MAT1005 Mathematics for Business and Economics
(Semester 1, 2020-21)
ASSIGNMENT 2
Deadline:
November 6, 2020 (Friday) noon
Topics 2 Linear Equation, Linear Inequalities & Linear Programming
(1)
Given the following Linear Programming problem:
Maximize
45𝑥$ + 30𝑥( + 24𝑥*
Subject to:
2𝑥$ + 3𝑥( + 4𝑥* ≤ 64
(Constraint 1)
2𝑥$ + 𝑥( + 𝑥* ≤ 32
(Constraint 2)
𝑥$ + 𝑥( + 𝑥* ≤ 20
(Constraint 3)
𝑥$ , 𝑥( , 𝑥* ≥ 0
(a) Prepare the initial Simplex tableau as discussed in the class.
(b) Solve the problem by using the Simplex method as discussed in the class. (You are
requested to show your full working to gain full credit)
(c) Find the shadow price for constraint 2 and 3
(2)
Alicia Liquide (Alicia), one of the top chemical manufacturer in the local region, is planning to
setup another production plant in Bougainvillea. The new plant will have 24/7 operation. Each
day of every week is divided into two 12-hour shift periods, 8am-8pm and 8pm-8am, denoted
by morning shift and night shift. The minimum number of workers required for each of these
shifts over any week is listed below
Sun.
Mon.
Tue.
Wed.
Thu.
Fri.
Sat.
Morning Shift
8
14
16
18
15
13
6
Night Shift
6
16
18
22
12
10
4
The code of practice of Alicia has the following requirements governing acceptable shifts for
workers
(1) Each worker works 4 consecutive days during any seven-day period
(2) Each worker is assigned to work either a morning shift or a night shift. Workers
assigned to a shift must remain on the same shift every day that they work
(3) Number of workers in the night shift is limited to a maximum of 40% of the total
workers in this plant
Workers are paid daily. The daily salaries are listed below:
Weekday (Mon. – Fri.)
Weekend (Sat. – Sun.)
Morning Shift
¥500 /day
¥600 /day
Night Shift
¥650 /day
¥800 /day
You, the production manager of Alicia, are attempting to develop the working schedule to
minimize the weekly salary expenses. Formulate this problem as a linear programming
problem. (You are NOT required to solve it)
(3)
Given the following Linear Programming problem:
Maximize
2𝑦$ + 5𝑦( + 9𝑦*
Subject to:
3𝑦$ + 3𝑦( + 𝑦* ≤ 90
5𝑦$ − 𝑦( + 𝑦* ≥ 120
4𝑦$ + 2𝑦( + 𝑦* ≤ 150
(a)
(b)
(c)
(Constraint 1)
(Constraint 2)
(Constraint 3)
𝑦$ , 𝑦( , 𝑦* ≥ 0
Prepare the initial Simplex tableau as discussed in the class.
Determine the entering and leaving variables in the first iteration.
Find out the Simplex tableau after the first iteration
Topics 3 Matrix & System of Linear Equations
(4)
Let 𝐴 =
𝑥
7 −1
and 𝐵 =
𝑦
2 4
1
, where 𝑥, 𝑦 and 𝑧 are non-zero constant
𝑧
Given that
(1)
𝑥
6𝑥
=𝐴 𝑦
6𝑦
(2) 6 + 𝑥 5𝑧 = 1 𝑧 𝐴8
(a) Find 𝑥, 𝑦 and 𝑧
(b) Determine if 𝐵 is a non-singular matrix.
(c) Evaluate 𝐵9$ 𝐴𝐵
(d)
By using (c), determine 𝐴$: (you may use
𝑚
0
0
𝑛
=
=
𝑚=
0
0
without proof)
𝑛=