# Facilities Delocation in the Retail Sector: A Mixed 0-1 Nonlinear Optimization Model and Its Linear Reformulation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Description

#### 2.1. Types of Stores and Management Policies

_{j}is considered and applied in case of delocation. Notice that $\mathcal{J}={\mathcal{J}}^{F}\cup {\mathcal{J}}^{\mathrm{NF}}$.

#### 2.2. Customers and Their Behavior

- At least one store in which they have a tendency to abandon is delocated.
- All stores in which they consume are delocated.

- ${\mathcal{I}}^{\mathrm{L}}$, customers who can leave the network if some of the stores in which they consume are delocated. Three different kinds of customers can be considered here:
- -
- ${\mathcal{I}}^{\mathrm{A}}$, customers with a tendency to abandon the network in every non-fixed store where they consume. A customer in ${\mathcal{I}}^{\mathrm{A}}$ leaves the network if at least one non-fixed store in which he/she consumes is delocated.
- -
- ${\mathcal{I}}^{\mathrm{BA}}$, customers with a tendency to abandon the network in some, but not all, non-fixed stores where they consume. A customer in ${\mathcal{I}}^{\mathrm{BA}}$ leaves the network if at least one store in which he/she has a tendency to abandon is delocated. However, the client does not leave the network in case the stores where they do not have a tendency to abandon are delocated. In the latter case, their consumption in those delocated stores will be distributed among the non-delocated stores in which they still consume.
- -
- ${\mathcal{I}}^{\mathrm{NS}}$, customers consuming only in non-fixed stores and without a tendency to abandon in any of them. A customer in ${\mathcal{I}}^{\mathrm{NS}}$ leaves the network if every store in which the client consumes is delocated. However, if at least one store in which they consume remains in the network, their consumption in those delocated stores will be distributed among the non-delocated stores in which he/she consumes.

- ${\mathcal{I}}^{\mathrm{S}}$, customers who will never leave the network whether or not any of the stores in which they consume are delocated. These customers consume in at least one fixed store and they do not have a tendency to abandon the network in any of the non-fixed stores where they consume. The customer’s consumption in the delocated stores is distributed among the rest of stores where he/she consumes.

- The network consists of a set of stores all over a specific region.
- At the present time, each store is managed by one of the four different management policies considered.
- The stores are divided into two groups regarding the decisions that can be made concerning them: non-fixed and fixed stores. Fixed stores are characterized by the impossibility of delocation or modification of management policy. On the other hand, both type of decisions can be made for non-fixed stores.
- The global profit is obtained from the consumption of goods by customers in the stores. Profit margins vary depending on the type of management policy.
- Each customer consumes in a certain set of stores due to customer-store agreements.
- We can distinguish two types of customer behavior: customers with a tendency to abandon a certain store will leave the entire network if that store is delocated, customers with a tendency to stay in a certain store will remain consuming in the network in case the store is delocated. In the latter behavior the consumption in the delocated stores is distributed among the remaining stores where the customer consumes.
- The tendency to abandon or to stay is given as a parameter and calculated by using machine learning techniques for all stores (fixed or non fixed) where the customer consumes.
- This behavior is particularly notable in this company chain since it depends on agreements and not on distances.
- Store capacities are not considered since the company is the one responsible of supplying the products to any store within the network.
- In order to maintain a service level, a minimum number of stores must remain open.
- The store delocation process has a fixed cost associated to it.

## 3. First Approach: A Mixed 0-1 NonLinear Model

#### 3.1. Notation

#### 3.1.1. Sets

- $\mathcal{J}$,
- set of stores, divided in two groups: ${\mathcal{J}}^{\mathrm{F}}$ and ${\mathcal{J}}^{\mathrm{NF}}$, set of fixed and non-fixed stores, respectively.
- $\mathcal{I}$,
- set of customers. Note that only customers that consume in at least one non-fixed store are considered here, since, customers consuming only in fixed stores do not imply any change in the company’s profit. Set $\mathcal{I}$ can be partitioned in the following subsets:
- ${\mathcal{I}}^{\mathrm{L}}$, customers who may leave the network if some of the stores in which they consume are delocated:
- -
- ${\mathcal{I}}^{\mathrm{A}}$, customers with a tendency to abandon the network in every non-fixed store where they consume.
- -
- ${\mathcal{I}}^{\mathrm{BA}}$, customers with a tendency to abandon the network in some, but not all, non-fixed stores where they consume.
- -
- ${\mathcal{I}}^{\mathrm{NS}}$, customers consuming only in non-fixed stores without a tendency to abandon any of them.

