answer the survey Quations .short 3 answers

20. a) xij = no. of students bused from district i to school j, where i = n, s, e, w, c and

j = c,w,s

minimize Z = 8xnc + 11xnw + 14xns + 12xsc

+ 9xsw + 0xss + 9xec + 16xew

+ 10xes + 8xwc + 0xww + 9xws

+ 0xcc + 8xcw + 12xcs

subject to

xnc + xnw + xns = 700

xsc + xsw + xss = 300

xec + xew + xes = 900

xwc + xww + xws = 600

xcc + xcw + xsc = 500

xnc + xsc + xec + xwc + xcc ≤ 1,200

xnw + xsw + xew + xww + xcw ≤ 1,200

xns + xss + xes + xws + xcs ≤ 1,200

xij ≥ 0

xnc = 700

xss = 300

xes = 900

xww = 600

xcc = 500

Z = 14,600

b) Add the following 3 constraints to the original formulation:

xss ≤ 150

xww ≤ 300

xcc ≤ 250

xnc = 700

xnw = 0

xsw = 150

xss = 150

xes = 900

xwc = 250

xww = 300

xws = 50

xcc = 250

xcw = 250

Z = 20,400

c) Change the 3 demand constraints in the

(a) formulation from ≤ 1,200 to = 1,000.

xnc = 400

xnw = 300

xsw = 150

xss = 150

xec = 50

xes = 850

xwc = 300

xww = 300

xcc = 250

xcw = 250

Z = 21,200

22. a) x1 = no. of lb of oats

x2 = no. of lb of corn

x3 = no. of lb of soybeans

x4 = no. of lb of vitamin supplement

minimize Z = .50x1 + 1.20x2 + .60x3 +2.00x4

subject to

x1 ≤ 300

x2 ≤ 400

x3 ≤ 200

x4 ≤ 100

x3/(x1 + x2 + x3 + x4) ≥ .30

x4/(x1 + x2 + x3 + x4) ≥ .20

x2/x1 ≤ 2/1

x1 ≤ x3

x1 + x2 + x3 + x4 ≥ 500

x1, x2, x3, x4 ≥ 0

b) x1 = 200

x2 = 0

x3 = 200

x4 = 100.00

Z = $420

24. a) xij = dollar amount invested in alternative i in year j, where i = p (product research and development), m

(manufacturing operations improvements), a (advertising and sales promotion) and j = 1,2,3,4 (denoting

year):

sj = slack, or uninvested funds in year j

maximize Z = s4 + 1.2xa4 + 1.3xm3 + 1.5xp3

subject to

xa1 ≥ 30,000

xm1 ≥ 40,000

xp1 ≥ 50,000

xa1 + xm1 + xp1 + s1 = 500,000

xa2 + xm2 + xp2 + s2 = s1 + 1.2xa1

xa3 + xm3 + xs3 = s2 + 1.2xa2 + 1.3xm1

xa4 + xs4 = s3 + 1.2xa3 + 1.3xm2 + 1.5xp1

xij, sj ≥ 0

Note: Since it is assumed that any amount of funds can be invested in each alternative—i.e., there is no

minimum investment required—and funds can always be invested in as short a period as one year yielding a

positive return, it is apparent that the sj variables for uninvested funds will be driven to zero in every period.

Thus, these variables could be omitted from the model formulation for this problem.

b) xa1 = 410,000 xm1 = 40,000

xa2 = 492,000 xp1 = 50,000

xa3 = 642,400 Z = $1,015,056

xa4 = 845,880

26. a) x1 = asparagus (acres)

x2 = corn (acres)

x3 = tomatoes (acres)

x4 = green beans (acres)

x5 = red peppers (acres)

Maximize Z = (1.90)(2,000)x1 + (0.10)(7,200)x2 +(3.25)(25,000)x3 + (3.40)(3,900)x4 + (3.45)(12,500)x5

Subject to

x1 + x2 + x3 + x4 + x5 = 1

1,800x1 + 1,740x2 + 6,000x3 + 3,000x4 + 2,700x5 ≤ 5,000

(2,000)x1 ≤ 1,200

(25,000)x3 ≤ 10,000

(3,900)x4 ≤ 2,000

(12,500)x5 ≤ 5,000

x1, x2, x3, x4, x5 ≥ 0

b) x1, x2 = 0, x3 = 0.4, x4 = 0.2, x5 = 0.4;

Z = $52,402