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Problem 1 – include all steps
A small winery manufactures 2 types of wine, Burbo’s Better (X) and Burbo’s Best (Y). Burbo’s Better results in profit of $4 per quart, whereas Burbo’s Best has profit of $5 per quart. Two production workers mix the 2 wines. It takes a production worker 2 hours to mix a quart of the Better and 3 hours to mix a quart of the Best. Each worker puts in a 9 hour day. The quantity of alcohol than can be used to fortify the wine is limited to 24 ounces daily. Six ounces of alcohol are added to each quart of Burbo’s Better and 3 ounces are added to Burbo’s Best. Give the LP Model and use both the graphical method and Excel to find the optimal solution. ( I attached powerpoint presentation to show lp model graph method and excel)
Problem 2
Last week’s sales of a product at a retail store are shown in the following table:
|
Day |
Sales (Dollars) |
|
200 |
|
|
250 |
|
|
180 |
|
|
190 |
|
|
175 |
|
|
170 |
|
include all steps to reach solution
(a)
Use the 3-day moving average method for forecasting days 4-7.
(b)
Use the 3-day weighted moving average method for forecasting days 4-7. Use Weight 1 day ago = 4, Weight 2 days ago = 3, and Weight 3 days ago = 2.
(c)
Compare the techniques in a table format using the Mean Absolute Deviation (MAD).
Problem 3
include all steps to reach solution
The revenue and cost functions for producing and selling quantity x for a certain company are given below.
R(x) = 12x – x2
C(x) = 21 + 2x
(a) Determine the profit function P(x).
(b) Compute the break-even quantities.
(c) Determine the average cost at the break-even quantities.
(d) Determine the marginal revenue R’(x).
(e) Determine the marginal cost C’(x).
(f) At what quantity is the profit maximized?
Copyright 2006 John Wiley & So, Inc.
Operations Management – 5th Edition
Chapter 13 Supplement
Roberta Russell & Bernard W. Taylor, III
Linear Programming
Copyright 2006 John Wiley & So, Inc.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Lecture Outline
Model Formulation
Graphical Solution Method
Linear Programming Model
Solution
Solving Linear Programming Problems with Excel
Sensitivity Analysis
Copyright 2006 John Wiley & Sons, Inc.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
A model consisting of linear relationships
representing a firm’s objective and resource constraints
Linear Programming (LP)
LP is a mathematical modeling technique used to determine a level of operational activity in order to achieve an objective, subject to restrictions called constraints
Copyright 2006 John Wiley & Sons, Inc.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Types of LP
Copyright 2006 John Wiley & Sons, Inc.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Types of LP (cont.)
Copyright 2006 John Wiley & Sons, Inc.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Types of LP (cont.)
Copyright 2006 John Wiley & Sons, Inc.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
LP Model Formulation
Decision variables
mathematical symbols representing levels of activity of an operation
Objective function
a linear relationship reflecting the objective of an operation
most frequent objective of business firms is to maximize profit
most frequent objective of individual operational units (such as a production or packaging department) is to minimize cost
Constraint
a linear relationship representing a restriction on decision making
Copyright 2006 John Wiley & Sons, Inc.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
LP Model Formulation (cont.)
Max/min z = c1x1 + c2x2 + … + cnxn
subject to:
a11x1 + a12x2 + … + a1nxn (≤, =, ≥) b1
a21x1 + a22x2 + … + a2nxn (≤, =, ≥) b2
:
am1x1 + am2x2 + … + amnxn (≤, =, ≥) bm
xj = decision variables
bi = constraint levels
cj = objective function coefficients
aij = constraint coefficients
Copyright 2006 John Wiley & Sons, Inc.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
LP Model: Example
Labor Clay Revenue
PRODUCT (hr/unit) (lb/unit) ($/unit)
Bowl 1 4 40
Mug 2 3 50
There are 40 hours of labor and 120 pounds of clay available each day
Decision variables
x1 = number of bowls to produce
x2 = number of mugs to produce
RESOURCE REQUIREMENTS
Copyright 2006 John Wiley & Sons, Inc.
*
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
LP Formulation: Example
Maximize Z = $40 x1 + 50 x2
Subject to
x1 + 2×2 40 hr (labor constraint)
4×1 + 3×2 120 lb (clay constraint)
x1 , x2 0
Solution is x1 = 24 bowls x2 = 8 mugs
Revenue = $1,360
Copyright 2006 John Wiley & Sons, Inc.
*
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Graphical Solution Method
Plot model constraint on a set of coordinates in a plane
Identify the feasible solution space on the graph where all constraints are satisfied simultaneously
Plot objective function to find the point on boundary of this space that maximizes (or minimizes) value of objective function
Copyright 2006 John Wiley & Sons, Inc.