- ${\mathcal{I}}^{\mathrm{S}}$, customers consuming in at least one fixed store and without a tendency to abandon any non-fixed stores.

- ${\mathcal{J}}_{i}$,
- set of stores where the customer i consumes in, $i\in \mathcal{I}$. ${\mathcal{J}}_{i}$ is divided in ${\mathcal{J}}_{i}^{\mathrm{F}}$ and ${\mathcal{J}}_{i}^{\mathrm{NF}}$, the sets of fixed and non-fixed stores, respectively.
- ${\mathcal{J}}_{i}^{\mathrm{A}}$,
- set of stores where customer i has a tendency to abandon the network (i.e., customer i leaves the network if a store in ${\mathcal{J}}_{i}^{\mathrm{A}}$ is delocated), $i\in {\mathcal{I}}^{\mathrm{A}}\cup {\mathcal{I}}^{\mathrm{BA}}$. Note: ${\mathcal{J}}_{i}^{\mathrm{A}}\subseteq {\mathcal{J}}_{i}^{\mathrm{NF}}$.
- $\mathcal{K}$,
- set of management policies, such as company ownership, the company operates the store or there is an external dealer for instance. The set of possible management policies for store j will be denoted as ${\mathcal{K}}^{j}$, $j\in \mathcal{J}.$

#### 3.1.2. Parameters

- $p,$ service level, given by the number of (fixed and non-fixed) stores that must remain open.
- ${c}_{j},$ delocation costs of any store j, $j\in {\mathcal{J}}^{\mathrm{NF}}$.
- ${e}_{jk},$ percentage of extra volume of goods consumed in store j with management policy k (if there is no profit, ${e}_{jk}=0$), $j\in \mathcal{J},k\in {\mathcal{K}}^{j}$.
- ${r}_{jk}$, profit margin to be applied to the extra volume of goods in store j with management policy k, $j\in \mathcal{J},k\in {\mathcal{K}}^{j}$.
- ${g}_{ij},$ initial amount of goods consumed by customer i in store j, $i\in \mathcal{I}$, $j\in {\mathcal{J}}_{i}$.
- ${m}_{ijk},$ unit profit obtained by customer i in store j with management policy k, $i\in \mathcal{I}$, $j\in {\mathcal{J}}_{i}$, $k\in {\mathcal{K}}^{j}$.
- ${M}_{i},$ upper bound of the volume of goods consumed by customer i, $i\in \mathcal{I}$.
- ${k}_{j}^{*},$ management policy of any fixed store $j\in {\mathcal{J}}^{\mathrm{F}}$. Note: ${k}_{j}^{*}\in {\mathcal{K}}^{j}$.

#### 3.1.3. Decision Variables

- ${\alpha}_{j}=1,$ if the (non-fixed) store j is delocated and, 0 otherwise, $j\in {\mathcal{J}}^{\mathrm{NF}}$.
- ${\gamma}_{jk}=1,$ if the (non-fixed) store j is managed with management policy k and, 0 otherwise, $j\in {\mathcal{J}}^{\mathrm{NF}},k\in {\mathcal{K}}^{j}$.
- ${\beta}_{i}=1,$ if customer i leaves the network and, 0 otherwise, $i\in {\mathcal{I}}^{\mathrm{L}}$.
- ${x}_{i},$ total amount of goods consumed by customer i in all the delocated stores (must be transferred to the non-delocated stores), $i\in \mathcal{I}\backslash {\mathcal{I}}^{\mathrm{A}}$.
- ${y}_{ij},$ increase in the consumption of goods of customer i in store j if at least one of the stores where i consumes is delocated, $i\in \mathcal{I}\backslash {\mathcal{I}}^{\mathrm{A}}$, $j\in {\mathcal{J}}_{i}$.
- ${z}_{ijk},$ increase in the consumption of goods of customer i in store j if the store is managed by policy k, $i\in \mathcal{I}\backslash {\mathcal{I}}^{\mathrm{A}}$, $j\in {\mathcal{J}}_{i}$, $k\in {\mathcal{K}}^{j}$.

