Exam 4: Sensitivity Analysis and the Simplex Method

Exhibit 4.2 The following questions correspond to the problem below and associated Risk Solver Platform (RSP) sensitivity report. Robert Hope received a welcome surprise in this management science class; the instructor has decided to let each person define the percentage contribution to their grade for each of the graded instruments used in the class. These instruments were: homework, an individual project, a mid-term exam, and a final exam. Robert's grades on these instruments were 75, 94, 85, and 92, respectively. However, the instructor complicated Robert's task somewhat by adding the following stipulations: -homewark can account far up to $25 \%$ of the grade, but must be at least $5 \%$ af the grade; - the praject can account for up to $25 \%$ of the grade, but must be at least $5 \%$ af the grade; - the mid-term and final must each accaunt far betwen $10 \%$ and $40 \%$ of the grade but cannot accaunt far mare than $7 \%$ of the grade when the percentages are cambined; and -the project and final exam grades may not collectively constitute more than $50 \%$ of the Iratade. The following LP model allows Robert to maximize his numerical grade. Let = weight as51gMed to hamewark = waight as5igned to the praject = weight assigned to the midi-term = waight as5igned to the final MAX: 75+94+85+92 Subject ta: +++=1 +\leq0.70 +\geq0.50 0. 05\leq\leq0.25 0. 05\leq\leq0.25 0.10 \leq\leq0.40 0.10\leq\leq0.40 $\text { Adjustaibla Calls}$ Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease \ F\ 5 Mid Term to grade 0.40 10.00 85 1+30 10 \ F\ 6 Final to grade 0.25 0.00 92 2 17 \ F\ 7 Project to grade 0.25 2.00 94 1+30 2 \ F\ 8 Homework to grade 0.10 0.00 75 10 1+30 $\text { Constraints }$ Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease \ E\ 14 Both Exams Total 0.65 0 0.7 1+30 0.05 \ E\ 15 Final \& Project Total 0.5 17 0.5 0.05 0.15 \ F\ 9 100\% to gracle 1.00 75.00 1 0.15 0.05 -Refer to Exhibit 4.2. Constraint cell F9 corresponds to the constraint, W₁ + W₂ + W₃ + W₄ = 1, and has a shadow price of 75. Armed with this information, what can Robert request of his instructor regarding this constraint?

The sensitivity analysis provides information about which of the following:

When the allowable increase or allowable decrease for the objective function coefficient of one or more variables is zero it indicates (in the absence of degeneracy) that

A farmer is planning his spring planting. He has 20 acres on which he can plant a combination of Corn, Pumpkins and Beans. He wants to maximize his profit but there is a limited demand for each crop. Each crop also requires fertilizer and irrigation water both of which are in short supply. The following table summarizes the data for the problem. Crap Profitper Acre ( Yied per Acre (lb) Meximum Denand (lb) Irripation (acreft) Fetilize (pounds/are) Carm 2,100 21,000 200,000 2 500 pumplin 900 10,000 180,000 3 400 Beans 1,050 3,500 80,000 1 300 Based on the following Risk Solver Platform (RSP) sensitivity output, how much can the price of Corn drop before it is no longer profitable to plant corn? Changing Calls Cell Name Final Value Reduced Cost Objective Caefficiant Allowable Inerease Allowable Decrease \8 \4 Acres of Carm 9.52 0 2100 1+30 350 Acres of Pumplan 0 -500.01 899.99 500.01 1+30 D \4 Acres of Beans 10.79 0 1050 210 375.00

Solve this problem graphically. What is the optimal solution and what constraints are binding at the optimal solution? $\mathrm { MAX } : \quad \mathbf { 8 \mathbf { X } _ { 1 } } + 4 \mathbf { X } _ { \mathbf { 2 } }$ Subject to: 5+5\leq20 6+2\leq18 ,\geq0

A binding less than or equal to ( $\le$ ) constraint in a maximization problem means

For a minimization problem, if a decision variable's final value is 0, and its reduced cost is negative, which of the following is true?

