1. The document provides steps for dividing decimals by decimals. It includes moving the decimal in the divisor to make it a whole number, moving the decimal in the dividend the same number of places, and bringing up the decimal to the quotient.
2. There are examples of dividing decimals with values such as 9.0 ÷ 0.9, 29.6 ÷ 3.7, and 10.81 ÷ 2.3.
3. Questions are asked such as how many weeks it will take Anna to save $6.35 per week to buy a $57.15 game, and how long it took Ben to run a 19.5 mile race at 7.8 miles per hour.
[This sheet must be completed and attached to the last page of.docxhanneloremccaffery
[This sheet must be completed and attached to the last page of your homework]
ISE 421
Operations Research II
Term 161
Homework #1
Student Name ID# Signature
Homework Guidelines
To receive full credit, you should make sure you follow the following guidelines.
Homework Presentation:
• Every main problem should be answered on a different page.
• You should submit the solutions for the first two problems only.
• All pages of your homework should be in chronological order.
• Your name, and the homework number should be clearly indicated.
Modeling Questions:
• Clearly define all the variables in one group. Then clearly define all the parameters in another group. Then display
the final model in the standard style (Objective, Constraints, Restriction on Domain). You can use ABCD, and
EVER OLD CARD mnemonic if desired.
ISE-421 HW-1
Problem #1
Suppose that the decision variables of a mathematical programming model are defined as:
xi,j,t := acers of land plot i allocated to crop j in year t
Ct := the funds in SAR donated by the governament at the begining of year t
Rj,t := the revenue generated from crop j in $ at the end of year t
where i = 1, . . . , 47; j = 1, . . . , 9; t = 1, . . . , 10.
Use summation (
∑
) and enumeration (∀) indexed notation to write expressions for each of
the following systems of constraints in terms of these decision variables, and determine how many
constraints belong to each system. You need to define additional variables to model the following
constraints. Assume $1 = 3.75SAR. In addition assume appropriate information wherever neces-
sary.
(a) The acres allocated in each plot i cannot exceed the available acreage (call it Ai) in any year.
(b) At least 1000 total acres must be devoted to corn (corp j = 4) in each year.
(c) At least one-third of the total acreage planted over 10 years must be in soybeans (corp j = 2).
(d) Either rice (corp j = 9) or wheat (corp j = 8) should be planted in a given year.
(e) Grapes (corp j = 7) should be planted in a year, when the current funds from the government
and the total revenue from the previous year is greater than or equal to 38000 SAR.
(f) In the odd years (t = 1, 3, . . . , 9) land plot 32 is unusable.
(g) On the same land plot, there should be at least a two years of difference between corn and rice
crops plantation.
(h) If soybeans are planted in a land plot, then no other crops should be planted on the same land
plot.
(i) Every plot must be used for planting in a given year.
(j) In every year, there should be at least 7 different crops.
1
ISE-421 HW-1
Problem #2
Consider the following IP problem.
maximize :
14 ∗ x1 + 22 ∗ x2 + 12 ∗ x3 + 10 ∗ x4
subject to :
50 ∗ x1 + 70 ∗ x2 + 40 ∗ x3 + 30 ∗ x4 ≤ 100
10 ∗ x1 + 60 ∗ x2 + 50 ∗ x3 + 60 ∗ x4 ≤ 80
6 ∗ x1 + 1 ∗ x2 + 3 ∗ x3 + 7 ∗ x4 ≤ 9
xi ∈ {0, 1} ∀ i = 1, . . . , 4
(a) Write the LP relaxation of the above model.
(b) Get the optimal objective function value of the LP relaxation from Tabl.
[This sheet must be completed and attached to the last page of.docxhanneloremccaffery
[This sheet must be completed and attached to the last page of your homework]
ISE 421
Operations Research II
Term 161
Homework #1
Student Name ID# Signature
Homework Guidelines
To receive full credit, you should make sure you follow the following guidelines.
Homework Presentation:
• Every main problem should be answered on a different page.
• You should submit the solutions for the first two problems only.
• All pages of your homework should be in chronological order.
• Your name, and the homework number should be clearly indicated.
Modeling Questions:
• Clearly define all the variables in one group. Then clearly define all the parameters in another group. Then display
the final model in the standard style (Objective, Constraints, Restriction on Domain). You can use ABCD, and
EVER OLD CARD mnemonic if desired.
ISE-421 HW-1
Problem #1
Suppose that the decision variables of a mathematical programming model are defined as:
xi,j,t := acers of land plot i allocated to crop j in year t
Ct := the funds in SAR donated by the governament at the begining of year t
Rj,t := the revenue generated from crop j in $ at the end of year t
where i = 1, . . . , 47; j = 1, . . . , 9; t = 1, . . . , 10.
Use summation (
∑
) and enumeration (∀) indexed notation to write expressions for each of
the following systems of constraints in terms of these decision variables, and determine how many
constraints belong to each system. You need to define additional variables to model the following
constraints. Assume $1 = 3.75SAR. In addition assume appropriate information wherever neces-
sary.
(a) The acres allocated in each plot i cannot exceed the available acreage (call it Ai) in any year.
(b) At least 1000 total acres must be devoted to corn (corp j = 4) in each year.
(c) At least one-third of the total acreage planted over 10 years must be in soybeans (corp j = 2).
(d) Either rice (corp j = 9) or wheat (corp j = 8) should be planted in a given year.
(e) Grapes (corp j = 7) should be planted in a year, when the current funds from the government
and the total revenue from the previous year is greater than or equal to 38000 SAR.
(f) In the odd years (t = 1, 3, . . . , 9) land plot 32 is unusable.
(g) On the same land plot, there should be at least a two years of difference between corn and rice
crops plantation.
(h) If soybeans are planted in a land plot, then no other crops should be planted on the same land
plot.
(i) Every plot must be used for planting in a given year.
(j) In every year, there should be at least 7 different crops.
1
ISE-421 HW-1
Problem #2
Consider the following IP problem.
maximize :
14 ∗ x1 + 22 ∗ x2 + 12 ∗ x3 + 10 ∗ x4
subject to :
50 ∗ x1 + 70 ∗ x2 + 40 ∗ x3 + 30 ∗ x4 ≤ 100
10 ∗ x1 + 60 ∗ x2 + 50 ∗ x3 + 60 ∗ x4 ≤ 80
6 ∗ x1 + 1 ∗ x2 + 3 ∗ x3 + 7 ∗ x4 ≤ 9
xi ∈ {0, 1} ∀ i = 1, . . . , 4
(a) Write the LP relaxation of the above model.
(b) Get the optimal objective function value of the LP relaxation from Tabl.
1. Name ____________________________ Date ___________________
Mrs. Labuski & Mrs. Portsmore Per ___ Unit 3 Lesson 6 Dividing Decimals
OC 1-5 / TB 3-7 & 3-8
Steps for Dividing Decimals by Decimals: Whatever you do outside the house,
you must do inside the house!
1. __________________________
3.2 )11.52
____________________________
2. __________________________
____________________________
32)11.52
3. __________________________ .
____________________________ 32)115.2
4. __________________________
Now you try:
1.) 9.0 ÷ 0.9 2.) 29.6 ÷ 3.7 3.) 10.81 ÷ 2.3
2. Evaluate the expression x ÷1.2 for the given value of x
1. x = 40.8 2. x = 1.8 3. x = 10.8
4. x = 14.4 5. x = 4.32 6. x = 0.06
7. Anna is saving $6.35 a week to buy a computer game that costs $57.15. How
many weeks will she have to save to buy the game?
8. Ben ran a 19.5-mile race last Saturday. His average speed during the race was 7.8
miles per hour. How long did it take Ben to finish the race?
3. Name ____________________________ Date ___________________
Mrs. Labuski & Mrs. Portsmore Per ___ Unit 3 Lesson 6 Dividing Decimals
OC 1-5 / TB 3-7 & 3-8
Steps for Dividing Decimals by Decimals: Whatever you do outside the house,
you must do inside the house!
1. Move the decimal in the divisor to
3.2)11.52
the right to make it a whole number
2. Move the decimal in the dividend
32)11.52
the same number of places .
3. Bring up the decimal to the quotient 32)115.2
4. Divide as usual.
Now you try:
1.) 9.0 ÷ 0.9 2.) 29.6 ÷ 3.7 3.) 10.81 ÷ 2.3
____ ____ ____
0.9 ) 9.0 3.7)29.6 2.3 )10.81
⇓ ⇓ ⇓
________ ________ ____________
9)90 37)296 23)108.1
10 8 4.7
9)90 37)296 23)108.1
-90 -296 - 92 ↓
0 0 161
-161
0
4. Evaluate the expression x ÷1.2 for the given value of x
1. x = 40.8 2. x = 1.8 3. x = 10.8
x ÷1.2 x ÷1.2 x ÷1.2
40.8÷1.2 1.8÷1.2 10.8÷1.2
____ ____ _____
1.2)40.8 1.2)1.8 1.2)10.8
⇓ ⇓ ⇓
__________ ________ __________
1.2)40.8 1.2)1.8 1.2)10.8
⇓ ⇓ ⇓
__________ ________ __________
12)408 12)18 12)108
34 1.6 9
12)408 12)18.0 12)108
-36↓ -12↓ -108
48 60 0
-48 -60
0 0
4. x = 14.4 5. x = 4.32 6. x = 0.06
x ÷1.2 x ÷1.2 x ÷1.2
14.4 ÷1.2 4.32 ÷1.2 0.06 ÷1.2
__________ ________ __________
1.2)14.4 1.2)4.32 1.2)0.06
⇓ ⇓ ⇓
__________ ________ __________
12)144 12)43.2 12)00.6
12 3.6 00.05
12)144 12)43.2 12)00.60
-12↓ -36↓ -60
24 92 0
-92
0
5. 7. Anna is saving $6.35 a week to buy a computer game that costs $57.15. How
many weeks will she have to save to buy the game?
_____
6.35)57.15
_____
635)5715
9
635)5715
-5715
0
She will have to save for 9 weeks.
8. Ben ran a 19.5-mile race last Saturday. His average speed during the race was 7.8
miles per hour. How long did it take Ben to finish the race?
_____
7.8)19.5
____
78)195
2.5
78)195.0
-156 ↓
390
-390
0
It took Ben 2.5 hours to complete the race.