#### 3.2. Mathematical Formulation

#### 3.2.1. Objective Function

#### 3.2.2. Constraints

## 4. Model Reformulation: A Mixed 0-1 Linear Optimization Model

#### 4.1. Linearization of the Nonlinear Equations

#### 4.1.1. Objective Function

#### 4.1.2. Constraints related to increase of goods consumption

- Constraints (10), defined for $i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}}$ and $j\in {\mathcal{J}}_{i}^{\mathrm{NF}}$:Constraints (10) contain a cubic term, $(1-{\alpha}_{j})(1-{\beta}_{i}){x}_{i}$. A new binary auxiliary variable ${\delta}_{ij}$ is introduced to represent the product $(1-{\alpha}_{j})(1-{\beta}_{i})$, such that, if either one of ${\alpha}_{j}$ or ${\beta}_{i}$ is equal to 1, then automatically ${\delta}_{ij}=0$. Then, ${\delta}_{ij}=1$ if customer i does not leave the network and store j is not delocated and 0 otherwiseVariables ${\delta}_{ij}$ are computed by using variables ${\alpha}_{j}$ and ${\beta}_{i}$ through the following constraints,$$\begin{array}{cc}\hfill {\delta}_{ij}+{\alpha}_{j}\u2a7d1& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}},\hfill \\ \hfill {\delta}_{ij}+{\beta}_{i}\u2a7d1& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}},\hfill \\ \hfill {\delta}_{ij}+{\alpha}_{j}+{\beta}_{i}\u2a7e1& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}}.\hfill \end{array}$$Replacing this new variable in ${x}_{i}(1-{\alpha}_{j})(1-{\beta}_{i})$, the result is ${x}_{i}{\delta}_{ij}$. The latter expression is a product of a continuous variable, ${x}_{i}$, and a $0-1$ variable ${\delta}_{ij}$, that can also be linearized using the Fortet inequalities scheme. Therefore, a new continuous variable has to be introduced, ${u}_{ij}^{2}={x}_{i}{\delta}_{ij}$, together with the following set of additional constraints:$$\begin{array}{cc}\hfill {u}_{ij}^{2}\u2a7d{x}_{i}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}},\hfill \\ \hfill {u}_{ij}^{2}\u2a7d{M}_{ij}{\delta}_{ij}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}},\hfill \\ \hfill {x}_{i}-{u}_{ij}^{2}\u2a7d{M}_{ij}(1-{\delta}_{ij})& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}},\hfill \end{array}$$
- Constraints (11), defined for $i\in {\mathcal{I}}^{\mathrm{S}}$ and $j\in {\mathcal{J}}_{i}^{\mathrm{NF}}$:These constraints contain a quadratic term, $(1-{\alpha}_{j}){x}_{i}$. By using the Fortet inequalities scheme, this product can be replaced by a new continuous variable, ${u}_{ij}^{3}={x}_{i}(1-{\alpha}_{j})$, and the following set of additional constraints:$$\begin{array}{cc}\hfill {u}_{ij}^{3}\u2a7d{x}_{i}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{S}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}},\hfill \\ \hfill {u}_{ij}^{3}\u2a7d{M}_{ij}(1-{\alpha}_{j})& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{S}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}},\hfill \\ \hfill {x}_{i}-{u}_{ij}^{3}\u2a7d{M}_{ij}{\alpha}_{j}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{S}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}},\hfill \end{array}$$
- Constraints (12), defined for $i\in {\mathcal{I}}^{\mathrm{BA}}$ and $j\in {\mathcal{J}}_{i}^{\mathrm{F}}$:These constraints contain a quadratic term, $(1-{\beta}_{i}){x}_{i}$. Yet again, the Fortet inequalities scheme can be used to linearize this quadratic term. Therefore, a new continuous variable has to be introduced, ${u}_{i}^{4}={x}_{i}(1-{\beta}_{i})$, and the following set of additional constraints:$$\begin{array}{cc}\hfill {u}_{i}^{4}\u2a7d{x}_{i}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}:{\mathcal{J}}_{i}^{\mathrm{F}}\ne \varnothing ,\hfill \\ \hfill {u}_{i}^{4}\u2a7d{M}_{ij}(1-{\beta}_{i})& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}:{\mathcal{J}}_{i}^{\mathrm{F}}\ne \varnothing ,\hfill \\ \hfill {x}_{i}-{u}_{i}^{4}\u2a7d{M}_{ij}{\beta}_{i}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}:{\mathcal{J}}_{i}^{\mathrm{F}}\ne \varnothing ,\hfill \end{array}$$
- Constraints (13), defined for $i\in {\mathcal{I}}^{\mathrm{S}}$ and $j\in {\mathcal{J}}_{i}^{\mathrm{F}}$:Once the left hand side of the constraints has been linearized, these constrains become linear.

