solution
Liberty Taxes is a Vancouver based firm specializing in preparing Canadian Income Tax forms for their
individual and corporate clients. Liberty processes only two types of income tax returns; the T-1 Return
for individual clients and the Corporate Return for corporate clients. Three separate departments within
Liberty Tax, whether it is a T-1 or a Corporate Return, check each return. These departments are called
the Deductions, Income and Arithmetic departments and have respectively 500 hours, 600 hours and
600 hours of labour hours available each processing cycle. Each T-1 Return needs 2 Deduction hour, 3
Income hours and 3.3 Arithmetic hours and each Corporate Return needs 4 Deduction hours, 4 Income
hours and 3 Arithmetic hours.
Returns not completed during one processing cycle are completed during the next processing cycle.
Management requires that the company process at least 50 T-1 Returns each processing cycle. The
profit contributions for each T-1 and Corporate Return processed by Liberty Tax are $30 and $50
respectively.
Considering the above equations answer the following questions:
a. What are the two decisions that need to be made? (2 marks)
b. What equations you will use to develop the algebraic model? (6 marks)
c. Please complete the graph below and find the Optimal Solution by drawing Iso-Profit
line. (Total 20 marks)
i. Add all constraints (except Non-neg.). (5 marks)
ii. Label all the constraints by name. (5 marks)
iii. Shade and identify the feasible region. (2 marks)
iv. Draw and identify the iso-profit line on this graph showing where the optimal
solution is located. (2 marks)
v. Identify at which intersection the optimal solution exists. (2 marks)
vi. Label all the corner points of feasible region with a, b, c, d, e, —- (4 marks)
Page 2
d. Identify amount of profits at the points a, b, c, d, e, – -? (5 marks)
e. The Boss at Liberty Taxes has said that the company should only prepare Corporate
returns since each Corporate return translates into $50 profit, whereas each T-1 Return
only provides $30 per return. Without referring to the, yet unknown, optimal solution, how
would you respond to this argument? (2 marks)
f. Liberty management is hoping to produce 120 T-1 Returns and 110 Corporate Returns
the next processing cycle. Comment on their plans. (2 marks)
g. What is the optimal solution? (2 marks)
h. If this problem was formulated in Excel, what would be the value of the target cell? (1 mark)
i. How many hours will Liberty use in the Arithmetic Department? (1 mark)
j. What is the allowable increase and allowable decrease for the Arithmetic constraint?
(2 marks)
k. FOR THIS QUESTION ONLY, suppose the Min Dem constraint was changed so that the
minimum was now 40 units. {Circle the correct responses.} (3 marks)
i. Would the feasible region change? Yes No
ii. Would the optimal solution change? Yes No
iii. Would the set of binding constraints change? Yes No
Min Demand
Page 3
l. Determine the allowable increase and allowable decrease on the objective function
coefficient for Corporate Returns. (4 marks)
m. Determine the allowable increase and allowable decrease on the objective function
coefficient for T-1 Returns. (5 marks)
n. Determine the allowable increase on the Income constraint. (4 marks)
o. Determine the allowable decrease on the Income constraint. (4 marks)
p. Set up, but, DO NOT SOLVE, all the equations you would need to determine the shadow
price for the Arithmetic constraint. DO NOT SOLVE. (3 marks)
q. Set up the above problem in Excel using the style that we used in class. Make sure that
your spreadsheet model is logical, well organized and easy to understand. In building
formulas, use the sumproduct function where appropriate. Use text boxes to identify the
changing cells, target cell and constraints. (15 marks)
r. Print out the cell formulas in landscape orientation and include it with your answers. (5
marks)
s. Which constraints are binding? non-binding? 4 marks)
t. The Liberty problem was correctly formulated and solved in Excel with the following
partial sensitivity report output. (5 marks)
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$E$5 Deductions Used 500 7.5 500 50 42.85714286
$E$7 Arithmetic Used 555 0 600 1E+30 45
Suppose Liberty can secure up to an additional 60 hours in either the Deductions or Arithmetic
department, but not both. In which department, Deductions or Arithmetic, would you recommend
purchasing extra time? Circle the correct response and provide convincing evidence.
Deductions Income Arithmetic
Evidence:
u. FOR THIS QUESTION ONLY, suppose revenue and cost savings could be realized that
would change the objective function to Max 50T + 30C (from the current Max 30T + 50C).
