Econ 172A Fall 2007 Midterm Solutions and Grading Notes What follows are the approximate rules used to assign points 1 21 points total a 4 points 1 point for writing a dual with the correct number of variables and constraints 1 point for objective function 1 point for constraints 1 point for nonnegativity constraints b 3 points 2 points for checking first three constraints and one for nonnegativity if they ignored nonnegativity in the first part do not make second deduction c 9 points 3 points for drawing correct inference from non binding constraints you can give one per constraint 3 points for drawing the correct inference from positive variable one point per variable 3 points for writing down the correct system to solve for dual guess if they show their reasoning full credit here for the system consistent with their CS deductions that is if they make a mistake selecting which constraints bind then do not punish them again one point for correct answer again no double punishment if they reason correctly following an earlier slip It is possible that you ll want to give some partial credit for students who write something sensible but fail to get points according to the rules above Talk to me if you are uncertain d 5 points I want the students to check the remaining constraints and draw the right conclusion If they do this I d give three points for doing the checking and 2 points for drawing the right conclusion Other answers are possible For example if a student exhibits a feasible x that gives value greater than 3 0 4 0 is an easy to spot example then they deserve full credit 2 They need to provide a definition of variables the correct objective function and the correct constraints I would give 3 points for the variables 3 points for the objective function 4 points for the constraints notice adding non negativity is ok and 6 points for the second part Partial credit for the second part is probably possible check with me if you are not sure about how to allocate it There is no need to simplify and it is ok to have a constant in the expression On this question I am happy if students come up with alternative explanations but 3 straightforward 4 straightforward as in the previous problem 5 34 points 4 4 1 point for the reason 4 1 point for the reason 4 1 point for reason 6 2 points for recognizing this is about a particular shadow price 2 points for checking allowable range 2 points for multiplication getting the right answer 6 4 points for figuring out the cost of the new product and 2 points for knowing what to compare it to 6 same as part f There are two forms They differ in small ways In the first question x1 and x2 are interchanged The second part of the second question is slightly different One of the corners in question 3 is different leading to a change in some of the details of the subsequent questions The order of the parts in Question 4 is different In Question 5 I changed the price of cheeseburgers which has an influence on some of the answers 1 1 Consider the linear programming problem Find x1 x2 and x3 to solve P max subject to x1 x1 2x1 x1 x2 2x2 x2 x2 x 4x3 x3 x3 0 4 4 3 You must provide justifications for your answers to the questions below In particular say what you need to do to check for feasibility and the basis for your inferences in part c a Write the dual of the problem P Find y1 y2 and y3 to solve D min subject to 4y1 y1 2y1 y1 4y2 2y2 y2 y2 y 3y3 y3 y3 1 1 4 0 b Verify that x1 x2 x3 2 1 0 is feasible for P Plainly x 0 The first and third constraints are binding The second one holds with slack equal to 1 c Assuming that 2 1 0 is a solution to P use Complementary Slackness to determine a candidate solution to the dual Since x1 and x2 are positive the first two dual constraints must bind Since the second primal constraint is not binding y2 0 Hence the solution to the dual must satisfy y1 y3 1 and 2y1 y3 1 so y1 0 and y3 1 and y2 0 is the candidate solution to the dual d Is 2 1 0 a solution to P Explain No While these values are non negative they fail to satisfy the third constraint of the dual Since the guess for the dual is not feasible the 2 1 0 cannot solve the primal 2 2 Convex Pizza is a producer of frozen pizza products The company makes a net income of 1 00 for each regular pizza and 1 50 for each deluxe pizza produced The firm currently has 150 pounds of dough mix and 50 pounds of topping mix Each regular pizza uses 1 pound of dough mix and 4 ounces 16 ounces 1 pound of topping mix Each deluxe pizza uses 1 pound of dough mix and 8 ounces of topping mix Based on the past demand per week Convex can sell at least 50 regular pizzas and at least 25 deluxe pizzas The problem is to determine the number of regular and deluxe pizzas the company should make to maximize net income Formulate this problem as an LP problem Your formulation should include a definition of the variables in words Let x1 and x2 be the number of regular and deluxe pizzas produced then the LP formulation is max subject to x1 x1 25x1 x1 1 5x2 x2 5x2 x2 150 50 50 25 The first constraint describes the constraint about dough mixing and the second constraint describes the resource constraint about topping The last two constraints interpret the problem as stating that you must produce a minimum of so many regular and deluxe pizzas If unused topping mix is worth 5 per pound then the objective function becomes x1 1 5x2 5 25x1 5x2 875x1 1 25x2 On the other form I asked what happens if unused dough mix was worth 75 per pound This makes the objective function 25x1 75x2 3 2 6 b Ab A b A bb b A b A b b A b b A b A b A A A A A A 2 2 3 3 3 3 3 The following questions relate to the triangle above and its interior which I call S You should think of S as the feasible set of a linear programming problem Remember any number of choices from zero to four can be correct As described on the answer sheet you receive credit for each correct choice you select and for each incorrect choice you do not select a The triangle above and its interior is described by which of the following sets of linear inequalities 3x1 x1 i 2x1 5x2 x2 x2 x2 24 0 2 0 ii 3x1 x1 2x1 5x2 x2 x2 24 0 …