#### 4.2. Mathematical Formulation

#### 4.2.1. Additional Auxiliary Binary Variables

- ${\delta}_{ij}=1$
- if customer $i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}}$ leaves the network or store $j\in {\mathcal{J}}_{i}^{\mathrm{NF}}$ is delocated and 0 otherwise.
- ${\sigma}_{ijk}=1$
- if customer $i\in {\mathcal{I}}^{\mathrm{L}}$ leaves the network or store $j\in {\mathcal{J}}_{i}^{\mathrm{NF}}$ is operated with management policy $k\in {\mathcal{K}}^{j}$ and 0 otherwise.

#### 4.2.2. Additional Auxiliary Continuous Variables

- ${u}_{ij{j}^{\prime}}^{1},$
- non-negative continuous variables for each customer $i\in \mathcal{I}\backslash {\mathcal{I}}^{\mathrm{A}}$ and stores $j\in {\mathcal{J}}_{i}$, ${j}^{\prime}\in {\mathcal{J}}_{i}^{\mathrm{NF}}$.
- ${u}_{ij}^{2},$
- non-negative continuous variables for each customer $i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}}$ and store $j\in {\mathcal{J}}_{i}^{\mathrm{NF}}$.
- ${u}_{ij}^{3},$
- non-negative continuous variables for each customer $i\in {\mathcal{I}}^{\mathrm{S}}$ and store $j\in {\mathcal{J}}_{i}^{\mathrm{NF}}$.
- ${u}_{i}^{4},$
- non-negative continuous variables for each customer $i\in {\mathcal{I}}^{\mathrm{BA}}:{\mathcal{J}}_{i}^{\mathrm{F}}\ne \varnothing $.

