You are given the following linear programming model in algebraic form, with X1 and X2 as the decision variables

3. [25 points] You are given the following linear programming model in algebraic
form, with X1 and X2 as the decision variables: Note: Each part is independent
(i.e., any change made in one problem part does not apply to any other parts).
Minimize 40X1+50X2
Subject to
2X1+3X2>=30
2 X1+ X2>=20
X1>=0, X2>=0
a) Graph the feasible region and label the corner point. Compute the optimal
solution using any method of your choice. Justify your answer and indicate the
optimal solution on your graph. [10 points]
b) How does the optimal solution change if the objective function is changed to
40 X1+70 X2? [10 points]

c) Now your boss tells you “I apologize, but I was just informed that the X2
coefficient is not reliable, Can you tell me how much the X2 coefficient may
change, both up and down, whereby the optimal solution you reported in problem
1(a) remains optimal?” Justify your answer. [5 points]

[35 points] The Ferguson Paper Company produces rolls of paper for cash
registers, adding machines, and desk calculators. They sell three widths—1.5,
2.5, and 3.5 inches—all the same diameter. The supplier provides a standard 10-
inch roll from which Ferguson must cut the various sizes. The cutting machine
allows 7 cutting alternatives, namely, 7 different ways that the 10-inch roll may be
divided into the various widths, as described in the table below.
Cutting Number of Rolls
Alternative 1.5 inch 2.5 inch 3.5 inch
1 6 0 0
2 0 4 0
3 2 0 2
4 0 1 2
5 1 3 0
6 1 2 1
7 4 0 1
For example, cutting alternative 4 consumes 9.5 inches with one 2.5-inch roll and
two 3.5-inch rolls and thus leaves ½ inch of waste that must be scrapped. Due to
demand requirements, the minimum production quantities for this period are
Roll Width (inches) 1.5 2.5 3.5
Units 1000 2000 4000
To minimize costs, the company wants to minimize the total number of 10-inch
rolls that are consumed during the manufacturing process.
1) Based on this information, explain what are (i) the decision variables, (ii) the
objective, (iii) and the constraints of the decision problem? Answer in words, not
math. Explain. [5 points]
2) Formulate the decision problem into a linear programming in mathematic
forms. [10 points]
3) Please solve your linear programming problem in Excel solver, and report the
optimal solution. [10 points]
4) Please identify which constraints are binding and which are non-binding. Why?
Explain. [10 points]
5. [40 points] Colonial Furniture produces hand-crafted colonial style furniture.
Plans are now being made for the production of rocking chairs, dining room
tables, and/or armoires over the next week. These products go through two
stages of production (assembly and finishing). The following table gives the time
required for each item to go through these two stages, the amount of wood
required (fine cherry wood), and the corresponding unit profits, along with the
amount of each resource available next week.
Rocking
Chair
Dining
Room Table
Armoire Available
Assembly (minutes) 100 180 120 3,600
Finishing (minutes) 60 80 80 2,000
Wood (pounds) 30 180 120 4,000
Unit Profit $240 $720 $600
A linear programming model has been formulated in a spreadsheet to determine
the production levels that would maximize profit. The solved spreadsheet model
and corresponding sensitivity report are shown below.
For each of the following parts, answer the question as specifically and
completely as is possible without resolving the problem with solver. Please show
all your steps. Note: Each part is independent (i.e., any change made in one
problem part does not apply to any other parts).
a. Suppose the profit per armoire decreases by $50. Will this change the
optimal production quantities? What can be said about the change in total
profit? [10 points]
b. Suppose the profit per table decreases by $60 and the profit per armoire
increases by $90. Will this change the optimal production quantities? What
can be said about the change in total profit? [10 points]

Suppose a part-time worker in the assembly department calls in sick, so
that now four fewer hours are available that day in the assembly
department. How much would this affect total profit? Would it change the
optimal production quantities? [10 points]
d. Suppose one of the workers in the assembly department is also trained to
do finishing. Would it be a good idea to have this worker shift some of his
time from the assembly department to the finishing department? Indicate
the rate at which this would increase or decrease total profit per minute
shifted. [10 points]