When a manager considers the effect of changes in an LP model's coefficients he/she is performing

When a solution is degenerate the reduced costs for the changing cells

Which of the following statements is false concerning either of the Allowable Increase and Allowable Decrease columns in the Sensitivity Report?

Constraint 3 is a non-binding constraint in the final solution to a maximization problem. Complete the following entry for the Risk Solver Platform (RSP) sensitivity report. Cell labels are included to ease of reference. Constraint Cell Name Final Value Shadow Price Constraint R.H. Side Allowable Inerease Allowable Decrease \D \8 Constraint 3 6 ?? 10 ?? ??

If the allowable increase for a constraint is 100 and we add 110 units of the resource what happens to the objective function value?

Exhibit 4.2 The following questions correspond to the problem below and associated Risk Solver Platform (RSP) sensitivity report. Robert Hope received a welcome surprise in this management science class; the instructor has decided to let each person define the percentage contribution to their grade for each of the graded instruments used in the class. These instruments were: homework, an individual project, a mid-term exam, and a final exam. Robert's grades on these instruments were 75, 94, 85, and 92, respectively. However, the instructor complicated Robert's task somewhat by adding the following stipulations: -homewark can account far up to $25 \%$ of the grade, but must be at least $5 \%$ af the grade; - the praject can account for up to $25 \%$ of the grade, but must be at least $5 \%$ af the grade; - the mid-term and final must each accaunt far betwen $10 \%$ and $40 \%$ of the grade but cannot accaunt far mare than $7 \%$ of the grade when the percentages are cambined; and -the project and final exam grades may not collectively constitute more than $50 \%$ of the Iratade. The following LP model allows Robert to maximize his numerical grade. Let = weight as51gMed to hamewark = waight as5igned to the praject = weight assigned to the midi-term = waight as5igned to the final MAX: 75+94+85+92 Subject ta: +++=1 +\leq0.70 +\geq0.50 0. 05\leq\leq0.25 0. 05\leq\leq0.25 0.10 \leq\leq0.40 0.10\leq\leq0.40 $\text { Adjustaibla Calls}$ Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease \ F\ 5 Mid Term to grade 0.40 10.00 85 1+30 10 \ F\ 6 Final to grade 0.25 0.00 92 2 17 \ F\ 7 Project to grade 0.25 2.00 94 1+30 2 \ F\ 8 Homework to grade 0.10 0.00 75 10 1+30 $\text { Constraints }$ Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease \ E\ 14 Both Exams Total 0.65 0 0.7 1+30 0.05 \ E\ 15 Final \& Project Total 0.5 17 0.5 0.05 0.15 \ F\ 9 100\% to gracle 1.00 75.00 1 0.15 0.05 -Refer to Exhibit 4.2. Based on the Risk Solver Platform (RSP) sensitivity report information, Robert has been approved by his instructor to increase the total weight allowed for the project and final exam to 0.50 plus the allowable increase. When Robert re-solves his model, what will his new final grade score be?

Solve this problem graphically. What is the optimal solution and what constraints are binding at the optimal solution? MIN: $\quad 7 \mathbf { X } _ { 1 } + 3 \mathbf { X } _ { 2 }$ Subject ta: 4+4\geq40 2+3\geq24 ,\geq0

A change in the right hand side of a constraint changes

The optimization technique that locates solutions in the interior of the feasible region is known as _____?

The allowable decrease for a changing cell (decision variable) is

If the shadow price for a resource is 0 and 150 units of the resource are added what happens to the optimal solution?

Given an objective function value of 150 and a shadow price for resource 1 of 5, if 10 more units of resource 1 are added (assuming the allowable increase is greater than 10), what is the impact on the objective function value?

Use slack variables to rewrite this problem so that all its constraints are equality constraints. $\mathrm { MAX } : \quad 2 \mathbf { X } _ { 1 } + 7 \mathbf { X } _ { \mathbf { 2 } }$ Subject to: +\leq90 +\leq144 \leq \geq0