Page 71 -

GRAPHICAL SOLUTION PROCEDURE 51
Can you identify the
slack associated with a and packaging time are referred to as slack for the two departments. In linear
constraint? Try Problem programming terminology, any unused capacity for a
13(e).
Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has
deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
4 Draw a line on the graph showing the values of the decision variables that will
give this value for the objective function.
5 Using a ruler or straightedge, move the objective function line as far from the
origin as possible until any further movement would take the line out of the
feasible region altogether.
6 The feasible solution on this objective function line is the optimal solution.
7 Confirm the solution point mathematically using simultaneous equations.
One of the convenient things about LP is that we literally have a programme of
steps to follow that are the same for every maximization problem.
Slack Variables
In addition to the optimal solution and its associated profit contribution, GulfGolf
management will want information about the production time requirements for each
production operation. How many hours do we actually need to produce the optimal
solution quantities? We can determine this information by substituting the optimal
solution values (S ¼ 540, D ¼ 252) into the constraints of the linear programme.
Hours Required for S = 540 Hours Unused
Constraint and D = 252 Available Hours
Cutting and 7 / 10 (540) + 1(252) ¼ 630 630 0
dyeing
5
Sewing 1 / 2 (540) + / 6 (252) ¼ 480 600 120
2
Finishing 1(540) + / 3 (252) ¼ 708 708 0
1
Inspection and 1 / 10 (540) + / 4 (252) ¼ 117 135 18
packaging
The complete solution tells management that the production of 540 standard bags
and 252 deluxe bags will require all available cutting and dyeing time (630 hours)
and all available finishing time (708 hours), while 600 480 ¼ 120 hours of sewing
time and 135 117 ¼ 18 hours of inspection and packaging time will remain
unused. The 120 hours of unused sewing time and 18 hours of unused inspection
constraint is referred to as
the slack associated with the constraint.
Earlier, we introduced the distinction between binding and non-binding con-
straints. By definition, for constraints taking the form , a binding constraint will
have a slack value of zero at the optimal solution point since by definition for a
binding constraint, both sides of the constraint will be equal. For a non-binding
constraint taking the form , however, the slack value will be positive since, again,
by definition, for a non-binding constraint the left-hand side of the constraint will
be less than the right-hand side. Looking at the slack values (unused hours) in the
table above we can confirm directly that cutting and dyeing is a binding constraint
as is finishing whilst the other two constraints are non-binding.
Often variables, called slack variables, are added to the formulation of a linear
programming problem to represent the slack, or idle capacity. Unused capacity
makes no contribution to profit; thus, slack variables have coefficients of zero in
the objective function. After the addition of four slack variables, denoted S 1 , S 2 , S 3
and S 4 , the mathematical model of the GulfGolf problem becomes: