Mathematical Optimization: Models, Methods and Applications

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Part 1

General about part 1

The purpose with this part is to analyze a Single-Sourcing Problem (SSP). A Single-Sourcing Problem of course both has benefits and risks, but I will discuss that furthermore through the assignment. During the assignment I will try to discuss and comment on everything that I do. My code and the answers I receive from www.neos-server.org can be seen in my appendices.

(i)

In the first question in part 1, I am asked to solve the SSP using the data in Figure 1. We have 4 facilities and 30 customers. In Figure 1 the demand of each customer is also given, and of course I will have to satisfy this. Therefore this will become one of my constraints.  It is also known that each facility has a capacity, and of course this will become a constraint as well. Because it is a SSP problem, we are also given the information that each customer has to be served by exactly one facility. When a facility delivers one unit to a customer it faces a cost. The purpose with the first question is to minimize the cost that the facility faces delivering the units. I will now show what the problem looks like:

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Now I have formulated the problem, and I will now use a Mixed Integer Linear Programming solver from www.neos-server.org. As I mentioned earlier my code and the whole answer from the website will be in my appendices.

With the constraints from the first question I get a minimum cost of: [pic 7]

(ii)

In the second question a new constraint has been given. It says that each facility must satisfy at least   of the customers. Therefore our problem will now like this:[pic 8]

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This is the new problem. As you can see we got a new constraint, which says that each facility must satisfy at least 6 customers. There are 30 customers, and therefore 6 is the 20%. With the new constraint I get a minimum total cost of:  [pic 15]

As we can see, the minimum total cost is a little bit higher in question 2 than in question 1. If we take a look at the appendices in the part about 1.(i), then we can see on the screenshot from the website, that facility 4 only had 5 customers. That was the optimal solution in that case, so therefore if we add another constraint that says every facility must have at least 6 customers, then of course the minimum total cost will be higher.