[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. [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:

2. • 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-

3. 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

4. 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 Table 1. Is it a lower or an
upper bound? Explain.
(c) Is x = [1, 0, 0, 0]T a feasible solution to the above problem.
If yes, then obtain its objective
function value from Table 1. Is it a lower or an upper bound?
Explain.
Solve the above problem using branch & bound method, and
build the enumeration tree using the
following strategies. You can use the information from Table 1.
Note: For every strategy that you
pick, you will generate one tree, i.e., one tree for Part (d), one
for Part (e) and one for Part (f).

5. (d) Strategy-1:
• Node Selection: Best First
Select the node with the best objective function value.
• Variable Selection: Nearest to integer
A fractional variable with fractional value nearest to an integer
will be used for branching.
• Branching Direction: Up
Select the branch of ≥ side (lower bound is increased.)
(e) Strategy-2:
• Node Selection: Depth First the Best Back
Select the most recently created child node to solve. If no child
exists, then backtrack to
the best bound node available in the entire tree.
• Variable Selection: Lowest fraction
A fractional variable with lowest fraction will be used for
branching.
• Branching Direction: Down
Select the branch of ≤ side (upper bound is decreased.)
(f) Strategy-3:
• Node Selection: Breadth First the Best Next
All nodes at one level of the search tree are processed before
any node at a deeper level.
In a given level, best node should be processed first.
• Variable Selection: Highest fraction
A fractional variable with highest fraction will be used for
branching.

6. • Branching Direction: You are free to pick any rule.
2
ISE-421 HW-1
Table 1: Optimal
Solution
s for All LP Relaxations, # Indicates Unfixed Variables
S. No x1 x2 x3 x4 Sol. Obj.
1 # # # # (0.44, 1, 0, 0.26) 30.8
2 # # # 0 (0.33, 1, 0.33, 0) 30.7
3 # # # 1 (0.29, 0.29, 0, 1) 20.3
4 # # 0 # (0.44, 1, 0, 0.26) 30.8
5 # # 0 0 (0.6, 1, 0, 0) 30.4
6 # # 0 1 (0.29, 0.29, 0, 1) 20.3
7 # # 1 # (0.65, 0.39, 1, 0) 29.7
8 # # 1 0 (0.65, 0.39, 1, 0) 29.7
9 # # 1 1 infeasible —
10 # 0 # # (1, 0, 1, 0) 26
11 # 0 # 0 (1, 0, 1, 0) 26

7. 12 # 0 # 1 (0.15, 0, 0.37, 1) 16.5
13 # 0 0 # (1, 0, 0, 0.43) 18.3
14 # 0 0 0 (1, 0, 0, 0) 14
15 # 0 0 1 (0.33, 0, 0, 1) 14.7
16 # 0 1 # (1, 0, 1, 0) 26
17 # 0 1 0 (1, 0, 1, 0) 26
18 # 0 1 1 infeasible —
19 # 1 # # (0.44, 1, 0, 0.26) 30.8
20 # 1 # 0 (0.33, 1, 0.33, 0) 30.7
21 # 1 # 1 infeasible —
22 # 1 0 # (0.44, 1, 0, 0.26) 30.8
23 # 1 0 0 (0.6, 1, 0, 0) 30.4
24 # 1 0 1 infeasible —
25 # 1 1 # infeasible —
26 # 1 1 0 infeasible —
27 # 1 1 1 infeasible —
28 0 # # # (0, 1, 0.4, 0) 26.8
29 0 # # 0 (0, 1, 0.4, 0) 26.8
30 0 # # 1 (0, 0.33, 0, 1) 17.3
31 0 # 0 # (0, 1, 0, 0.33) 25.3
32 0 # 0 0 (0, 1, 0, 0) 22
33 0 # 0 1 (0, 0.33, 0, 1) 17.3
34 0 # 1 # (0, 0.5, 1, 0) 23
35 0 # 1 0 (0, 0.5, 1, 0) 23
36 0 # 1 1 infeasible —

