Ryerson University
Department of Chemical Engineering
Process and Engineering Optimization - CHE 425 and CE 8210 - Midterm Test - February 25, 2013
Time allowed: 2 hours
-wn·te you r name ::ind the following code on vour test paper: CH600
P-1 (15) -
a) Find l and global optima of the following function on (i) (0, oo), (ii) (-oo, 0), (iii) (5, 10).
f (x) =x + llx
b) Determine interval(s) on which/(x) is concave, strictly concave, convex or strictly convex.
*o
P-2 (25) - 1pany manufactures .m;.w;ts r y week. ThwuapJity of:e odi.ici: prndpced d!J.cing the week is(i = 1, 2. 3).and L?• Ci = 1.2.3) js thv Wffii[jce peumi t of eaJ5rodi1ct Wlier.e:
p1 = 12 - q1 p2 = 18 -2q2 p3 = 20 -3q3
Weekly production cost is C = 7 + 2 (q1 + q2 + q3). The company's goal is to optimize its profit.
a) Setup the objective function and the constraints. Indicate the type of optintization problem.
b) In order to achieve company's goal how many of each product should be produced weekly?
c) Show that the profit is maximized.
d) Classify the type of profit function as concave, strictly concave, convex or strictly convex.
.1::.
P-3 (30) - A man!Uac. 1 · r produces difforent m1Jij ducts which al] m11st hp machi ned. .s.bed, and a · T regients p3dut, v lablstime Qfycli.pD).Cest>, Q.W.S.- .
Machining (hours) Polishing (hours) Assembling (hours) Unit profits (S)
Product I 3 1 2 7
Product II 2 1 1 5
Product III 2 2 2 8
Product IV 4 3 1 9
Available hours per week 580 450 400
Th_e finn hs a. ontract with .a _distrbutor tq_ prov.i_de 70 uni& of product I and @ units.J.?:f.Jwy ction of products II and IU each week. Th£9ugh other customers, the firm can sell each k as mauy units of products I, II, and III as it can .i;roduce, but can sell only 45 units op oduct
·Th-!l.tSJo unitU2.Leach qduct should manufacture eacp we to meet
.fill G.o. nwctual oblig'!ti.Q.us..and ma.x.i,wi e ·t tutal ·
a) Wri te the objective function and all the constraints but do not solve the problem.
b) Indicate the type of optimization problem . 2 :J
P-4 (35) - Solve the following linear programming optimization problem. Show all of your work.
Maximize: Subject to:
Z == X1 + 3x2 + 4x:.i
3x1 + 2x2
13
x2 + 3x3
I 7
