Tutor Tues Apr 2, 2-4 PM since no classes Apr 1

Set 6 due Apr 9

Set 7 due April 11

C-3 due April 18

Typo: Summer Schedule

• Page 27, MSCI 306M

• Session 1: June 3-July 12 (6 weeks)

Answers to Practice Problems II

Problem (1)

(1) 4X1 + 3X2 >96

X1 X2

0 96/3=32

96/4=24 0

X1

X2

0,32

24,0

(1)

(2) X1 + X2 <50

X1 X2

0 50

50 0

X1

X2

0,32

24,0

0,50

50,0

(1)(2)

(3) X1 +3X2 >60

X1 X2

0 60/3=20

60 0

X1

X2

0,32

24,0

0,50

50,0

(1)(2)

0,20

60,0

(3)

Corner Point: (1),(3)

• (1) 4X1 + 3X2 = 96

• (3) X1 + 3X2 = 60

• (1)-(3) 3X1 = 36

• X1 = 12

• (3) 12 + 3X2 = 60, so X2 = 16

Corner Point: (2),(3)

• (2) X1 + X2 = 50

• (3) X1 + 3X2 = 60

• (3)-(2) 2X2= 10

• X2 = 5

• (2) X1 + 5 = 50

• X1 = 45

X1

X2

0,32

24,0

0,50

50,0

(1)(2)

0,20

60,0

(3)

Corner points

Not feasible

Notfeasible 12,16 45,5

MINIMUM COST

X1 X2 2X1+7X2

12 16 136

45 5 125=MIN

0 32 224

0 50 350

EXAM FORMAT

MAKE 45 UNITS OF ALASKA AND 5 UNITS ARABIAN FOR

$ 125 COST

ANSWER: (2)

SIMULTANEOUS EQUATIONS

• (1) 2X1 + X2 = 12

• (2) X1 – X2 = -3

• (1)+(2) 3X1 = 9

• X1 =3

• 3-X2 = -3

• X2 = 6

0,12

6,0

0,3

3,6

INFEASIBLE

INFEASIBLE

MINIMIZE COST

X1 X2 COST=

4X1+3X2

0 12 36

3 6 30=MIN

ORIGINAL PROBLEM

• MAKE BOTH X1 AND X2

NEW PROBLEM

• CHANGE (1) 2X1+X2 >8

0,12

6,0

0,3

3,6

INFEASIBLE

0,8

4,0

1.7,4.7

MIN COST

X1 X2 COST=

4X1+3X2

0 8 24

1.7 4.7 20=MIN

OUTPUT INSENSITIVE

• NEW: MAKE BOTH X1 AND X2

• ALSO INPUT INSENSITIVE SINCE BOTH ORIGINAL PROBLEM AND NEW PROBLEM HAVE NO SLACK

(3) ALGEBRA

STEP 1: OBJECTIVE FUNCTION

OLD C1 = 5

Z= C1X1 +12X2

• 12X2 = Z – 5X1

• X2 = (1/12)Z – (C1/12)X1

• COEFFICIENT OF X1 IS –C1/12

STEP 2: CONSTRAINT (1)

• 2X1 + X2 = 200

• X2 = 200 –2X1

• COEF OF X1 IS -2

STEP 3: STEP 1 = STEP 2

• -C1/12 = -2

• C1 = 24

• OLD C1 = 5

• C1 < 24

STEP 4: CONSTRAINT (2)

• X1 + 2X2 = 300

• 2X2 = 300 –X1

• X2 = 150 –0.5 X1

• COEF OF X1 IS –0.5

STEP 5: STEP 1 = STEP 4

• -C1/12 = -0.5

• C1 = 6

• OLD C1 = 5

• C1 < 6

• STEP 3: C1 < 24

• ANSWER: C1 < 6

(4) MAX X1+3X2+2X3+4X4

• X1=AM PHONE, X2=AM RIDE, X3=AFT PHONE, X4=AFT RIDE

• CONSTRAINTS

• AM: X1+20X2 < 12(60)=720

• AFT: 2X3+30X4 < 14(60) = 840

• GAS: X2+X4 < 100

(5)MIN 3X1+4X2+2X3+X4, X1=FISH,X2=BEEF,

X3=LETTUCE, X4=BEANS

• CAL: 1000<400X1+700X2+40X3+500X4<1800

• FAT: 15<4X1+80X2+2X4<40

• PROT: 90X1+100X2+2X3+20X4>25

• CHOL: 100X1+300X2<19

(6)MIN 100000X1+50000X2, X1=TV, X2= RADIO

• X2 X1 > 2

• 1000000X1+ 400000X2 > 3000000

• .6(1000000)X1+(1-.7)(40000)X2 1000000X1+400000X2 > .4

• X1 < 5