Consider an investment problem where X is the number of shares of a pharmaceutic
ID: 3142560 • Letter: C
Question
Consider an investment problem where X is the number of shares of a pharmaceutical stock that are purchased and Y is the number of shares of a bank stock that are purchased. The LP has several constraints, but one is a budget constraint that is of the form: 25X + 50Y = 10,000. Thus, a share of X costs $25 and a share of Y costs $50 and a sum of $10,000 is available for investment. Consider the following information from the LP solution:
Optimal value of Objective Function: $12,500
Optimal value of Decision Variables: X = 128.5; Y= 135.75
Allowed increase in the RHS of Budget Constraint = $1 E + 30
Allowed decrease in the RHS of Budget Constraint = $2,750
Shadow Price of Budget Constraint= $2.50
Answer the following questions considering the information above. The only answers possible for the following questions are "$5,000, "$7,500", "$10,000", "$12,500", "$15,000", or "$17,500":
1.) If you discover that you have an additional $2000 to add to the budget constraint, what will be the new value of the Objective Function?
2.) If you decrease the RHS of the budget constraint to $9000, what will be the new value of the Objective Function?
State if the following question is True or False:
3.) If the RHS of the Budget Constraint is changed to $7000, the Shadow Price will change to some value other than $2.50. TRUE OR FALSE?
Explanation / Answer
SOLUTION:-
1) The shadow price is the instantaneous change, per unit of the constraint, in the objective
value of the optimal solution of an optimization problem obtained by relaxing the constraint.
In other words, it is the marginal utility of relaxing the constraint, or, equivalently, the marginal
cost of strengthening the constraint.
Here
Shadow Price of Budget Constraint= $2.50
so adding 2000 will increase the Objective Function by 2.5*2000 = 5000
Hence the new value of the Objective Function = 12500+5000 = 17500.
2) If you decrease the RHS of the budget constraint to $9000, what will be the new value of the Objective Function?
decrease= 10000-9000 = -1000
Hence decease in the Objective Function by 2.5*1000 = 2500
hence the new value of the Objective Function = 12500-2500 = 10000
3) if the RHS of the Budget Constraint is changed to $7000, the Shadow Price will change to some value other than $2.50.
it is true
as Allowed decrease in the RHS of Budget Constraint = $2,750 ,
since 7000> 2750
the Shadow Price will change.