*
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Graphical Solution: Example
4 x1 + 3 x2 120 lb
x1 + 2 x2 40 hr
Area common to
both constraints
50 –
40 –
30 –
20 –
10 –
0 –
|
10
|
60
|
50
|
20
|
30
|
40
x1
x2
Copyright 2006 John Wiley & Sons, Inc.
*
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Computing Optimal Values
24
8
x1 + 2×2 = 40
4×1 + 3×2 = 120
4×1 + 8×2 = 160
-4×1 – 3×2 = -120
5×2 = 40
x2 = 8
x1 + 2(8) = 40
x1 = 24
4 x1 + 3 x2 120 lb
x1 + 2 x2 40 hr
40 –
30 –
20 –
10 –
0 –
|
10
|
20
|
30
|
40
x1
x2
Z = $50(24) + $50(8) = $1,360
Copyright 2006 John Wiley & Sons, Inc.
*
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Extreme Corner Points
x1 = 224 bowls
x2 =8 mugs
Z = $1,360
x1 = 30 bowls
x2 =0 mugs
Z = $1,200
x1 = 0 bowls
x2 =20 mugs
Z = $1,000
A
B
C
|
20
|
30
|
40
|
10
x1
x2
40 –
30 –
20 –
10 –
0 –
Copyright 2006 John Wiley & Sons, Inc.
*
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
4×1 + 3×2 120 lb
x1 + 2×2 40 hr
40 –
30 –
20 –
10 –
0 –
B
|
10
|
20
|
30
|
40
x1
x2
C
A
Z = 70×1 + 20×2
Objective Function
Optimal point:
x1 = 30 bowls
x2 =0 mugs
Z = $2,100
Copyright 2006 John Wiley & Sons, Inc.
*
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Minimization Problem
Minimize Z = $6×1 + $3×2
subject to
2×1 + 4×2 16 lb of nitrogen
4×1 + 3×2 24 lb of phosphate
x1, x2 0
CHEMICAL CONTRIBUTION
Brand Nitrogen (lb/bag) Phosphate (lb/bag)
Gro-plus 2 4
Crop-fast 4 3
Copyright 2006 John Wiley & Sons, Inc.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
14 –
12 –
10 –
8 –
6 –
4 –
2 –
0 –
|
2
|
4
|
6
|
8
|
10
|
12
|
14
x1
x2
A
B
C
Graphical Solution
x1 = 0 bags of Gro-plus
x2 = 8 bags of Crop-fast
Z = $24
Z = 6×1 + 3×2
Copyright 2006 John Wiley & Sons, Inc.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Simplex Method
A mathematical procedure for solving linear programming problems according to a set of steps
Slack variables added to ≤ constraints to represent unused resources
x1 + 2×2 + s1 =40 hours of labor
4×1 + 3×2 + s2 =120 lb of clay
Surplus variables subtracted from ≥ constraints to represent excess above resource requirement. For example
2×1 + 4×2 ≥ 16 is transformed into
2×1 + 4×2 – s1 = 16
Slack/surplus variables have a 0 coefficient in the objective function
Z = $40×1 + $50×2 + 0s1 + 0s2
Copyright 2006 John Wiley & Sons, Inc.
*
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Solution Points with
Slack Variables
Copyright 2006 John Wiley & Sons, Inc.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Solution Points with
Surplus Variables
Copyright 2006 John Wiley & Sons, Inc.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Solving LP Problems with Excel
Click on “Tools”
to invoke “Solver.”
Objective function
Decision variables – bowls
(x1)=B10; mugs (x2)=B11
=C6*B10+D6*B11
=C7*B10+D7*B11
=E6-F6
=E7-F7
Copyright 2006 John Wiley & Sons, Inc.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Solving LP Problems with Excel (cont.)
After all parameters and constraints have been input, click on “Solve.”
Objective function
Decision variables
C6*B10+D6*B11≤40
C7*B10+D7*B11≤120
Click on “Add” to insert constraints
Copyright 2006 John Wiley & Sons, Inc.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Solving LP Problems with Excel (cont.)
Copyright 2006 John Wiley & Sons, Inc.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Sensitivity Analysis
Copyright 2006 John Wiley & Sons, Inc.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Sensitivity Range for Labor Hours
Copyright 2006 John Wiley & Sons, Inc.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Sensitivity Range for Bowls
Copyright 2006 John Wiley & Sons, Inc.
Copyright 2006 John Wiley & Sons, Inc.
Supplement 13-*
Copyright 2006 John Wiley & Sons, Inc.
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Copyright 2006 John Wiley & Sons, Inc.
