2- Integer Programming Problem (20 points) Printed circuit boards that are commonly used for many electronic devices often have a set of holes that are used for mounting computer chips. The position of these holes is a part of the design and is assumed to be known with certainty before the manufacturing process starts. One of the first stages of printed circuit board manufacturing involves using an automated drilling machine to drill these required set of holes. The process goes as follows: Tool of an automated drilling machine moves from an initial housing (denoted by a blue dot in the Figure) then one-by-one drills all the holes on the board (denoted by black dots in the Figure), then returns back to the initial housing. The set of holes is denoted as H = {1,2,..., Nu}, where N indicate the number of holes. The drilling tool moves according to the following rules: The drilling tool cannot travel diagonally: i.e. it can only move in x-direction or y-direction. The drilling tool can travel in x-direction twice as fast as it can move in y-direction. For example, if traveling an inch in x-direction takes t amount of time, traveling an inch in y- direction would take 2t amount of time. Location of the initial housing Locations where a drilling operation is required y-direction x-direction Figure: Quick Schematic for the Operation (locations of the holes are provided to illustrate the process and does not represent the actual location of the holes) Assume that you know the locations of the holes. For each hole i, the terms Xi and Y indicate the distance to the initial housing location. For instance, for hole 1, X; = 3in and Y; = lin would indicate the drilling tool would need to travel 3in in x-direction and lin in y- direction to go from initial housing to the location of the first hole. As a manufacturing engineer, your objective is to minimize the time it takes for one of the printed circuit boards to be drilled. Define the necessary variables and additional parameters as required. If you introduce additional parameters, make sure you define them as a function of Xi and Y. Formulate the objective function and constraints clearly. 2- Integer Programming Problem (20 points) Printed circuit boards that are commonly used for many electronic devices often have a set of holes that are used for mounting computer chips. The position of these holes is a part of the design and is assumed to be known with certainty before the manufacturing process starts. One of the first stages of printed circuit board manufacturing involves using an automated drilling machine to drill these required set of holes. The process goes as follows: Tool of an automated drilling machine moves from an initial housing (denoted by a blue dot in the Figure) then one-by-one drills all the holes on the board (denoted by black dots in the Figure), then returns back to the initial housing. The set of holes is denoted as H = {1,2,..., Nu}, where N indicate the number of holes. The drilling tool moves according to the following rules: The drilling tool cannot travel diagonally: i.e. it can only move in x-direction or y-direction. The drilling tool can travel in x-direction twice as fast as it can move in y-direction. For example, if traveling an inch in x-direction takes t amount of time, traveling an inch in y- direction would take 2t amount of time. Location of the initial housing Locations where a drilling operation is required y-direction x-direction Figure: Quick Schematic for the Operation (locations of the holes are provided to illustrate the process and does not represent the actual location of the holes) Assume that you know the locations of the holes. For each hole i, the terms Xi and Y indicate the distance to the initial housing location. For instance, for hole 1, X; = 3in and Y; = lin would indicate the drilling tool would need to travel 3in in x-direction and lin in y- direction to go from initial housing to the location of the first hole. As a manufacturing engineer, your objective is to minimize the time it takes for one of the printed circuit boards to be drilled. Define the necessary variables and additional parameters as required. If you introduce additional parameters, make sure you define them as a function of Xi and Y. Formulate the objective function and constraints clearly