	Total number of returns received by the warehouse each year	3,118
$n_{del}$	Total number of delivered/ordered products each year, $n_{del} = \sum_{k = 1}^{11} δ_{n} (q_{k})$	11,204
$c_{rec}$	Receiving handling fee per item for incoming returns [EUR]	2.25
$c_{res}$	Handling fee per item for quality inspecting and restocking the returns [EUR]	4.50
$q_{k}$	Quantity of items in an order, where $q_{k} \in N : q_{k} \in [1,11]$ ( $q_{k}$ is an integer between 1 and 11)	[1,11]
$μ_{s}$	Fraction of single-item ( $q_{k} = 1$ ) customer orders	0.67
$r$	Return rate, assumes any of the values of $r_{1}$ , $r_{2}$ , or $r_{3}$	–
$r_{1}$	Overall return rate used in Situation 1	0.28
$r_{2}$	Return rate for fit-related returns used in Situation 2. $r_{2} = 0.55 r_{1}$	0.15
$r_{3}$	Return rate derived from pilot test used in Situation 3. $r_{3} = (1 - 0.8) r_{2}$	0.03
$c_{pick}^{order}$	Cost per order [EUR]	1.4
$c_{pick}^{item}$	Cost per item [EUR] (includes cost per order line, €0.56)	0.94
$p$	The retailer's average purchase price of an item [EUR]	30
$t$	Lead time for which the items are in customers' evaluation loops, i.e. the time between the order having left the warehouse until it is back at the warehouse [days]	15
$h$	Inventory holding interest rate	0.25
$c_{tra} (q_{k})$	Transportation fee to pick-up points for parcels containing $q_{k}$ items [EUR] $c_{tra} (q_{k}) = {\begin{matrix} 12.3, \| 1 \leq q_{k} \leq 2 \\ 14.2, \| 3 \leq q_{k} \leq 4 \\ 16.2, \| 5 \leq q_{k} \leq 6 \\ 18.7, \| 7 \leq q_{k} \leq 8 \\ 21.6, \| 9 \leq q_{k} \leq 10 \end{matrix}$	–
$c_{cus}$	Customer's cost to return, paid to the retailer [EUR]	4.9
$δ_{n} (q_{k})$	$n$ th distribution where $n \in N : n \in [1, 5]$ ( $n$ is an integer between 1 and 5)	–
$δ_{1} (q_{k})$	Customer order distribution based on $μ_{s}$ for single-item orders ( $q_{k} = 1$ ), and $1 - μ_{s}$ for multi-orders evenly distributed on orders for $1 < q_{k} \leq 10$ $δ_{1} (q_{k}) = {\begin{matrix} n_{del} μ_{s}, \| q_{k} = 1 \\ n_{del} (1 - μ_{s}) / 10, \| 1 < q_{k} \leq 10 \end{matrix}$	–
$δ_{2} (q_{k})$	Customer order distribution based on $μ_{s}$ for single-item orders and based $1 - μ_{s}$ for multi-orders ( $2 \leq q_{k} \leq 10$ ) exponentially declining by $2$ for each $k$ in $q_{k}$ $δ_{2} (q_{k}) = {\begin{matrix} n_{d e l} μ_{s}, \| q_{k} = 1 \\ n_{d e l} (1 - μ_{s}) / 2, \| q_{k} = 2 \\ δ_{2} (q_{k - 1}) / 2, \| 3 \leq q_{k} \leq 10 \end{matrix}$	–
$δ_{3} (q_{k})$	Customer order distribution based on $μ_{s}$ for single-item orders and based $1 - μ_{s}$ for multi-orders ( $2 \leq q_{k} \leq 10$ ) exponentially declining by $3$ for each $k$ in $q_{k}$ $δ_{3} (q_{k}) = {\begin{matrix} n_{del} μ_{s}, \| q_{k} = 1 \\ n_{del} (1 - μ_{s}) / 3, \| q_{k} = 2 \\ δ_{3} (q_{k - 1}) / 3, \| 3 \leq q_{k} \leq 10 \end{matrix}$	–
$δ_{4} (q_{k})$	Customer order distribution based on $μ_{s}$ for single-item orders and based $1 - μ_{s}$ for multi-orders ( $2 \leq q_{k} \leq 10$ ) exponentially declining by $4$ for each $k$ in $q_{k}$ $δ_{4} (q_{k}) = {\begin{matrix} n_{del} μ_{s}, \| q_{k} = 1 \\ n_{del} (1 - μ_{s}) / 4, \| q_{k} = 2 \\ δ_{4} (q_{k - 1}) / 4, \| 3 \leq q_{k} \leq 10 \end{matrix}$	–
$δ_{5} (q_{k})$	Customer order distribution based on $μ_{s}$ for single-item orders and based $1 - μ_{s}$ for multi-orders ( $2 \leq q_{k} \leq 10$ ) exponentially declining by $5$ for each $k$ in $q_{k}$ $δ_{5} (q_{k}) = {\begin{matrix} n_{del} μ_{s}, \| q_{k} = 1 \\ n_{del} (1 - μ_{s}) / 5, \| q_{k} = 2 \\ δ_{5} (q_{k - 1}) / 5, \| 3 \leq q_{k} \leq 10 \end{matrix}$	–

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