Consider the following Excel sensitivity report and it's accompanying problem:

Minimize cost = X + 2Y
subject to X + 3Y ≥ 90
8X + 2Y ≥ 160
3X + 2Y ≥120
Y ≤ 70
X, Y≥ 0

Variable Cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$B$3 X 25.71 0 1 2 0.333333333
$C$3 Y 21.43 0 2 1 1.333333333

Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$J$6           LHS 90 0.57 90 62 50
$J$7           LHS 248.57 0 160 88.57142857 1E+30
$J$8           LHS 120 0.14 120 150 28.18181818
$J$9           LHS 21.43 0 70 1E+30 48.57142857

Suppose we add another variable, x3, with an objective function coefficient of 9, and constraint coefficients of 8, 3, and 5 for the first three constraints, respectively. What is the marginal impact of this new variable on the objective function? (In your calculations round all the numbers to 2 decimals.)

Question

Consider the following Excel sensitivity report and it's accompanying problem:

Minimize cost = X + 2Y 	 
subject to 	X + 3Y ≥ 90
 	8X + 2Y ≥ 160 
 	3X + 2Y ≥120 
 	Y ≤ 70
 	 X, Y≥ 0

Variable Cells	 	 	 	 	 	 	 
 	 	 	Final	Reduced	Objective	Allowable	Allowable
 	Cell	Name	Value	Cost	Coefficient	Increase	Decrease
 	$B$3	X	25.71	0	1	2	0.333333333
 	$C$3	Y	21.43	0	2	1	1.333333333
 	 	 	 	 	 	 	 
Constraints	 	 	 	 	 	 	 
 	 	 	Final	Shadow	Constraint	Allowable	Allowable
 	Cell	Name	Value	Price	R.H. Side	Increase	Decrease
 	$J$6	          LHS	90	0.57	90	62	50
 	$J$7	          LHS	248.57	0	160	88.57142857	1E+30
 	$J$8	          LHS	120	0.14	120	150	28.18181818
 	$J$9	          LHS	21.43	0	70	1E+30	48.57142857

Suppose we add another variable, x3, with an objective function coefficient of 9, and constraint coefficients of 8, 3, and 5 for the first three constraints, respectively. What is the marginal impact of this new variable on the objective function? (In your calculations round all the numbers to 2 decimals.)

BlackTom AI · Accepted Answer

To assess the impact of adding a new variable x3 on the objective, we use the concept of reduced cost.
First, recall that for a new decision variable, the marginal effect on the objective when moving a unit of that variable from 0 to positive is given by its reduced cost rc_j = c_j - sum_i (y_i * a_ij), where c_j is the variable's objective coefficient, y_i are the dual prices (shadow prices) for the constraints, and a_ij are the constraint coefficients for the variable.
From the sensitivity report, the dual prices (shadow prices) for the three constraints are:
- y1 (for constraint 1) = 0.57
- y2 (for constraint 2) = 0
- y3 (for constraint 3) = 0.14
The new variable x3 has objective coefficient c_j = 9, and its coefficients in the three constraints are:
- a_13 = 8 (coefficient in constraint 1)
- a_23 = 3 (coefficient in constraint 2)
- a_33 = 5 (coefficient in constraint 3)
Next, compute the dot product of the dual prices with the variable's coefficients:
sum_i (y_i * a_ij) = (0.57 * 8) + (0 * 3) + (0.14 * 5) = 4.56 + 0 + 0.70 = 5.26
Now determine the reduced cost:
rc_j = c_j - sum_i (y_i * a_ij) = 9 - 5.26 = 3.74
This rc_j value represents the per-unit increase in the objective if x3 were introduced at a positive level.
Since the reduced cost is positive (3.74), adding x3 would raise the objective by 3.74 for each additional unit of x3, all else equal.
For completeness, if you round the intermediate calculations, 0.57 * 8 = 4.56 and 0.14 * 5 = 0.70, summing to 5.26, and 9 - 5.26 = 3.74, which matches the result above.
In summary, the marginal impact (per unit) of the new variable x3 on the objective function is 3.74 units.

BU.520.601.T2.FA25 Final Exam

查看解析

登录即可查看完整答案

类似问题

Solving a linear program can never result in integer values for the decision variables.

Consider the following optimization problem and the constraint boundary lines given below. Maximize profit = 4X + 4Y Constraints 3X + 2Y ≤ 150 X - 2Y ≤ 10 2X + 3Y ≤150 X, Y ≥ 0 When the constraint coefficient of x in the blue constraint changes from 1 to 3, the optimal solution changes.

A constraint is binding if it left hand side of a constraint is equal to the right hand side after the optimal solution is plugged into the constraint function.

A zero in the allowable increase or decrease columns of the variable cells table in the sensitivity report indicates that

It is not possible to have more than one optimal solution to a linear programming problem.

The objective function for a LP model is 3 X 1 + 2 X 2. If X 1 = 20 and X 2 = 30, what is the value of the objective function?

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单