Example is based on Taylor (1999), Chapter 2. Assume you own a furniture company that produces two
different types of wood products: x1 = chair and x2 = stool, and want to figure out how many of each unit
you should produce to maximize profit over the next day (assume the production cycle is one day).
The two products have the following resource requirements per item.

Each chair yields a profit of $40 and each stool yields a profit of $50

There are 40 total hours of labor available per day. Each chair requires 1 hours of labor and each
stool requires 2 hours of labor for production

There are 80 board feet of wood available for production each day. Each chair requires 4 ft. of
wood and each stool requires 3 ft. of wood for production

There are 60 gallons of polyurethane clear coat available to coat chairs and stools after they are
made. Each chair requires 3 gallons and each stool requires 2 gallons

Assume for contractual reasons, you must produce at least 10 stools a day.
Write out both the original and the standard form of the problem
Solve this problem graphically
 Graph all constraints
 Graph the OF
 Identify all extreme points
 Find optimal solution
1
Graph all four constraints and shade in the feasible region. How many potential optimal points (critical
points) are in the feasible region? What are the X1 and X2 values (the x and y coordinates) for each
critical point?
2
What are the X1 and X2 values associated with the optimal solution and what is the final profit
maximizing value?
Graph the Objective Function and list the endpoints that you use.
How many surplus variables are in the problem? How many slack variables are in the problem?
Calculate the values of S1, S2, S3, and S4 using the optimal solution and the constraints in standard
form. What do the values of zero mean? What do the positive values mean in the context of this
problem?
Using MS Excel, set up and solve the problem. Generate both the Answer Report and Sensitivity Report
3
Variable Cells
Cell
Name
$B$1 X1 (# of chairs)
$B$2 X2 (# of stools)
Final Reduced Objective Allowable Allowable
Value Cost Coefficient Increase Decrease
8
0
40 26.66666667
15
16
0
50
30
20
Constraints
Cell
$B$8
$B$9
$B$10
$B$11
Final Shadow Constraint Allowable Allowable
Name
Value Price
R.H. Side
Increase Decrease
1) Constraint#1 (labor) LHS
40
16
40 13.33333333
7.5
2) Constraint #2 (wood) LHS
80
6
80
5
20
3) Constraint #3 (polyurethane) LHS
56
0
60
1E+30
4
4) Constraint #4 (minimum stool) LHS
16
0
10
6
1E+30
If the per unit profit associated with chairs were to decrease from $40 to $20, does the optimal solution
mix change? Does the total profit change, and if so, by how much?
If the per unit profit associated with stools were to increase to $70/per stool, is the current solution still
optimal? What is the impact on the total profit?
If you added 10 hours of labor, what is the impact on the total profit?
4
If you increased the number of stools you were required to produce from 10 to 15, what would the
impact on profit be?
If you could increase your weekly supply of wood to 100 board feet, would you do it? Justify your
answer.
What would happen if you were to decrease the quantity of polyurethane from 60 gallons to 50 gallons
per week?
Provide the range of feasibility for constraint #2 (wood) and provide general interpretation of the
shadow price for that range.
5