#### 4.2.3. New Linear Constraints

- Equation (2a) must be substituted by its equivalent set of linear constraints:$$\begin{array}{cc}\hfill PRS=& \sum _{i\in {\mathcal{I}}^{\mathrm{A}}\cup {\mathcal{I}}^{\mathrm{BA}}}\left(\sum _{j\in {\mathcal{J}}_{i}^{\mathrm{F}}}{m}_{ij{k}_{j}^{*}}{g}_{ij}(1-{\beta}_{i})+\sum _{j\in {\mathcal{J}}_{i}^{\mathrm{NF}}}\sum _{k\in {\mathcal{K}}^{j}}{m}_{ijk}{g}_{ij}{\sigma}_{ijk}\right)+\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \sum _{i\in {\mathcal{I}}^{\mathrm{NS}}}\sum _{j\in {\mathcal{J}}_{i}^{\mathrm{NF}}}\sum _{k\in {\mathcal{K}}^{j}}{m}_{ijk}{g}_{ij}{\sigma}_{ijk}+\sum _{i\in {\mathcal{I}}^{\mathrm{S}}}\left(\sum _{j\in {\mathcal{J}}_{i}^{\mathrm{F}}}{m}_{ij{k}_{j}^{*}}{g}_{ij}+\sum _{j\in {\mathcal{J}}_{i}^{\mathrm{NF}}}\sum _{k\in {\mathcal{K}}^{j}}{m}_{ijk}{g}_{ij}{\gamma}_{jk}\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill {\sigma}_{ijk}\u2a7d{\gamma}_{jk}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{L}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}},k\in {\mathcal{K}}^{j}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\sigma}_{ijk}\u2a7d1-{\beta}_{i}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{L}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}},k\in {\mathcal{K}}^{j}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\sigma}_{ijk}\u2a7e{\gamma}_{jk}-{\beta}_{i}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{L}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}},k\in {\mathcal{K}}^{j}\hfill \end{array}$$
- Constraints (11)–(13) must be substituted by their equivalent set of linear constraints:$$\begin{array}{cc}\hfill {u}_{ij{j}^{\prime}}^{1}\u2a7d{y}_{ij}& \phantom{\rule{2.em}{0ex}}\forall i\in \mathcal{I}\backslash {\mathcal{I}}^{\mathrm{A}},j\in {J}^{{\left(}_{\right)}}i,{j}^{\prime}\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {u}_{ij{j}^{\prime}}^{1}\u2a7d{M}_{ij}(1-{\alpha}_{{j}^{\prime}})& \phantom{\rule{2.em}{0ex}}\forall i\in \mathcal{I}\backslash {\mathcal{I}}^{\mathrm{A}},j\in {J}^{{\left(}_{\right)}}i,{j}^{\prime}\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {y}_{ij}-{u}_{ij{j}^{\prime}}^{1}\u2a7d{M}_{ij}{\alpha}_{{j}^{\prime}}& \phantom{\rule{2.em}{0ex}}\forall i\in \mathcal{I}\backslash {\mathcal{I}}^{\mathrm{A}},j\in {J}^{{\left(}_{\right)}}i,{j}^{\prime}\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill \sum _{{j}^{\prime}\in {\mathcal{J}}_{i}^{\mathrm{F}}}{g}_{i{j}^{\prime}}{y}_{ij}+\sum _{{j}^{\prime}\in {\mathcal{J}}_{i}^{\mathrm{NF}}}{g}_{i{j}^{\prime}}{u}_{ij{j}^{\prime}}^{1}={g}_{ij}{u}_{ij}^{2}\phantom{\rule{1.em}{0ex}}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill \sum _{{j}^{\prime}\in {J}^{{\left(}_{\right)}}i}{g}_{i{j}^{\prime}}{u}_{ij{j}^{\prime}}^{1}={g}_{ij}{u}_{ij}^{2}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{NS}},j\in {J}^{{\left(}_{\right)}}i\hfill \end{array}$$$$\begin{array}{cc}\hfill {\delta}_{ij}\u2a7d1-{\alpha}_{j}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\delta}_{ij}\u2a7d1-{\beta}_{i}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\delta}_{ij}\u2a7e1-{\alpha}_{j}-{\beta}_{i}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {u}_{ij}^{2}\u2a7d{x}_{i}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {u}_{ij}^{2}\u2a7d{M}_{ij}{\delta}_{ij}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {x}_{i}-{u}_{ij}^{2}\u2a7d{M}_{ij}(1-{\delta}_{ij})& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill \sum _{{j}^{\prime}\in {\mathcal{J}}_{i}^{\mathrm{F}}}{g}_{i{j}^{\prime}}{y}_{ij}+\sum _{{j}^{\prime}\in {\mathcal{J}}_{i}^{\mathrm{NF}}}{g}_{i{j}^{\prime}}{u}_{ij{j}^{\prime}}^{1}={g}_{ij}{u}_{ij}^{3}\phantom{\rule{1.em}{0ex}}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{S}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {u}_{ij}^{3}\u2a7d{x}_{i}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{S}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {u}_{ij}^{3}\u2a7d{M}_{ij}(1-{\alpha}_{j})& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{S}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {x}_{i}-{u}_{ij}^{3}\u2a7d{M}_{ij}{\alpha}_{j}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{S}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill \sum _{{j}^{\prime}\in {\mathcal{J}}_{i}^{\mathrm{F}}}{g}_{i{j}^{\prime}}{y}_{ij}+\sum _{{j}^{\prime}\in {\mathcal{J}}_{i}^{\mathrm{NF}}}{g}_{i{j}^{\prime}}{u}_{ij{j}^{\prime}}^{1}={g}_{ij}{u}_{i}^{4}\phantom{\rule{1.em}{0ex}}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}},j\in {\mathcal{J}}_{i}^{\mathrm{F}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {u}_{i}^{4}\u2a7d{x}_{i}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}:{\mathcal{J}}_{i}^{\mathrm{F}}\ne \varnothing \hfill \end{array}$$$$\begin{array}{cc}\hfill {u}_{i}^{4}\u2a7d{M}_{ij}(1-{\beta}_{i})& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}:{\mathcal{J}}_{i}^{\mathrm{F}}\ne \varnothing \hfill \end{array}$$$$\begin{array}{cc}\hfill {x}_{i}-{u}_{i}^{4}\u2a7d{M}_{ij}{\beta}_{i}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}:{\mathcal{J}}_{i}^{\mathrm{F}}\ne \varnothing \hfill \end{array}$$$$\begin{array}{cc}\hfill \sum _{{j}^{\prime}\in {\mathcal{J}}_{i}^{\mathrm{F}}}{g}_{i{j}^{\prime}}{y}_{ij}+\sum _{{j}^{\prime}\in {\mathcal{J}}_{i}^{\mathrm{NF}}}{g}_{i{j}^{\prime}}{u}_{ij{j}^{\prime}}^{1}={g}_{ij}{x}_{i}\phantom{\rule{1.em}{0ex}}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{S}},j\in {\mathcal{J}}_{i}^{\mathrm{F}}\hfill \end{array}$$
- The model can be tightened by adding new constraints that allow certain feasible solutions to be cutoff its linear relaxation without eliminating any other feasible solution from the original model:$$\begin{array}{cc}\hfill {\delta}_{ij}=\sum _{k\in {\mathcal{K}}^{j}}{\sigma}_{ijk}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\sigma}_{ijk}\u2a7d{\delta}_{ij}& \phantom{\rule{2.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}},k\in {\mathcal{K}}^{j}\hfill \end{array}$$
- Variables’ domain:$$\begin{array}{cc}\hfill {\sigma}_{ijk}\in \{0,1\}& \phantom{\rule{1.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{L}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}},k\in {\mathcal{K}}^{j}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\delta}_{ij}\in \{0,1\}& \phantom{\rule{1.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {u}_{ij{j}^{\prime}}^{1}\u2a7e0& \phantom{\rule{1.em}{0ex}}\forall i\in \mathcal{I}\backslash {\mathcal{I}}^{\mathrm{A}},j\in {J}^{{\left(}_{\right)}}i,{j}^{\prime}\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {u}_{ij}^{2}\u2a7e0& \phantom{\rule{1.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}\cup {\mathcal{I}}^{\mathrm{NS}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {u}_{ij}^{3}\u2a7e0& \phantom{\rule{1.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{S}},j\in {\mathcal{J}}_{i}^{\mathrm{NF}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {u}_{i}^{4}\u2a7e0& \phantom{\rule{1.em}{0ex}}\forall i\in {\mathcal{I}}^{\mathrm{BA}}:{\mathcal{J}}_{i}^{\mathrm{F}}\ne \varnothing \hfill \end{array}$$

