管理数学

A
retailer has warehouses in Lima and Canton, from which two stores—one in Tiffin and one in Danville—place orders for bicycles. Tiffin orders 38 and Danville orders 46.
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Matrix
2 3 -4 7 5 -1
7 1 0 5 -8 3
3 5 0 -2 8 9
3 5 6 0 -2 5 8 9 12 5 -2 9
3 5 0 -2
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8 9 12
Matrix Notation
a11x1 + a12x2 +....+ a1nxn = b1 a21x1 + a22x2 +....+ a2nxn = b2 am1x1 + am2x2 +....+ amnxn = bm Transfer to be : AX = B
管理數學 Chapter 2: System of Linear Equations
XX. XX, XXX by XXXX 1
Agenda
Linear
Systems as Mathematical Models Linear Systems Having One or No Solutions Linear Systems Having Many Solutions
18
A Solution
To a equation: a1x1 + a2x2 + .... + a nx n = b ( t1, t2,...., tn )
To a linear system : A solution to each of linear equation simultaneously ps. Solution Set
4
Example 1
A
firm produces bargain and deluxe TV sets by buying the components, assembling them, and testing the sets before shipping.
5
Resources
The
Each
serving of soup has 15 calories and 1 gram of protein, each serving of tuna has 160 calories and 12 gram of protein, and each serving of green beans has 20 calories and 1 gram of protein.
7
Problem Formulation
Define
decision variables (unit of scale) Define the linear relation between variables (write the linear equations)
8
Example 2
1 -1 -2 A = 2 -3 -5 -1 3 5 AX = B x1 1 x2 B = 3 X= x3 -2
32
Example 2
3x1 + 2x2 - 5x3 = 7 (I) x1 - 8x2 + 4x3 = 9 2x1 + 6x2 - 7x3 = -2 Transfer (I) to matrix format.
26
Final Iteration
x1
x2
= -1 ------(1) = 1 ------(2) x3 = 2 ------(3)
Final answer: (x1, x2, x3) = (-1, 1, 2)
27
ty
It
is very hard to carry variables, xi’s, through the calculation process when applying elementary transformation
22
The Objective
A D
x1 + x2 + x3 = 2 --------(1) 2x1 + 3x2 + x3 = 3 ------(2) x1 - x2 - 2x3 = -6 --------(3) x1
x2
= ? ------(1) = ? ------(2) x3 = ? ------(3)
23
Solve the problem by elementary transformation
x1 + x2 + x3 = 2 ------(1) 2x1 + 3x2+ x3 = 3 ------(2) x1 - x2 - 2x3= -6 ------(3) Add a -2 multiple (1) to (2). Add a -1 multiple (1) to (3).
10
Question
If
these three foods are to furnish 380 calories and 27 grams of protein in casserole, how many servings of each should be used?
11
Example 3
6
Question
Use
a system of linear equations to model the number of each type of TV set that the company can produce each week while using all of its available labor.
30
Example 1
(I) x1 + x2 + x3 = 2 ------(1) 2x1 + 3x2 + x3 = 3 ------(2) x1 - x2 - 2x3 = -6 ------(3)
Transfer (I) to matrix format.
31
Matrix Representation
24
Iteration 1
x1 + x2 + x3 = 2 ------(1) x2 - x3 = -1 ------(2) - 2x2 -3x3 = -8 ------(3) Add a -1 multiple (2) to (1). Add a 2 multiple (2) to (3).
25
Problem
formulation: variable definition and equations Algorithms or formula Interpretation of solutions
15
System of 2 Linear Equations
(1)
{
x + 3y = 9 -2x + y = -4
12
Limitations and Question
Each
warehouse has enough to supply all orders but twice as many are to be shipped from Lima to Danville as from Canton to Tiffin. Write the linear equations.
13
Agenda
Linear
Systems as Mathematical Models Linear Systems Having One or No Solutions Linear Systems Having Many Solutions
14
Solving a System of Linear Equations
21
Result of operations
A
B
x1 + x2 + x3 = 2 -------------(1) x1 + 1.5x2 + 0.5x3 = 3 -----(2) x1 - x2 - 2x3 = -6 ------------(3) x1 - x2 - 2x3 = -6-----------(3) x1 +1.5x2 + 0.5x3 = 1.5--(2) -0.5x2 + 0.5x3 = 0.5---(1)
(1)
(2)
(3)
17
A System of Linear Equations ( A Linear System )
A finite collection of linear equations a11 x1 + a12 x2 +....+ a1n xn = b1 a21 x1 + a22 x2 +....+ a2n xn = b2 .... am1 x1 + am2 x2 +....+ amn xn = bm
3 2 A= 1 -8 2 6 -5 4 -7 x1 X= x2 x3 7 B= 9 -2
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Matrix of Coefficients
Coefficients
of the system
or Matrix A
x1+x2+x3=2 2x1+3x2+x3=3 x1-x2-2x3=-6
-2x + y = 3 (2) -4x + 2y = 2
{
4x - 2y = 6 (3) 6x - 3y = 9