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Unformatted text preview: CO 227 - Homework assignment 1 Fall 10 Page 1 CO 227 - Fall 10 Homework assignment #1: (Due at the beginning of the class, 10:30AM, on Monday, Oct 4th) Instructions: Please show all your work and justify your answers. Answers without proper justification will not be considered. Please list who you collaborated with. Clearly define what are your decision variables and write down your final model. Total value = 100 points Question 1- Shark feeding problem (20 points) The shark tank at Captain Nemos aquarium contains a number NS of sharks. Every day, each shark requires at least enough, but not too much, of each of NN nutrients. Let l ij denote the daily minimum requirement of nutrient i (in pounds) for shark j , and let u ij denote the maximum amount of nutrient i (in pounds) that shark j can ingest per day (for all i = 1 ,...,NN , j = 1 ,...,NS ). The sharks are fed a total number of NF different foods. Let n ik denote the amount of nutrient i (in pounds) in each pound of food k given to the sharks, and let c k be the per-pound cost of food k (for all i = 1 ,...,NN , k = 1 ,...,NF ). All sharks are hand-fed, so that each shark will get exactly the food that it is supposed to. Moreover, there is a huge supply of food, so we can assume there is no limit on the maximum amount of food that can be used to feed the sharks. Formulate a linear programming problem to determine what is the minimum cost diet that satisfies all nutritional requirements. Solution: Decision variables: We need to decide how much of each kind of food is fed to a shark. For each j = 1 ,...,NS and each k = 1 ,...,NF , let x jk denote the decision variable that indicates the number of pounds of food k that we feed to shark j . Each x jk should be a non-negative real number. (Correct decision variables: 10pts) Constraints: We need to satisfy all the nutritional requirements. Each shark j needs at least l ij pounds of nutrient i and cannot take in more than u ij pounds of nutrient i . The amount of nutrient given to shark j contained in all different food we feed to it is NF k =1 n ik x jk . So for each i = 1 ,...,NN and each j = 1 ,...,NS , we have the constraint l ij NF k =1 n ik x jk u ij . (Correct constraints: 6pts. 1 pt for nonnegativity, 5 points for the above constraint) Objective function: We want to minimize the cost of the diet. The total cost of all food fed to a particular shark j is NF k =1 c jk x jk . Summing these over all sharks, we get the total cost of the diet NS j =1 NF k =1 c jk x jk . (Correct objective function: 3pts) min NS j =1 NF k =1 c jk x jk st l ij NF k =1 n ik x jk u ij , i = 1 ,...,NN,j = 1 ,...,NS x CO 227 - Homework assignment 1 Fall 10 Page 2 (Writing down the final model: 1pt) (Failing to write the correct conditions on constraints or summation indices: -1pt each) (Partial credit should be given to models with incorrect decision variables but constraints that do make some sense. However, the maximum that can be earned in this case is 7pts. Models with nomake some sense....
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- Spring '10