BUAD 831

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BUAD
831 Operations Management/Management Science
Linear Programming
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5-1 Linear Programming - Definition
Linear Programming is a procedure for solving
problems that involve a set of decision
variables. The goal is to maximize or minimize
an objective function, which is defined as a
linear function of the decision variables. The
maximization or minimization is restricted by a
set of constraints that limit values that the
decision variables can take on. Each constraint
is defined as a linear function of the decision
variables.
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5-2 A Linear Programming Problem
A furniture company manufactures tables and
chairs, and it is trying to determine how many of
each to produce during the next production
scheduling period. Each table generates a profit
of 50 and each chair generates a profit of 10.
There are resource limitations on how many of
each might be produced. There is only enough
hardware available to make 30 tables and 140
chairs. None of the hardware is used in common
in tables and chairs. In addition, each table
uses 20 units of wood and each chair uses 5 units
of wood, with a total of 1,000 units of wood
being available. Assuming that any mix of output
can be sold, how many tables and chairs should be
produced to maximize profit?
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5-3 Graphic Solution to a Linear Programming
Problem
Chair Hardware Constraint C lt 140
Not Feasible
Feasible
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5-4 Graphic Solution to a Linear Programming
Problem
Table Hardware Constraint T lt 30
Feasible
Not Feasible
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5-5 Graphic Solution to a Linear Programming
Problem
Wood Constraint 20T 5C lt 1000
20(0) 5(200) 1000
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Not Feasible
20(50) 5(0) 1000
Feasible
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5-6 Graphic Solution to a Linear
Programming Problem
Both Table and Chair Hardware Constraints T lt 30
and C lt 140
Feasible for Table Hardware Only
Not Feasible for Either
Feasible for Both Hardware Constraints
Feasible for Chair Hardware Only
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5-7 Graphic Solution to a Linear
Programming Problem
All Three Constraints
T lt 30 C lt 140 and 20T 5C lt 1000
Not Feasible for any Constraints
Feasible for Table Hardware Only
Feasible for Chair Hardware Only
Feasible for all Constraints
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5-8 Graphic Solution to a Linear Programming
Problem
Profit 50T 10C
Profit 2,200 ?
T 30 20T 5C 1000 C 80
Profit 2,300
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50(0) 10(220) 2,200
50(44) 10(0) 2,200
2,000
2,200
1,000
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5-9 Linear Programming - Example 1
A firm produces three different products A,B,C
that are all processed through three different
work centers X,Y,Z. The amount of time that
each different product takes to be processed
through each of the work centers is given by
Product _
Work Center
A B C
X 7 4 9 Y 5 1 8 Z 0 2 1 There are
limitations on the number of work hours that are
available to the work centers on a weekly basis,
with the limits given by Work Center
Weekly Hours Available
X 770 Y 620 Z 280 The marketing
department indicates that all of the output of A
and B during the next week can be sold, but no
more than 95 units of C can be sold. The unit
profits for the three products are given by
Product Unit Profit A
13 B 11 C
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produced to maximize profit?
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5-11 Linear Programming - Example
2
A firm has to determine a distribution plan to
minimize shipping costs for getting finished
product to four cities A,B,C,D from three
different production facilities X,Y,Z. The
demand requirements at the four cities, and the
production capacities at the three facilities are
given by Demand Requirements
Capacity Restrictions A 90,000 X
110,000 B 80,000 Y 120,000 C
90,000 Z 130,000 D 70,000 Due to
differences in distances between cities, and the
availability of different forms of
transportation, the shipping costs depend upon
the origin and destination. The unit costs of
shipping from sources to destinations are given
by A B C D X 7 11 8 13 Y 20 17 13 25
Z 9 25 11 14 How should the product be
distributed to minimize total shipping cost?
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5-13 Linear Programming - Example
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A firm mixes different fuels (A and B) for use
in company trucks, to minimize cost. The firm
requires a total of 3,000 gallons of fuel for use
next month. The maximum storage capacity is
4,000 gallons, and the current availability of
fuels is 2,500 gallons of A and 3,750 gallons of
B. The mix of fuels to be used in the trucks
must have an octane rating of at least 80. When
fuels are mixed, the total amount of fuel that is
obtained is the sum of the amounts of A and B put
into the mixture. The octane rating of the
resulting mix is the weighted average of the
individual octane ratings of A and B, weighted in
proportion to the respective volumes that were
used. Fuel A has an octane rating of 90, and it
costs 1.15 per gallon. Fuel B has an octane
rating of 74 and it has a cost of 0.95 per
gallon.
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5-15 Solving Linear Programming Problems with
LINDO
Type the formulation with any word processor, or
input it directly into LINDO
? Do not go beyond column 72 in any line. For
longer formulations, break before column 72 and
continue on the next line.
? The entries lt and gt will be interpreted as lt
and gt.
? Rewrite all constraints so that all variables
appear on the left hand side of lt, or gt, with a
simple constant on the right hand side.
? No algebraic entries will be recognized. For
example, you can not write 9(AB). This must be
entered as 9A 9B.
? No simple constant terms are allowed in the
objective function. If you want to add a
constant term 123 to the objective function, you
can enter it as 123ONE in the objective function
and then add an additional constraint ONE 1.
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5-16 Critical LINDO Statements
? When you are prompted for sensitivity analysis,
respond No.
? The optimal value of the objective function is
given directly.
? The VALUE entry for each VARIABLE term
gives the resulting variable values in the
optimal solution.
? A response of UGH! means that LINDO can not
interpret your input as a valid formulation.
? A response of NO FEASIBLE SOLUTION means that
your combined constraints are not consistent, and
that you have defined a situation for which there
is no possible answer.
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