i want CPLEX mod & dat code,the objective function $,$ constrains and sheet data are clear $in$ the picture....Solving the Vehicle Routing Problem $(VRP)$ Using CPLEX OPL and External Data

Experiment Overview

This experiment introduces you $to$ solving a Vehicle Routing Problem $(VRP)$ using CPLEX OPL. The goal $isto$ minimize the total travel cost for a fleet $of$ vehicles serving multiple customers, considering vehicle capacities and customer demands. You will also learn how $to$ integrate external data stored $inan$ Excel sheet into your OPL model.

Prerequisites

Before starting this experiment, you should have:

Familiarity with Linear Programming and Optimization.

Basic understanding $of$ CPLEX Optimization Studio.

Knowledge $of$ the $VRP$ problem structure.

Basic experience with Microsoft Excel.

Mathematical Formulation

Indices and Sets:

$i$

$,$

$j$

$in$

$$

$i,jinN$: Set $of$ nodes, where

$$

$=$

$\{$

$0$

$,$

$1$

$,$

$2$

$,$

$...$

$,$

$n$

$\}$

$N=\{0,1,2,...,n\},$ with $0$ representing the depot.

$v$

$in$

$$

$vinV$: Set $of$ vehicles.

Parameters:

$$

$i$

$j$

$C$

$ij$

: Distance $orcostof$ traveling from node

$i$

$ito$ node

$j$

$j.$

$d$

$i$

$d$

$i$

: Demand $of$ customer

$i$

$i.$

$$

$Q$: Capacity $of$ each vehicle.

$$

$M$: A sufficiently large number $(used$ for subtour elimination$).$

Decision Variables:

$x$

$i$

$j$

$v$

$in$

$\{$

$0$

$,$

$1$

$\}$

$x$

ijv

$in\{0,1\}$: Binary variable indicating whether vehicle

$v$

$v$ travels from node

$i$

$ito$ node

$j$

$j.$

$y$

$i$

$v$

$in$

$\{$

$0$

$,$

$1$

$\}$

$y$

$iv$

$in\{0,1\}$: Binary variable indicating whether customer

$i$

$iis$ served $by$ vehicle

$v$

$v.$

$u$

$i$

$u$

$i$

: Continuous variable representing the cumulative load $at$ node

$i$

$i(used$ for subtour elimination$).$

Building and Running the Model

$1-$ Combine the $.$mod and $.$dat files $in$ a CPLEX project.

$2-$ Load the Excel data into the model.

$3-$ Solve the model and analyze the results

Minimize $={\displaystyle \underset{i}{\overset{?}{}}}{\displaystyle \underset{j}{\overset{?}{}}}{\displaystyle \underset{v}{\overset{?}{}}}{C}_{ij}{x}_{\text{i}\text{j}\text{v}}$

Constraints

$1-$ Vehicle Flow Conservation: Each vehicle that enters a node must also

leave $it.$

${\displaystyle \underset{\text{j}\text{i}\text{n}\text{N}\text{:}j}{\overset{?}{}}}{x}_{\text{i}\text{j}\text{v}}-{\displaystyle \underset{\text{j}\text{i}\text{n}\text{N}\text{:}tj}{\overset{?}{}}}{x}_{\text{j}\text{i}\text{v}}=0,$AAiinN,vinV

$2-$ Customer Visit Constraints: Each customer $is$ visited exactly once $byon$

vehicle.

${\displaystyle \underset{\text{j}\text{i}\text{n}\text{N}\text{:}j}{\overset{?}{}}}{\displaystyle \underset{\text{v}\text{i}\text{n}\text{V}}{\overset{?}{}}}{x}_{\text{i}\text{j}\text{v}}=1,$AAiin$\frac{N}{0}$

$3-$ Depot Constraints: Each vehicle starts and ends $at$ the depot.

${\displaystyle \underset{\text{j}\text{i}\text{n}\frac{N}{\frac{?}{?}}(0)}{\overset{?}{}}}{x}_{0jv}=1,$AAvinV

${\displaystyle \underset{\text{j}\text{i}\text{n}\text{N}(0)}{\overset{?}{}}}{x}_{j0v}=1,$AAvinV

$4-$ Capacity Constraints: Total demand $on$ each route must not exceed the

vehicle's capacity.

${\displaystyle \underset{i}{\overset{?}{}}}{\displaystyle \underset{j}{\overset{?}{}}}{d}_i{x}_{\text{i}\text{j}\text{v}}Q,$AAvinV

$5-$ Subtour Elimination $(MTZ$ Constraints$)$: Ensure $no$ subtours exist using

cumulative demand variables.

$u_i-u_j+M*x_{\text{i}\text{j}\text{v}}M-d_i,$AAi,jin$\frac{M}{\frac{?}{?}}\{0\},$vinV,$ij$

$6-$ Binary and non$-$negativity Constraints.

$y_{iv}in\{0,1\}$

$x_{\text{i}\text{j}\text{v}}in\{0,1\}$

$u_{i}0$

Customers: Contains customer demands.

Distance Matrix: Contains pairwise distances $or$ costs.

Vehicles: Contains vehicle capacities.