What effect(s), if any, would this change have on the optimal solution? No calculations
are required. (4 marks)
individual and corporate clients. Liberty processes only two types of income tax returns; the T-1 Return
for individual clients and the Corporate Return for corporate clients. Three separate departments within
Liberty Tax, whether it is a T-1 or a Corporate Return, check each return. These departments are called
the Deductions, Income and Arithmetic departments and have respectively 500 hours, 600 hours and
600 hours of labour hours available each processing cycle. Each T-1 Return needs 2 Deduction hour, 3
Income hours and 3.3 Arithmetic hours and each Corporate Return needs 4 Deduction hours, 4 Income
hours and 3 Arithmetic hours.
Returns not completed during one processing cycle are completed during the next processing cycle.
Management requires that the company process at least 50 T-1 Returns each processing cycle. The
profit contributions for each T-1 and Corporate Return processed by Liberty Tax are $30 and $50
respectively.
Considering the above equations answer the following questions:
a. What are the two decisions that need to be made? (2 marks)
b. What equations you will use to develop the algebraic model? (6 marks)
c. Please complete the graph below and find the Optimal Solution by drawing Iso-Profit
line. (Total 20 marks)
i. Add all constraints (except Non-neg.). (5 marks)
ii. Label all the constraints by name. (5 marks)
iii. Shade and identify the feasible region. (2 marks)
iv. Draw and identify the iso-profit line on this graph showing where the optimal
solution is located. (2 marks)
v. Identify at which intersection the optimal solution exists. (2 marks)
vi. Label all the corner points of feasible region with a, b, c, d, e, —- (4 marks)
Page 2
d. Identify amount of profits at the points a, b, c, d, e, – -? (5 marks)
e. The Boss at Liberty Taxes has said that the company should only prepare Corporate
returns since each Corporate return translates into $50 profit, whereas each T-1 Return
only provides $30 per return. Without referring to the, yet unknown, optimal solution, how
would you respond to this argument? (2 marks)
f. Liberty management is hoping to produce 120 T-1 Returns and 110 Corporate Returns
the next processing cycle. Comment on their plans. (2 marks)
g. What is the optimal solution? (2 marks)
h. If this problem was formulated in Excel, what would be the value of the target cell? (1 mark)
i. How many hours will Liberty use in the Arithmetic Department? (1 mark)
j. What is the allowable increase and allowable decrease for the Arithmetic constraint?
(2 marks)
k. FOR THIS QUESTION ONLY, suppose the Min Dem constraint was changed so that the
minimum was now 40 units. {Circle the correct responses.} (3 marks)
i. Would the feasible region change? Yes No
ii. Would the optimal solution change? Yes No
iii. Would the set of binding constraints change? Yes No
Min Demand
Page 3
l. Determine the allowable increase and allowable decrease on the objective function
coefficient for Corporate Returns. (4 marks)
m. Determine the allowable increase and allowable decrease on the objective function
coefficient for T-1 Returns. (5 marks)
n. Determine the allowable increase on the Income constraint. (4 marks)
o. Determine the allowable decrease on the Income constraint. (4 marks)
p. Set up, but, DO NOT SOLVE, all the equations you would need to determine the shadow
price for the Arithmetic constraint. DO NOT SOLVE. (3 marks)
q. Set up the above problem in Excel using the style that we used in class. Make sure that
your spreadsheet model is logical, well organized and easy to understand. In building
formulas, use the sumproduct function where appropriate. Use text boxes to identify the
changing cells, target cell and constraints. (15 marks)
r. Print out the cell formulas in landscape orientation and include it with your answers. (5
marks)
s. Which constraints are binding? non-binding? 4 marks)
t. The Liberty problem was correctly formulated and solved in Excel with the following
partial sensitivity report output. (5 marks)
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$E$5 Deductions Used 500 7.5 500 50 42.85714286
$E$7 Arithmetic Used 555 0 600 1E+30 45
Suppose Liberty can secure up to an additional 60 hours in either the Deductions or Arithmetic
department, but not both. In which department, Deductions or Arithmetic, would you recommend
purchasing extra time? Circle the correct response and provide convincing evidence.
Deductions Income Arithmetic
Evidence:
u. FOR THIS QUESTION ONLY, suppose revenue and cost savings could be realized that
would change the objective function to Max 50T + 30C (from the current Max 30T + 50C).
What effect(s), if any, would this change have on the optimal solution? No calculations
are required. (4 marks)
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