Chapter 7 of Quantitative Methods for Business by
Anderson, Sweeney and Williams
Read sections 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 7.7 and
appendix 7.1
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One of the most widely used M.S. tools
Used to solve a wide variety of problems
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Human Resource Scheduling
Routing of Delivery Vehicles
Selection of Advertising Media
Planning Product Levels in Manufacturing
Blending in Oil Refineries
Selection of Investment Opportunities
and many, many more
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Most widely used solution technique: The Simplex
Method
• Developed by George Dantzig in the 1940’s
• First used to solve military operations problems during
WWII
• Coincided with the development of the first widely used
computers
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Objective is to minimize or maximize some linear
function of the decision variables
Constraints are linear equations or inequalities
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Before the next class, you should complete
the following homework problems in chapter
7
 #1
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Extremely large models easily solvable using
existing computer programs
Consider using L.P. Models whenever you are
faced with allocating scarce resources among
competing activities
3 Steps to Formulate A Linear Programming Problem:
1. DEFINE DECISION VARIABLES
2. FORMULATE OBJECTIVE
3. IDENTIFY CONSTRAINTS
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A glass company is considering using its excess
capacity to manufacture two new products:
• wood frame windows, which earn $3 profit each, and
• aluminum frame doors which earn $5 profit, each.
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There are
• 4 man-hours / day available in plant 1 (Plant 1 is used for
wood frames)
• 12 man-hours / day available in plant 2 (Plant 2 for
aluminum frames)
• 18 man-hours / day available in plant 3 (Plant 3 for glass
and assembly)
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Each unit of the new products would require the
following plant resources (in man-hours):
WINDOWS
PLANT 1
1
PLANT 2
-PLANT 3
3
DOORS
-2
2
Formulate the problem:
1. DEFINE DECISION VARIABLES
2. OBJECTIVE FUNCTION
3. IDENTIFY CONSTRAINTS
1.
2.
3.
Associate the horizontal axis with one decision
variable, and associate the vertical axis with
the other.
Draw the constraints.
Identify the feasible region as the area where
all the constraints intersect (are “satisfied”).
4.
Find the optimal solution (the feasible point
which gives the best value of the objective).
a. Graph the objective function line for any 2 arbitrary
values of Z
b. Identify the improving direction for Z
c. Move a pen parallel to the Z lines in the improving
direction as far as possible
d. Last feasible point pen touches is the optimal
solution
How to find the coordinates of the optimal point?
1. Identify the (2) constraints which go through the point
2. Solve those (2) constraint equations simultaneously

Before the next class, you should complete
the following homework problems in chapter
7
 #3, #7, #8, #11 (by hand), #24 (by hand), and #31 (by hand)
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A unique (extreme point) optimal solution
Alternate optimal solutions
An unbounded solution
An infeasible problem
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LINEARITY OF OBJECTIVE AND CONSTRAINTS
• i.e. they can be written in the form:
C1X1+C2X2+…+CnXn  RHS
where the Xis are the decision variables, the Cis are (constant)
coefficients, and RHS is the (constant) RHS
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DIVISIBILITY OF DECISION VARIABLES
• i.e. they may take fractional values
1. Label one row for each of these:
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Objective function coefficients
Values of the decision variables
Each constraint.
2. Label one column for each of these:
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Each decision variable
Total value of the left hand side
Right hand side value.
3. Key in coefficients from constraints (blue border)
4. Key in right-hand-side values for constraints (blue
border)
5. Key in objective function coefficients (blue border)
6. Designate cells for decision values (red border)
7. Designate cell for objective function value (double
black border)
8. Specify formula for left-hand-sides of constraints,
using the SUMPRODUCT function
9. Specify formula of Objective Function Value cell, using
the SUMPRODUCT function
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(Key in any values for decision variables
Try different values, to see what happens to left-handside values and objective value)
10. Click Data  Analysis  Solver
11. Specify “Target Cell” as the objective function value
(cell with double black border)
13. Click MAX (or MIN)
14. Specify “changing cells” as decision variable
value cells (cells with red border)
15. Click “Add” to add constraints, one (or
several) at a time by selecting each:
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Total Left-hand-side cell(s)
Appropriate inequality/equality sign
Right-hand-side cell(s)
16. Be sure the box next to “Make
Unconstrained Variables Non-negative” is
checked.
17. By “Select a Solving Method”, choose
“Simplex LP”
18. Click SOLVE
19. Study the solution to be sure it is
reasonable (including feasible) and
modify/correct the model and resolve if it is
not.
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See Solver Hints file (on Blackboard, under
Module2 Linear Programming) if you need help.
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Before the next class, you should complete
the following homework problems in chapter
7
 #11 (on Excel), #24 (on Excel), #31 (on Excel)