APPM 4120/5120, Spring 2015 HW 3

Transcription

1 APPM 4120/5120, Spring 2015 HW 3 1. (Transportation Problem) Mr. Cupid, a lonely gentleman, does not want to spend Valentine s day alone in As one of his New Year s resolutions, he intends to send roses to nine women. Being an avid procrastinator, however, he waits until the last minute and finds out that only eight flower shops in his city still have roses available. For lack of better names, let s call those flower shops F 1, F 2, F 3,..., F 8. The number of bouquets of roses available at each shop are, respectively, 4, 4, 3, 2, 2, 2, 2, and 1. Based on how well he knows each of his potential valentines, whom we re going to call V 1, V 2, V 3,..., V 9, he calculates how many bouquets he needs to send to each of them to increase his chances of going on at least one date. The numbers are, respectively, 3, 2, 2, 2, 2, 2, 2, 2, and 3. At this point, a light bulb goes off in Mr. Cupid s head, and he remembers from his Operations Research class that this is a transportation problem. But there s something missing. Ah! The costs! He calls each of the eight flower shops and asks how much it would cost to ship one bouquet of roses to the addresses of each of his nine love interests. He then compiles the following table of costs (in dollars): V 1 V 2 V 3 V 4 V 5 V 6 V 7 V 8 V 9 F F F F F F F F Table 1: Costs to Transport Roses Assume that, for each flower shop, the number of bouquets that it ships has to be equal to the number of bouquets that it has. Similarly, he needs one constraint for each valentine saying that the number of bouquets that they receive has to be equal to the number that they want (according to his estimates above). To avoid suspicion, he also decides that it s better for each flower shop to send no more than one bouquet to the same person. Solve this balanced transportation problem (in Excel) by minimizing his cost. Solution: See the spreadsheet.

2 2. (H& L, ) You need to take a trip by car to another town that you have never visited before. You are studying a map to determine the shortest path. Depending on which route you choose, there are 5 other towns that you may pass through (call them A, B, C, D, and E) on the way. The map shows the mileage along each road that directly connects two towns. These numbers are summarized in table (2). (a) Formulate this problem as a shortest path problem by drawing a network where roads represent towns, links represent roads, and numbers indicate the length of each link in miles. (b) Find the shortest path. (c) Formulate and solve an Excel model of this problem.

3 (d) If each number in the table represented your cost (in dollars) for driving your car from one town to the next, would your route give you the minimum cost route? (e) If each number in the table represented your time (in minutes) for driving your car from one town to the next, would your route give you the minimum time route? Town A B C D E Destination Origin A B C 50 D E 80 Solution: (a) Table 2: Miles Between Adjacent Towns A 70 D O B T C 50 E (b) See below for details. The shortest path from the origin to the destination is O A B E D T with a total distance of 160 miles. (c) See below for spreadsheet. (d) Yes. (e) Yes.

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5 3. Let C be the cost matrix for a transportation problem and a i and b i be the supply and demand constraints, respectively. For C = and s 1 = 20, s 2 = 40, s 3 = 80, d 1 = 10, d 2 = 20, d 3 = 10, d 4 = 40 and d 5 = 60. (a) Find a feasible solution to the following transportation problem using the Northwest Corner method. (b) Find a feasible solution to the following transportation problem using the Lowest Cost Entry method. (c) Find an optimal solution using Hungarian algorithm to the initial feasible solution found in part (a). Solution: 4. Consider the following transportation tableau and solution. From To A B C Supply M Dummy Demand ,300 (a) Is this a balanced or unbalanced transportation problem? Explain. (b) Is this solution degenerate? Explain. If it is degenerate, show how it would be put into proper form. (c) Is there a prohibited route in this problem?

6 (d) Compute the total cost of this solution. (e) What is the value of x 2B in this solution? Solution: 5. (Nonlinear model): Network Flow Problem Safety Trans is a trucking company that specializes transporting extremely valuable and extremely hazardous materials. Due to the nature of the business, the company places great importance on maintaining a clean driving safety record. This not only helps keep their reputation up but also helps keep their insurance premium down. The company is also conscious of the fact that when carrying hazardous materials, the environmental consequences of even a minor accident could be disastrous. Safety Trans likes to ensure that it selects routes that are least likely to result in an accident. The company is currently trying to identify the safest routes for carrying a load of hazardous materials from Los Angeles to Amarillo, Texas. The following network summarizes the routes under consideration. The numbers on each arc represent the probability of having an accident on each potential leg of the journey. Set up and solve this problem (using Excel or Matlab) using the data in figure (1). You may want to use the fact the probability of something happening is 1 minus the probability of it not happening. The constraints will be linear. Note: Let s say, hypothetically, you wanted to make an app that calculated the safest path from one intersection of a city to another, this would be a good initial model. Figure 1: Network of danger

7 Solution:

8 students 1. A network analysis of Ms. Pac Man In 1981 Ms. Pac Man was released forever giving people like your instructor a reason to get up in the morning. You will apply a shortest-path algorithm from class and model the board presented in figure (2). In this figure, the small yellow boxes are pills Ms. Pac Man eats for points and the 4 super pills allow Ms. Pac Man to eat the ghosts. The 4 tunnels on the left and right edges, wrap the board. For example, if Ms. Pac Man enters the tunnel in the top left she will come out on the top right. Figure 2: Initial Ms. Pac Man Board (a) First, let s consider minimizing the time it takes to eat all of the pills. For our model assume it takes 1 second to move from one pill to the next where they are adjacent. Also, we need to assume the connection between pills where there is blank space has a certain

9 cost. Assume the time it takes to move between pills is the same as the distance between them and that we can only visit pills connected directly. Find the shortest time it takes Ms. Pac Man to clear the board when she begins at node (13, 7) assuming no ghosts. Do this by using Excel or Matlab. On the website a zip file for Dijkstra s algorithm will be posted if you choose to use this. (b) Now, let s add an element of danger. Let us assume there is a cost associated with each edge that is based on the euclidean distance at that node to a ghost. The cost then is given by 4 cost source,target = (X source X k ) 2 + (X target X k ) k=1 where the index runs from 1 to 4 because there are 4 ghosts. Here X is represents a node. Assume the four ghosts are at nodes (1, 3), (26, 2), (21, 21) and (9, 27). Find the safest path to complete the board. What is the cost? Solution: Representation of Starting Ms. Pac Man board