## 5. Study Case Analysis

^{2}. On average, each customer consumed in 2.1 different stores. Figure 3a shows the percentage of customers in the network consuming in one store, two stores, etc. Note that almost 60% of the customers visited only one store. There were some customers who visited more than six stores, although they can be considered outliers. Figure 3b shows the percentage of customers that were served by each store. There were five stores attending a high number of customers (more than 3000), but the other 15 had a number of customers, approximately between 1000 and 3000. Table 1 reports the distribution of the customers in the different categories.

#### Extended Computational Experiment

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Laporte, G.; Nickel, S.; da Gama, F.S. (Eds.) Location Science; Springer: Berlin/Heidelberg, Germany, 2015. [Google Scholar] [CrossRef]
- Bhaumik, P. Optimal shrinking of the distribution chain: The facilities delocation decision. Int. J. Syst. Sci.
**2010**, 41, 271–280. [Google Scholar] [CrossRef] - ReVelle, C.; Murray, A.; Serra, D. Location models for ceding market share and shrinking services. Int. J. Manag. Sci.
**2007**, 35, 533–540. [Google Scholar] [CrossRef] [Green Version] - Murray, T.; Wu, X. Accessibility tradeoffs in public transit planning. J. Geogr. Syst.
**2003**, 5, 93–107. [Google Scholar] [CrossRef] - Bruno, J.; Andersen, P. Analytical methods for planning educational facilities in an era of declining enrollments. Socio-Econ. Plan. Sci.
**1982**, 16, 121–131. [Google Scholar] [CrossRef] - Morrison, P.; O’Brien, R. Bank branch closures in New Zealand: The application of a spatial interaction model. Appl. Geogr.
**2001**, 21, 301–330. [Google Scholar] [CrossRef] - Wang, Q.; Batta, R.; Bhadury, J.; Rump, C. Budget constrained location problem with opening and closing of facilities. Comput. Oper. Res.
**2003**, 30, 2047–2069. [Google Scholar] [CrossRef] - Monteiro, M.R.; Fontes, D. Locating and sizing bank-branches by opening, closing or maintaining facilities. In Proceedings of the Operations Research, Bremen, Germany, 7–9 September 2005; Springer: Berlin/Heidelberg, Germany, 2005; pp. 303–308. [Google Scholar] [CrossRef]
- Ruiz-Hernández, D.; Delgado-Gómez, D.; López-Pascual, J. Restructuring bank networks after mergers and acquisitions: A capacitated delocation model for closing and resizing branches. Comput. Oper. Res.
**2015**, 62, 316–324. [Google Scholar] [CrossRef] - Ruiz-Hernández, D.; Delgado-Gómez, D. The stochastic capacitated branch restructuring problem. Ann. Oper. Res.
**2016**, 246, 77–100. [Google Scholar] [CrossRef] - Shields, M.; Kures, M. Black out of the blue light: An analysis of Kmart store closing decisions. J. Retail. Consum. Serv.
**2007**, 14, 259–268. [Google Scholar] [CrossRef] - Yavari, M.; Mousavi-Saleh, M. Restructuring hierarchical capacitated facility location problem with extended coverage radius under uncertainty. Oper. Res.
**2019**. [Google Scholar] [CrossRef] - Fortet, R. Application de l’algebre de boole en recherche operationelle. Rev. Fr. Rech. Oper.
**1960**, 13, 205–213. [Google Scholar] [CrossRef] - Hammer, P.L.; Rudeau, S. Boolean Methods in Operations Research and Related Areas, 1st ed.; Springer: Berlin/Heidelberg, Germany, 1968; Volume 7. [Google Scholar] [CrossRef] [Green Version]
- Alonso-Ayuso, A.; Escudero, L.F.; Martín-Campo, F.J. Multiobjective optimization for aircraft conflict resolution. A metaheuristic approach. Eur. J. Oper. Res.
**2016**, 248, 691–702. [Google Scholar] [CrossRef] - Fourer, R.; Gay, D.M.; Kernighan, B.W. A Modeling Language for Mathematical Programming. Manag. Sci.
**1990**, 36, 519–554. [Google Scholar] [CrossRef] [Green Version] - Gurobi Optimization, Incorporate. Gurobi Optimizer Reference Manual. 2018. Available online: http://www.gurobi.com (accessed on 5 November 2020).

