Boxes & Lights (B&L) is a private corporation, which recently joined the list of fortune 500. To become largest and most valuable corporation in the coming years, B&L studied potential areas for future investment. The research and development team identified many possible areas that will be profitable in the future. Among these areas, B&L corp. selected to study possible avenues for handling high human population in the future. One of the directions of study is to identify potential planets that can provide basic needs for humans in the future. Human beings can be deployed in such foreign planets with constant support until they establish sustainable ecosystem on the foreign planets. Once such system is successfully established, then the foreign planets can become autonomous. In fact, the indigenous resources of the foreign planet can enhance the current potential of the human beings. The organization is looking for Industrial Engineers (IEs) to help in decision-making of this project, called Wall-Z. The aim of Wall-Z project is to send spacecrafts with robots to explore space. The main objectives in Wall-Z are described as follows: 1. Design the optimal tour for the spacecrafts. 2. Identify the hub planets and the location of satellites/relay stations. Relevant data and details are provided in the following pages of this document. Since the organization is unaware of the methodologies that you will be using to design the plan, you are allowed to make reasonable assumptions for the missing data. You are expected to use the standards wherever applicable. Your goal is to provide the organization with the best possible solution.

**Prologue**

Human population on earth is rising at alarming rates. Consequently, the resources available on earth are at the verge of being drained. It is predicted that the available resources on earth soon will not be able to meet the basic human needs. The Boxes & Lights (B&L) private corporation is planning to invest on identifying possible other planets that are capable for human survival. All the activities under this direction are termed as Wall-Z project. The goal of Wall-Z is to visit the planets, study the ecosystem and environment, and estimate sustainability score for each project. The shortlisted planets based on sustainability score and travel years will be considered for possible future settlements. Wall-Z project might take hundreds of years. Thus, B&L plans to start this process soon, so that the earth will be ready for expansion, when required, in the future. The researchers at B&L have discovered around N planets that can be reached in reasonable time from earth. Also, there is a high probability to establish continuous communication between earth and these planets. However, the planets may not provide earth like atmosphere (air, land and water). In such cases, the easiness of artificially generating earth like atmosphere will be the key part of sustainability score. On the other hand, there could be indigenous species on the planet. In such cases, the co-existence will be the key part of sustainability score. The bottleneck of Wall-Z project is visiting the planets and returning to earth in shortest possible time. Furthermore, the duration of space travel and the analysis to be done on planets may not be viable with the involvement of human beings on the spacecraft. Thus, it is decided to send robots into the space craft. The primary challenge of Wall-Z is to visit the N planets with enough resources from earth. The primary challenge can be divided into the following objectives:

**Objective-1: Design the optimal tour for the spacecrafts**

The aim of this objective is to find a closed tours for the spacecrafts (starting and ending on earth) such that the maximum distance travelled by each spacecraft is minimized (the planets should be visited only once by any ship). Thus, the first objective is to find the length of the tours for the above criterion when ? ships are used, where ? = 1, … ,6. Based on the optimal length of the tours for various values of ?, the decision makers at B&L will decide on the number of ships and tours to be used in the implementation phase of the project. The coordinates of all the planets are provided in excel sheet titled Coordinates under the excel workbook DATA-xx.xlsx (where “xx” stands for your group number).

**Objective-2: Identify the hub planets and the location of satellites/relay stations**

This objective can be divided into the following sub-objectives:

(2a) In order to establish continuous communication with the planets, B&L would like to install satellites and relay stations. The aim is to build communication among all the planets including earth. Thus, a relay station on each planet must be built. In addition to that, some planets will be selected to maintain a satellite. That is, a satellite will be associated with a planet (say hub planet). This satellite can receive (or send) signals from (or to) relay station of the other planets, if the distance between the relay station and the hub planet is less than ? ??. The objective is to find the set of hub planets such that the total maintenance cost is minimized. The cost of maintenance of the satellite at each planet (CMp) is given in the excel sheet titled Satellites-1, under the workbook DATA-xx.xlsx.

(2b) In addition to that, finding the locations of the satellite and relay stations is very crucial. In this sub-objective, the focus is on identifying the location of the satellites. The idea is to keep the satellite near the hub planet, but also to minimize the total distance from the other planets. Following figure illustrates the scenario. The objective is to find the optimal location of the satellite for the following configuration of 1 hub planet and 10 neighbor planets. The data related for the following configuration is given in the excel sheet titled Satellites-2, under the workbook DATA-xx.xlsx.

(2c) Finally, the location of the relay stations on the planet can be obtained from the location of the associated satellite. The focus of this sub-objective is to identify the location of the relay station based on the location of the satellite. For example, based on the location of the satellite the nearest point on the planet can be taken as the spot for building the relay stations. For the following configuration of 1 hub planet and 10 neighbor planets, identify the location of the relay station on all the planets in the configuration. Ignore the rotations of the planets for this sub-objective.

**Deliverables:**

Objective-1:

1. Build the mathematical model, for given value of N and n. Clearly state the assumptions, notations before writing the model. Number (label) the objective function, and all the constraint of the model. At the end of the model, explain the objective and all the constraints referring to their numbers.

2. Build a corresponding software model for the above mathematical model. No need to solve the model. The model code should be error free, and you should be able to read the given data into the model.

3. Suggest a greedy with immediate neighborhood heuristic for the above objective. Show ONE iteration by hand.

Objective-2:

1. Build the mathematical model, for any TWO of the subobjectives. Clearly state the assumptions, notations before writing the model. Number (label) the objective function, and all the constraint of the models. At the end of the models, explain the objective and all the constraints referring to their numbers.

2. Build corresponding software models for the above TWO mathematical models. No need to solve the models. The model code should be error free, and you should be able to read the given data into the models.

3. Suggest any heuristic for any ONE of the sub-objectives. Show ONE iteration by hand.

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