The last line of the problem statement for question 5 should be changed to:

"The company wishes to determine how many windows of each type to produce per week to maximize total revenue."

. Problem 5. The Whitt Window Company, a company with three plants, makes two different kinds of hand-crafted windows: a wood-framed window and an aluminum-framed window. Plant 1 makes wood frames. Plant 2 makes aluminum frames. Plant 3 assembles both wood-framed windows and aluminum-framed windows. Suppose Plant I can spend 6 hours per week in making wood-frames, Plant 2 can spend 4 hours per week in making aluminum frames, and Plant 3 can spend 48 hours per week to assemble windows. It takes one hour of Plant 1 and six hours of Plant 3 to produce one thousand wood-framed windows. It takes one hour of Plant 2 and eight hours of Plant 3 to produce one thousand aluminum-framed windows. The company earns $300 revenue for selling one thousand wood-framed window and $150 revenue for selling one thousand aluminum-framed window. The company wishes to determine how many windows of each type to produce per day to maximize total revenue. (a) (5pt] Formulate a linear program for this problem. (6) 2pt] Let 13, 14, and as denote the slack variables for the constraints of the primal LP in part (a). After we apply the simplex method, the final simplex tableau is: Basic Var. z 11 12 13 14 25 RHS Z 1 0 0 187. 5 0 18.75 2025 2 0 1 0 1 0 0 6 10 0 0 0.75 1 -0.125 2.5 220 01 -0.750 0.125 1.5 What is the optimal solution? (c) (5pt] Suppose the number of available working hours of Plant 3 increases from 48 to 54. Write down the new optimization problem and update the corresponding simplex tableau, using the matrix form of LP. (d) [2pt] Does the updated simplex tableau give an optimal CPF solution of the new optimization problem? If so, explain why, and write down the optimal CPF solution; if not, find the new optimal solution by utilizing the matrix form of LP. (e) [2pt] After using Excel to solve the problem, the sensitivity report is the following: Variable Cells Cell Name $B$10 Wood-Framed Windows (in thousands) $B$11 Aluminum-Framed Windows (in thousands) Final Reduced Objective Allowable Value Cost Coefficient Increase 6 0 3001E+30 1. 5 0 150 250 Allowable Decrease 187.5 150 Constraints Name Cell $B$19 Plant 1 Hours $B$20 Plant 2 Hours $B$21 Plant 3 Hours Final Shadow Constraint Allowable Allowable Value Price R.H. Side Increase Decrease 6 187.5 6 2 3.333333333 1.5 0 4 1E+30 2.5 48 18.75 48 20 1 2 Please explain the meaning of the two "allowable increase in the variables" part and "constraints" part respectively. () [2pt] If the number of available working hours of Plant 3 increases from 48 to 50. What will be the optimal value of the new problem? Describe how you compute this new value not allowed to use computer, but just by looking at the sensitivity report. (9) 2pt] What business insights can you obtain based on this table? Write at least two of them. Your answer may include the analysis, but you should highlight the sentences that can be understood by someone who has not learned any linear program before add quotation marks to these sentences) these sentences are the ones that appear in a report to the manager. Remark: No computer/software is needed in this problem. . Problem 5. The Whitt Window Company, a company with three plants, makes two different kinds of hand-crafted windows: a wood-framed window and an aluminum-framed window. Plant 1 makes wood frames. Plant 2 makes aluminum frames. Plant 3 assembles both wood-framed windows and aluminum-framed windows. Suppose Plant I can spend 6 hours per week in making wood-frames, Plant 2 can spend 4 hours per week in making aluminum frames, and Plant 3 can spend 48 hours per week to assemble windows. It takes one hour of Plant 1 and six hours of Plant 3 to produce one thousand wood-framed windows. It takes one hour of Plant 2 and eight hours of Plant 3 to produce one thousand aluminum-framed windows. The company earns $300 revenue for selling one thousand wood-framed window and $150 revenue for selling one thousand aluminum-framed window. The company wishes to determine how many windows of each type to produce per day to maximize total revenue. (a) (5pt] Formulate a linear program for this problem. (6) 2pt] Let 13, 14, and as denote the slack variables for the constraints of the primal LP in part (a). After we apply the simplex method, the final simplex tableau is: Basic Var. z 11 12 13 14 25 RHS Z 1 0 0 187. 5 0 18.75 2025 2 0 1 0 1 0 0 6 10 0 0 0.75 1 -0.125 2.5 220 01 -0.750 0.125 1.5 What is the optimal solution? (c) (5pt] Suppose the number of available working hours of Plant 3 increases from 48 to 54. Write down the new optimization problem and update the corresponding simplex tableau, using the matrix form of LP. (d) [2pt] Does the updated simplex tableau give an optimal CPF solution of the new optimization problem? If so, explain why, and write down the optimal CPF solution; if not, find the new optimal solution by utilizing the matrix form of LP. (e) [2pt] After using Excel to solve the problem, the sensitivity report is the following: Variable Cells Cell Name $B$10 Wood-Framed Windows (in thousands) $B$11 Aluminum-Framed Windows (in thousands) Final Reduced Objective Allowable Value Cost Coefficient Increase 6 0 3001E+30 1. 5 0 150 250 Allowable Decrease 187.5 150 Constraints Name Cell $B$19 Plant 1 Hours $B$20 Plant 2 Hours $B$21 Plant 3 Hours Final Shadow Constraint Allowable Allowable Value Price R.H. Side Increase Decrease 6 187.5 6 2 3.333333333 1.5 0 4 1E+30 2.5 48 18.75 48 20 1 2 Please explain the meaning of the two "allowable increase in the variables" part and "constraints" part respectively. () [2pt] If the number of available working hours of Plant 3 increases from 48 to 50. What will be the optimal value of the new problem? Describe how you compute this new value not allowed to use computer, but just by looking at the sensitivity report. (9) 2pt] What business insights can you obtain based on this table? Write at least two of them. Your answer may include the analysis, but you should highlight the sentences that can be understood by someone who has not learned any linear program before add quotation marks to these sentences) these sentences are the ones that appear in a report to the manager. Remark: No computer/software is needed in this