Production Scheduling 4'3

1
Production Scheduling4.3 58
A small manufacturing plant makes three types of
inflatable boats one person, two person and
four person models. Each boat requires the
services of the cutting, assembly and packaging
departments.
What is the objective?
Objective operate at full capacity use all
resources.
What controls the decision? These are usually the
control variables.
X1 of one person boats, X2 two person, X3
4 person, nonnegative
What resources are available? Constraints? At
what rate are the resources used?
Type One Two Four Avail. Dept Cut .5 1.0
1.5 350 Ass .6 0.9 1.2 330 Pack .2 0.3 0.5
115
. . X1 X2 X3 X1
X2 X3 X1 X2 X3
2
3.3 Systems of Linear Equations and Gauss-Jordan
Method

• What a generalization of the augmented matrix
  method
• Why means to solve system of any number of
  linear equations or determine that the system is
  inconsistent

3
Idea
Start
a1x1 a2x2 a3x3 d b1x1 b2x2 b3x3 e c1x1
c2x2 c3x3 f
Matrix row operations
Equivalent matrix corresponds to easy to solve
equivalent system Reduced Row-Echelon form
x1 x2 X3
Finish
4
Elementary Row Operations

• Switch any two rows
• Multiply or divide one of the rows by a non-zero
  number
• Replace a row by its sum or difference with a
  non-zero multiple of another row

Keep a record of operations
Important
5
Reduced Row-Echelon form

• All zero rows are below every non-zero row
• The first non-zero element of every non-zero row
  is 1 (leftmost 1 or leading 1)
• Every leftmost 1 is the only non-zero element
  of its column
• Each leftmost 1 appears to the right of the
  leftmost 1s in the rows above

6
Order of Steps
4.3 Even 26-44, p. 212
If possible
7
Applications

• 4.3 58, 60, 62 p. 213

Hint Read and Re-read the statement of the
problem to identify the objective, the control
variables, variable limitations (nonnegative,
integer, ), resource constraints. Organize and
use all the information use table or diagram.
Interpret the results when finished, dont just
compute.
8
Production Scheduling4.3 58
A small manufacturing plant makes three types of
inflatable boats one person, two person and
four person models. Each boat requires the
services of the cutting, assembly and packaging
departments.
What is the objective?
Objective operate at full capacity use all
resources.
What controls the decision? These are usually the
control variables.
X1Number of one person boats, X2two person, X3
4 person
What resources are available? Constraints? At
what rate are the resources used?
Type One Two Four Avail. Dept Cut .5 1.0
1.5 350 Ass .6 0.9 1.2 330 Pack .2 0.3 0.5
115
. . X1 X2 X3 X1
X2 X3 X1 X2 X3
9
4.3 58 revisited
Cut .5 X1 1.0 X2 1.5 X3 350 Ass .6
X1 0.9 X2 1.2 X3 330 Pack .2 X1
0.3 X2 0.5 X3 115 X1, X2, X3
nonnegative integers
58a
58b No Packaging
58c Four person boat discontinued
.5 X1 1.0 X2 350 .6 X1 0.9 X2 330 .2 X1
0.3 X2 115 X1, X2, X3 nonnegative integers
.5 X1 1.0 X2 1.5 X3 350 .6 X1 0.9 X2
1.2 X3 330 X1, X2, X3 nonnegative integers
May not have a feasible solution
Really do NOT have to use all the resources