IENG 2201- Assignment Problem in Linear Programming

Assignment Task

For this assignment, you will be given a series of scenarios explained in writing, and asked to translate them into a mathematical model, both as written equations, and in Excel. You will be asked to find at least 1 VALID solution to each model, but DO NOT try to solve for the optimal solution (unless you really want to). You can also guess and test a few solutions if you wish to get a better understanding of your model.

1. A truck driver is going to be driving across country from point 1 to point 25 on the following network. They want to take the shortest route possible. The arcs are all one way with distance measured in Km.

1. Formulate this problem as a linear network model. Write out the equations of the objective function and all constraints. Remember to define your decision variables and parameters and explain what the objective function and each constraint is meant to do.

2. Represent this linear model on an Excel spreadsheet.

3. Using the Excel model, guess and test a valid solution (i.e., doesn’t break any constraints). You do not need to find the optimal solution.

4. Interpret your solution. What is the route? How long is it? Write it out and highlight the path on a drawing of the network. Do you think it is close to the optimum?

2. Craig’s Canadian Cargo Company recently performed an audit on the inventory of all their warehouses. To their surprise, they found that some warehouses had far too much product and some had far too little. They want to fix this problem by redistributing product across the network efficiently as possible. They’ve provided a matrix with the distance between all their warehouses below, along with the surplus or deficit of inventory each is currently experiencing. Due to limits on the trucks that will be carrying the product, the company doesn’t want to send any more than 80,000 product between any two warehouses.

1. Formulate this problem as a linear network model. Write out the equations of the objective function and all constraints. Remember to define your decision variables and parameters and explain what the objective function and each constraint is meant to do.

2. Represent this linear model on an Excel spreadsheet.

3. Using the Excel model, guess and test a valid solution (i.e., doesn’t break any constraints). You do not need to find the optimal solution.

4. Interpret your solution. How much is sent from each warehouse to each other warehouse?

5. Draw a network diagram of the solution and indicate how much is being sent along each arc and in which direction. You don’t need to draw arcs that have zero flow along them.

3. A cargo train company is looking to move coal from mining centers (green nodes) to manufacturing centers (orange nodes) through their distribution network. The quantity of coal is represented in Tonnage and different rail lines will have different costs per Tonnage associated with using them (From the diagram assume 4 = $4000/Ton, 5 = $5000/Ton, etc.). Each line has a maximum capacity of Tonnage they can handle. Some lines can handle trains traveling in either direction, but others can only handle a one-way flow of trains. Additionally, several lines have been granted special contracts that guarantee they will see at least a certain Tonnage of coal pass through them. It doesn’t matter which way the coal travels on the line, it just must go through that line at some point. For travel between nodes 4 and 11, for example, 13 is the cost and [5,15] means that the minimum allowable flow must be 5 and the maximum flow through that arc would be 15.

4. A driver has been given a series of deliveries they need to make today. They were given the coordinates of each delivery site and told that once they make all the deliveries they are done for the day. Wanting to finish as quickly as possible so they can go home to catch the big game, they want to find the shortest possible path they could take that, starting from the depot, would visit all delivery sites and then return them back to the depot.

1. Calculate the distances between each site. Assume that Euclidean distances can be used for the distance between each delivery location.

2. Formulate this problem as a linear network model. Write out the equations of the objective function and all constraints. Remember to define your decision variables and parameters and explain what the objective function and each constraint is meant to do.

3. Represent this linear model on an Excel spreadsheet

4. Using the Excel model, guess and test a valid solution (i.e., doesn’t break any constraints). You do not need to find the optimal solution.

5. Using the coordinates as a rough guideline, place the depot and each delivery point, then draw your path.

6. Interpret your solution. What is the delivery sequence? How long is the path?

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