Homework 1 Solutions

# Homework 1 Solutions - Math 171A Mathematical Programming...

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Math 171A: Mathematical Programming Instructor: Philip E. Gill Winter Quarter 2009 Homework Assignment #1 Due Friday January 16, 2009 I know that you are all aware of the importance of doing the homework assignments. This is the best way to keep up with the class and do well in the midterm and ﬁnal examinations. Unfortunately, our TA (Joey Reed) will not have the time to grade every exercise. Instead, he will grade two or three exercises (including at least one Matlab exercise) and give a ﬁxed score for every other exercise attempted. The starred exercises require the use of Matlab . Remember that it is necessary to do all the Matlab assignments to obtain credit for the class. Exercise 1.1. In the juice mixture problem considered in class, x represents gallons of cranapple, y gallons of appleberry. (a) How many gallons of each juice mixture are represented by each corner point? Evaluate the proﬁt formula 3 x + 4 y at each corner point of the feasible region. From the discussion in class, the constraints are 3 x + 2 y 200 x + 2 y 100 x 0 y 0 . The feasible region is pictured below. y x 20 30 40 50 60 70 80 90 100 110 120 100 20 30 40 50 60 70 80 90 110 120 feasible region D C B A

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2 Mathematics 171A The vertices are A (0 , 0) , B (66 2 3 , 0) , C (50 , 25) , and D (0 , 50) . We ﬁnd P ( A ) = 0 , P ( B ) = 200 , P ( C ) = 250 , and P ( D ) = 200 , where P denotes the proﬁt at the point ( x,y ) . (b) Which corner point represents the maximum proﬁt for each of the following proﬁt formulas: (i) proﬁt = 3 x + 4 y (the deﬁnition used in class); From part (a) , the point C gives the maximum. (ii) proﬁt = 2 x + 5 y ; In this case P ( A ) = 0 , P ( B ) = 133 1 3 , P ( C ) = 225 , and P ( D ) = 250 , whence D gives the maximum. (iii) proﬁt = 5 x + 3 y . For this part, P ( A ) = 0 , P ( B ) = 333 1 3 , P ( C ) = 325 , and P ( D ) = 150 , from which we conclude that B is the maximizer with associated maximum 333 1 3 . Exercise 1.2. Consider the following six constraints in two variables: (1) x 1 +2 x 2 6; (2) x 1 - x 2 2; (3) x 2 1; (4) x 1 - x 2 4; (5) x 1 0; and (6) x 2 0. (a) Deﬁne the matrix A and vector b that express these constraints in the form Ax b . Inequalities (1), (3), and (4) are multiplied by - 1 giving A = - 1 - 2 1 - 1 0 - 1 - 1 1 1 0 0 1 and b = - 6 2 - 1 - 4 0 0 . (b) Draw the feasible region deﬁned by the six constraints. Outline the feasible region on your graph and label each of the vertices.
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## This note was uploaded on 10/23/2010 for the course MATH 171a taught by Professor Staff during the Winter '08 term at UCSD.

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Homework 1 Solutions - Math 171A Mathematical Programming...

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