Category of Customers | ${\mathcal{I}}^{\mathbf{A}}$ | ${\mathcal{I}}^{\mathbf{BA}}$ | ${\mathcal{I}}^{\mathbf{NS}}$ | ${\mathcal{I}}^{\mathbf{S}}$ |
---|---|---|---|---|

Distribution | 63.0 | 12.5 | 19.6 | 4.9 |

Result | Value |
---|---|

Final profit (Obj. Func.) | 1262.1 |

Initial profit | 1000.0 |

Amount of lost sales (%) | 28.3 |

Churn rate (% customers) | 19.9 |

Number of delocated stores | 3 |

Store | Management Policy | Consumption | Profit | Churn | ||||
---|---|---|---|---|---|---|---|---|

ID | Fixed | Initial | Final | Initial | Final | Initial | Final | Rate (%) |

1 | No | Type D | Type D | 39.8 | 36.3 | 12.2 | 11.4 | 11.4 |

2 | No | Type D | Type D | 108.1 | 89.9 | −55.5 | −45.0 | 15.3 |

3 | No | Type D | Type D | 41.1 | 36.6 | −23.4 | −20.3 | 15.1 |

4 | No | Type D | Delocated | 65.3 | 0.0 | −56.8 | 0.0 | 85.6 |

5 | Yes | Type D | Type D | 25.3 | 21.1 | −21.2 | −16.1 | 18.4 |

6 | No | Type D | Type D | 31.9 | 27.9 | −1.4 | −1.5 | 18.4 |

7 | Yes | Type D | Type D | 59.4 | 49.9 | −36.3 | −30.0 | 9.8 |

8 | No | Type D | Type D | 18.4 | 16.6 | 2.9 | 3.5 | 15.9 |

9 | No | Type D | Type D | 38.6 | 32.2 | −24.0 | −19.1 | 18.9 |

10 | No | Type A | Type B | 131.5 | 120.4 | 394.6 | 460.0 | 9.7 |

11 | Yes | Type D | Type D | 39.5 | 30.7 | 2.0 | 2.7 | 8.8 |

12 | Yes | Type C | Type C | 8.8 | 8.0 | 31.7 | 29.5 | 15.5 |

13 | Yes | Type C | Type C | 24.3 | 21.0 | 103.5 | 87.7 | 5.7 |

14 | Yes | Type C | Type C | 26.3 | 23.7 | 100.5 | 89.3 | 5.9 |

15 | No | Type D | Delocated | 68.8 | 0.0 | −50.8 | 0.0 | 84.0 |

16 | No | Type D | Delocated | 53.3 | 0.0 | −59.4 | 0.0 | 91.8 |

17 | No | Type A | Type B | 89.5 | 80.7 | 297.9 | 308.8 | 10.6 |

18 | No | Type A | Type B | 74.6 | 69.3 | 246.6 | 264.5 | 11.6 |

19 | No | Type A | Type B | 34.0 | 31.6 | 119.6 | 120.7 | 11.6 |

20 | No | Type D | Type D | 21.4 | 20.8 | 17.3 | 16.0 | 23.1 |

Network | 1000.0 | 716.7 | 1000.00 | 1262.1 | 19.9 |

#Stores | #Custm. (in Thousands) | #Constr. | #Var. | z${}_{\mathbf{IP}}$ | z${}_{\mathbf{LP}}$ | GAP (%) | Computing Time (s) | Profit Var. (%) | Consumption Var. (%) | Churn Rate (%) | #Deloc. Stores | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

