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## Short Assignment Requirements

I need to model the following question in pdf which will be on integer programming I need to clearly identify decision variables, objective function and set of constraints related to that. Finally I'll need to solve the model by using excel solver.

## Assignment Description

IE 342 - FALL 2016

Assignment

(Due: January 13, 2017 at 11:55pm)

You can work in groups of 2 or 3 people (not more than 3)

An electronics manufacturing company collects its outdated/end-of-life products from its customers in order recycle them. Currently, the company is focusing on two products and wants to design the collection plan for these two products. After these two products are collected from the customers, they are transported to recycling facilities by collection vehicles. The manufacturer has two collection vehicles used in this collection activity. Each vehicle collects one type of product from the customers and delivers them to the relevant recycling facility. The manufacturer has two recycling facilities, one for each product. Vehicle 1 collects product 1 from the customers, and delivers them to the Recycling Facility 1, and similarly, Vehicle 2 collects product 2 from the customers and delivers them to the Recycling Facility 2. Both vehicles start their travel from the depot, and stop at a recycling facility. Because of the small size of the products, you can assume that vehicles re uncapacitated. The company’s goal is to collect the products from the customers and deliver them to the recycling facilities with minimum total transportation cost. Note that a customer may have both products to be collected and recycled. In order to decrease the total transportation cost, one vehicle may “help” the other vehicle by collecting its products from the customers and drop them off at another customer location in order for the other vehicle to collect it.  Hence, vehicle i does not have to visit a certain customer if that customer does not have any product i to be collected or those products are collected by the other vehicle and dropped at another customer location.

You are given the following notation in order to solve the company’s problem:

N: number of customers

S = {1, 2, …, N}: set of customers

0: node representing the depot

N+1: node representing the recycling facility for the first product

N+2: node representing the recycling facility for the second product

Using the notation above, the problem can be defined on a graph G = (V,E) where V = {0,1,2,…,N, N+1, N+2} is the set of nodes and E is the set of edges. Assume that the graph is a complete graph, that is, it is possible to go from any node i to any other node j directly. The cost of going from node i to node j is denoted by cij. Customer k (k ϵ S) has 𝑑𝑘𝑖 units of product i to be collected.

In order to further explain the problem, consider the following instance with two customers (nodes 1 and 2).  Nodes 3 and 4 are the recycling facilities. Node 0 is the depot. The amount of products 1 and 2 to be collected from each customer are given next to the node of the customer. The travel costs are given on each edge.

In this example, a feasible solution is the one where the first vehicle follows the path 0-1-2-3 and the second vehicle follows the path 0-1-2-4. In this case both vehicles collect their own products and the total cost of the solution is 540 (= 260+280). However, a better solution may be obtained if the first vehicle follows the same path and collects the second product from the first customer and drops them off at the second customer, and the second vehicle follows the path 0-2-4. In this case, the total cost is going to be 480 (=260+220).

(a)    (80 points) Develop a mathematical model in order to determine the collection schedule with minimum cost.  You have to develop a general model using the notation provided above. I am not asking you to solve the instance above.

(b)    (20 points) Using the model developed in part (a) and a solver (GAMS, CPLEX, Excel Solver, etc.), solve the following small instance of the problem. In this instance, there are 6 customers, the x and y coordinates of the nodes are given. Assume that the travel cost between a pair of nodes is equal to the Euclidean distance between those nodes. The amount of each product type to be collected from each node is also given.

 Amount to be Collected x - coordinate y - coordinate Product 1 Product 2 Depot 0 0 Customer 1 5 40 7 6 Customer 2 13 46 5 0 Customer 3 14 58 4 3 Customer 4 60 82 0 6 Customer 5 19 81 3 2 Customer 6 15 65 3 0 Recycling Facility 1 22 78 Recycling Facility 2 55 90

Please provide both the code you developed and the optimal solution you found. Do not find the optimal solution manually.

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