Econ 172A, Fall 2007: Midterm Solutions and Grading NotesWhat 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; 1point for objec tive 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 theyignored nonnegativity in the first part, do not make second deduction).(c) 9 points: 3 points for drawing correct inference from non binding constraints (you cangive one per constraint); 3 p oints for drawing the correct inference from positive variable(one point per variable); 3 p oints for writing down the correct system to solve for dualguess (if they s how their reasoning, full credit here for the system consistent with theirCS deductions – that is if they make a mistake selecting which constraints bind, thendo not punish them again); one point for correct answer (again, no double punishment ifthey reason correctly following an earlier slip). It is possible that you’ll want to give somepartial credit for students who write something sensible, but fail to get points accordingto 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 rightconclusion. If they do this, I’d give three points for doing the checking and 2 pointsfor drawing the right conclusion. Other answers are possible. For example, if a studentexhibits 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 correctconstraints. I would give 3 points for the variables, 3 points for the objective function, 4points for the constraints (notice adding non-negativity is ok), and 6 points for the secondpart. Partial credit for the second part is probably possible (check with me if you are not sureabout how to allocate it). There is no need to simplify and it is ok to have a constant in theexpression. On this question, I am happy if students come up with alternative explanations,but3. straightforward4. straightforward (as in the previous problem)5. 34 points: 4, 4 (1 point for the reason), 4 (1 point for the reason), 4 (1 p oint for reason), 6 (2points for recognizing this is about a particular shadow price, 2 points for checking allowablerange, 2 points for multiplication – getting the right answer), 6 (4 points for figuring out thecost 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 x1and x2are interchanged.The second part of the second question is slightly different. One of the corners in question 3 isdifferent (leading to a change in some of the details of the subsequent questions). The order of theparts in Question 4 is different. In Question 5, I changed the price of cheeseburgers, which has aninfluence on some of the answers.11. Consider the linear programming problem:Find x1, x2and x3to solve P:max x1+ x2+ 4x3subject to x1+ 2x2+ x3≤ 42x1− x2+ x3≤ 4x1+ x2≤ 3x ≥ 0You must provide justifications for your answers to the questions below. In particular, saywhat 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, y2and y3to solve D:min 4y1+ 4y2+ 3y3subject to y1+ 2y2+ y3≥ 12y1− y2+ y3≥ 1y1+ y2≥ 4y ≥ 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 (withslack equal to 1).(c) Assuming that (2, 1, 0) is a solution to P, use Complementary Slackness to determine acandidate solution to the dual.Since x1and x2are positive, the first two dual constraints must bind. Since the secondprimal 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 solutionto 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 thedual. Since the “guess” for the dual is not feasible, the (2, 1, 0) cannot solve the primal.22. 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 has150 pounds of dough mix and 50 pounds of topping mix. Each regular pizza uses 1 poundof dough mix and 4 ounces (16 ounces = 1 pound) of topping mix. Each deluxe pizza uses1 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 todetermine the number of regular and deluxe pizzas the company should make to maximizenet income. Formulate this problem as an LP problem. Your formulation should include adefinition of the variables (in words).Let x1and x2be the number of regular and deluxe pizzas produced, then the LP formulationis:max x1+ 1.5x2subject to x1+ x2≤ 150.25x1+ .5x2≤ 50x1≥ 50x2≥ 25The first constraint describes the constraint about dough mixing and the second constraintdescribes the resource constraint about topping. The last two constraints interpret the problemas stating that you must produce a minimum of so many regular and deluxe pizzas.If unused topping mix is worth $.5 per p ound, then the objective function becomes:x1+ 1.5x2− .5(.25x1+ .5x2) = .875x1+ 1.25x2On the other form, I asked what happens if unused dough mix was worth $.75 per pound. Thismakes the objective function:.25x1+ .75x2.3(-2,6)(3, 3)(23,23)bbbbbbbbbbbbbAAAAAAAAAAAAAA3. The following questions relate to the triangle above (and its interior), which I call S. Youshould think of S as the feasible s et of a linear programming problem.Remember, any numbe r of choices (from zero to four) can be correct. As described on theanswer sheet, you receive cre dit for each correct choice you se lect and for each incorrect choiceyou do not select.(a) The triangle above (and its interior) is described by which of the follow ing sets of linearinequalities.i.3x1+ 5x2≤ 24x1− x2≤ 02x1+ x2≥ 2x2≥ 0ii.3x1+ 5x2≤ 24x1− x2≥ 02x1+ x2≥ 2iii.3x1+ 5x2≤ 24x1− x2≥ 02x1+ x2≥ 2x1≥ 0iv.3x1+ 5x2≥ 24x1− x2≤ 02x1+ x2≥ 2The