10 | (1,0,0,9) | 10–15 | 227,750 | 95,774 | 2858.3 | 4144.5 | 45.00 | 44.31 | 185.83 | −65.42 | 10.18 | 5 |

12 | (1,0,1,10) | 15–20 | 273,971 | 114,724 | 2272.7 | 3526.1 | 55.15 | 220.45 | 127.27 | −55.48 | 8.34 | 5 |

14 | (1,0,3,10) | 20–25 | 379,001 | 163,313 | 1239.6 | 1686.4 | 36.04 | 867.73 | 23.96 | −48.78 | 5.75 | 5 |

16 | (1,9,3,12) | 20–25 | 429,712 | 183,070 | 1408.3 | 2023.9 | 43.72 | 692.23 | 40.83 | −56.40 | 6.46 | 7 |

18 | (3,0,3,12) | 20–30 | 589,940 | 251,692 | 1224.6 | 1863.4 | 52.17 | 3687.44 | 22.46 | −22.48 | 1.61 | 3 |

20 | (4,0,3,13) | 20–30 | 638,280 | 271,088 | 1199.9 | 1863.2 | 55.28 | 4714.70 | 19.99 | −21.89 | 1.48 | 3 |

22 | (5,0,3,14) | 20–30 | 710,482 | 304,345 | 1233.5 | 2038.6 | 65.27 | 7664.40 | 23.35 | −20.28 | 1.29 | 3 |

24 | (6,0,3,15) | 20–30 | 804,612 | 337,805 | 1259.8 | 2036.3 | 61.64 | 7951.16 | 25.98 | −6.55 | 0.23 | 1 |

26 | (6,7,4,16) | 20–30 | 864,425 | 360,838 | 1263.0 | 2177.2 | 72.38 | 10,279.33 | 26.30 | −15.06 | 0.70 | 2 |

28 | (7,1,4,16) | 20–30 | 942,775 | 392,050 | 1265.1 | 2158.3 | 70.61 | 12,173.89 | 26.51 | 0.00 | 0.00 | 0 |

30 | (8,2,4,16) | 30–35 | 1,079,980 | 445,472 | 1246.7 | 2154.8 | 72.84 | 13,246.83 | 24.67 | 0.00 | 0.00 | 0 |

35 | (11,3,4,17) | 30–35 | 1,342,280 | 547,812 | 1224.9 | 2247.4 | 83.48 | 25,757.67 | 22.49 | 0.00 | 0.00 | 0 |

40 | (12,4,5,19) | 30–35 | 1,416,861 | 653,290 | * | 2453.8 | * | 16,693.94 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sierra-Paradinas, M.; Alonso-Ayuso, A.; Martín-Campo, F.J.; Rodríguez-Calo, F.; Lasso, E.
Facilities Delocation in the Retail Sector: A Mixed 0-1 Nonlinear Optimization Model and Its Linear Reformulation. *Mathematics* **2020**, *8*, 1986.
https://doi.org/10.3390/math8111986

**AMA Style**

Sierra-Paradinas M, Alonso-Ayuso A, Martín-Campo FJ, Rodríguez-Calo F, Lasso E.
Facilities Delocation in the Retail Sector: A Mixed 0-1 Nonlinear Optimization Model and Its Linear Reformulation. *Mathematics*. 2020; 8(11):1986.
https://doi.org/10.3390/math8111986

**Chicago/Turabian Style**

Sierra-Paradinas, María, Antonio Alonso-Ayuso, Francisco Javier Martín-Campo, Francisco Rodríguez-Calo, and Enrique Lasso.
2020. "Facilities Delocation in the Retail Sector: A Mixed 0-1 Nonlinear Optimization Model and Its Linear Reformulation" *Mathematics* 8, no. 11: 1986.
https://doi.org/10.3390/math8111986