8. 37 0 0 # # (0, 0, 1, 0.5) 17
38 0 0 # 0 (0, 0, 1, 0) 12
39 0 0 # 1 (0, 0, 0.4, 1) 14.8
40 0 0 0 # (0, 0, 0, 1) 10
41 0 0 0 0 (0, 0, 0, 0) 0
42 0 0 0 1 (0, 0, 0, 1) 10
43 0 0 1 # (0, 0, 1, 0.5) 17
44 0 0 1 0 (0, 0, 1, 0) 12
45 0 0 1 1 infeasible —
46 0 1 # # (0, 1, 0.4, 0) 26.8
47 0 1 # 0 (0, 1, 0.4, 0) 26.8
48 0 1 # 1 infeasible —
49 0 1 0 # (0, 1, 0, 0.33) 25.3
50 0 1 0 0 (0, 1, 0, 0) 22
51 0 1 0 1 infeasible —
52 0 1 1 # infeasible —
53 0 1 1 0 infeasible —
54 0 1 1 1 infeasible —
55 1 # # # (1, 0.57, 0, 0.35) 29.9
56 1 # # 0 (1, 0.71, 0, 0) 29.7
57 1 # # 1 infeasible —
58 1 # 0 # (1, 0.57, 0, 0.35) 29.9
59 1 # 0 0 (1, 0.71, 0, 0) 29.7
60 1 # 0 1 infeasible —
61 1 # 1 # (1, 0, 1, 0) 26

9. 62 1 # 1 0 (1, 0, 1, 0) 26
63 1 # 1 1 infeasible —
64 1 0 # # (1, 0, 1, 0) 26
65 1 0 # 0 (1, 0, 1, 0) 26
66 1 0 # 1 infeasible —
67 1 0 0 # (1, 0, 0, 0.43) 18.3
68 1 0 0 0 (1, 0, 0, 0) 14
69 1 0 0 1 infeasible —
70 1 0 1 # (1, 0, 1, 0) 26
71 1 0 1 0 (1, 0, 1, 0) 26
72 1 0 1 1 infeasible —
73 1 1 # # infeasible —
74 1 1 # 0 infeasible —
75 1 1 # 1 infeasible —
76 1 1 0 # infeasible —
77 1 1 0 0 infeasible —
78 1 1 0 1 infeasible —
79 1 1 1 # infeasible —
80 1 1 1 0 infeasible —
81 1 1 1 1 infeasible —
3

10. ISE-421 HW-1
Practice Problem #P1
Al-Mobile Junoob (AMJ) can build communication tower on
any of the 8 available mountains
in the central forest. The cost of building tower on any
mountain in dollars is given in the following
table:
M1 M2 M3 M4 M5 M6 M7 M8
cost 300 500 100 400 200 450 380 200
Based on the time and labor availability, AMJ wants to select
only three mountain sites in total,
which minimizes the total cost of building the towers. Due the
coverage requirements, at least 2
of the first 5 mountains must be selected. Also, towers should
not be build on both mountain M3
and M8. In addition to that, a tower can be built on mountain
M4 if and only if a tower is built
on mountain M1. Write the model clearly in the format learned
in the class. You can use ABCD,
and EVER OLD CARD mnemonic if desired.

11. 4
ISE-421 HW-1
Practice Problem #P2
Write each of the following expressions as compactly as
possible using summation and enumeration
symbols.
(a)
x1 + 4x2 + 9x3 + 16x4 + 25x5 ≥ 30
(b)
x1,1,1 + x1,1,2 + x1,1,3 + x2,1,1 + x2,1,2 + x2,1,3 ≤ a1
x1,2,1 + x1,2,2 + x1,2,3 + x2,2,1 + x2,2,2 + x2,2,3 ≤ a2
(c)
x1 + x2 + x3 ≥ 1
x2 + x3 + x4 ≥ 1
x3 + x4 + x5 ≥ 1

12. x4 + x5 + x6 ≥ 1
x5 + x6 + x7 ≥ 1
(d)
x1,1 ≥ 1, x1,2 ≥ 2, x1,3 ≥ 3
x2,1 ≥ 4, x2,2 ≥ 5, x2,3 ≥ 6
(e)
max :
x1 + x2 + x3 + x4
s.t. :
x1 + x2 ≤ 4
x2 + x3 ≤ 4
x3 + x4 ≤ 4
x4 + x5 ≤ 4
x1, x2, x3, x4, x5 ≥ 0
5

13. ISE-421 HW-1
Practice Problems from the Text Book
Pick the rules for variable selection, branching direction and
node selection. Solve the following
problem using Branch & Bound method. Use Simplex to solve
initial LP. Then use Dual Simplex
to solve the further LPs.
max :
3x1 + 2x2
s.t. :
2x1 + 5x2 ≤ 18
4x1 + 2x2 ≤ 18
x1, x2 ≥ 0 and integers
In addition to the above solve the following problems from the
text book:
• Problem Set 9.1A (Q. 6, Q. 18) – pages 352-354

14. • Problem Set 9.1C (Q. 3) – page 362
• Problem Set 9.1D (Q. 8) – page 368
6