Math 462 / 562Exam 1Fall 2012
1.(25 points) An office supply store sells packages of paper for copy machines and printers. Each day they sell exactly 100 packages. Whenever they run out they receive a new shipment of x packages where x is a quantity that would have to be determined in practice (but not in this problem). The cost s of the new shipment (called the restocking cost) is composed of two parts.
1.A fixed cost of $50 per shipment no matter how many packages are in the shipment.
2.A variable cost of $2 per package for each package in the shipment.
In addition to the restocking costs there is another type of cost that the store incurs. This is the cost of keeping unsold packages of paper on hand. This cost (called the holding cost) is incurred at a rate of 20 cents per package per day. So the total cost of selling the paper is composed of the cost of restocking and the cost of holding. Let
T=the time in days between new shipments.
Find formulas for
1.(5 points) T in terms of x, but no other variables.
2.(4 points) s in terms of x, but no other variables.
3.(5 points) the total holding cost h incurred during the time between shipments in terms of x and T, but no other variables. Hint: Suppose a new shipment is received at time t=0. Find a formula for the number n of packages on hand at time t days after a new shipment arrives. Make a graph of n vs t. The area A under this graph between 0 and T is the total number of package-days that packages were on hand during the period of one shipment. h is computed as if there were A packages on hand for a single day.
4.(3 points) the total cost c1 (restocking and holding) incurred during the time period of one shipment in terms of x and T and no other variables.
5.(5 points) the average daily cost c2 (including restocking and holding) during the time period of one shipment in terms of x and T and no other variables.
6.(3 points) c2 in terms of x and no other variables.
2.Let z = + + xy and consider the problem of finding x and y that minimize z over the set S of real numbers x and y where x and y are positive.
a.(5 points) Find a set of equations that the pair of values (x, y) that minimizes z must satisfy.
b.(7 points) Solve these equations to find the values of x and y that minimize z.
c.(6 points) Suppose one is minimizing z = + + xy over S where a is a positive parameter. Repeat parts a and b to find the values of x* = x*(a) and y*=y*(a) that minimize z.
d.(7 points) Compute the sensitivity of x* with respect to a when a = 1, i.e. find S(x*,a) when a=1.
3.Let z = (cos x)(cos y) + x2y2 + 5x2 + 6y2 + 2x and consider the problem of finding x and y to minimize z.
a.(5 points) Find a set of equations that a pair of values (x, y) that minimizes z must satisfy.
b.(20 points) Suppose one makes an initial guess of xo = 0 and yo = 0 for the values of x and y that minimize z. Go through one iteration of Newton's method to find a revised estimate of the values of x and y that minimize z.
4.Let z = x4+ 2y4 and consider the problem of finding x and y that are both positive and minimize z subject to the constraint x2+ 2y2 = 1.
a.(5 points) Suppose one is going to use Lagrange multipliers to solve this problem. Find a set of three equations that a pair of values (x, y) that minimizes z along with the multiplier λ must satisfy.
b.(13 points) Solve these equations to find the pair (x, y) that minimize z along with the value of the associated multiplier λ.
c.(7 points) Suppose the constraint is x2+ 2y2 = a as well as x > 0 and y > 0. Let z = z*(a) be the minimum value of z subject to these constraints. Find when a = 1.