Quant Test 3
Assuming all other parameters remain unchanged, if the objective function coefficient associated with "Large Rooms" increases by $15, what will be the change in the objective function value?
increases by $900 Allowable increase is 10^30 (infinit number). $15 is within the allowable increase. The optimal solution (60,40) will not change. Let's compute new maximum of earnings 65X+20Y=65(60)+20(40)=3900+800=4700. 4700-3800=900. Or 15*60=900
Assuming all other parameters remain unchanged, if another firm is ready to lease out its Storage Space to Personal Mini Warehouses, what is the maximum monthly rental per unit of storage space below which Personal Mini Warehouses should accept the offer?
$0.4 $0.40 because shadow price is 0.40. The shadow price is the maximum amount a manager would pay for one additional unit of resource.
If the "Advertising Budget" available were only $370, what would Personal Mini Warehouses' new optimal profit be?
$3800 Maximize monthly earnings = 50X + 20Y Subject to 2X + 4Y ≤ 400 (monthly advertising budget available) 100X + 50Y ≤ 8,000 (storage space) X ≤ 60 (rental limit expected), X, Y ≥0 RHS value of "monthly advertising budget available" constraint is 370 now. That represents decrease by 30 units in the value of RHS. Allowable decrease is 120 (table "Constraints"). 30 is within the allowable increase of 120. The shadow price 0 is operative. The optimal profit is $3800.
If the "Rental Limit" constraint's right-hand side were 70, what would have been optimal profit of the Personal Mini Warehouses?
$3900 Maximize monthly earnings = 50X + 20Y Subject to 2X + 4Y ≤ 400 (monthly advertising budget available) 100X + 50Y ≤ 8,000 (storage space) X ≤ 60 (rental limit expected) , X, Y ≥0. X ≤ 75. RHS value of "Rental Limit" is 70 now. That represents increase by 10 units in the value of RHS. Allowable increase is 20 (table "Constraints"). 10 is within the allowable increase of 20. The shadow price 10 is operative. Let's compute the increase in the value of optimal profit: 10*10=100.New value of the optimal profit is 3800+100=3900.
Assuming all other parameters remain unchanged, if another firm leases out 400 units its Storage Space to Personal Mini Warehouses at monthly rental $12 per 100 units of storage space, what will the new total profit be?
$3912 100X + 50Y ≤ 8,000+400 100X + 50Y ≤ 8,400 Let's find an improve of the optimal solution value: 400*0.4(shadow price) =$160 12*4=48 (we should pay for leasing) $160-$48=$112 $3,800+$112=$3,912
Consider the following linear programming problem: Maximize 5X+6Y Subject to: 4X+2Y<=420 1X+2Y<=120 all variables>=0 Which of the following points (X,Y) is not a feasible corner point?
(120,0)
Consider the following linear programming problem: Maximize 5X+6Y Subject to: 4X+2Y ≤ 420 1X+2Y ≤ 120 All variables≥0 Which of the following points (X,Y) is not feasible?
(50,40)
The feasible corner points are (48,84), (0,120), (0,0), (90,0). What is the maximum possible value for the objective function?
1200
Consider the following linear programming problem: Maximize 12Y+10Y Subject to: 4X+3Y<=480 2X+3Y<=360 all variables>=0 The maximum possible value for the objective function is
1520
Based on the optimal solution, what is the optimal monthly earnings?
60*50 + 40*20 = 3,800
Which of the following functions is not linear?
A) 5X + 3Z B) 3X + 4Y + Z - 3 C) 2X + 5YZ D) Z E) 2X - 5Y + 2Z
Which of the following is not one of the steps in formulating a linear program?
A) Graph the constraints to determine the feasible region. B) Define the decision variables. C) Use the decision variables to write mathematical expressions for the objective function and the constraints. D) Identify the objective and the constraints. 12 E) Completely understand the managerial problem being faced
Sensitivity analyses are used to examine the effects of changes in
A) contribution rates for each variable. B) technological coefficients. C) available resources. D) All of the above
Typical resources of an organization include
A) machinery usage. B) labor volume. C) warehouse space utilization. D) raw material usage. ALL OF THE ABOVE
Sensitivity analysis may also be called
A) postoptimality analysis. B) parametric programming. C) optimality analysis. D) All of the above
Assuming all other parameters remain unchanged, what is the range of the objective function coefficient associated with Small Rooms for which the current optimal solution still remains optimal?
Between 0 and 25 Let's compute sensitivity range for objective function coefficient associated with Small Rooms: 20-current objective function coefficient, 5-allowable increase, 20-allowable decrease (table "Variable Cells). Upper limit 20+5=25 Low limit: 20-20=0
Which of the following is not a part of every linear programming problem formulation?
D) a redundant constraint WHAT IS THO: A) an objective function B) a set of constraints C) non-negativity constraints E) maximization or minimization of a linear function
An objective function is necessary in a maximization problem but is not required in a minimization problem.
FALSE
Any linear programming problem can be solved using the graphical solution procedure.
FALSE
In a linear program, the constraints must be linear, but the objective function may be nonlinear.
FALSE
In some instances, an infeasible solution may be the optimum found by the corner point method.
FALSE
The term slack is associated with ≥ constraints.
FALSE
There are no limitations on the number of constraints or variables that can be graphed to solve an LP problem
FALSE
If the objective coefficient for large rooms changed to 45 and that for small rooms changed to 25, the optimal solution for the original problem will remain optimal.
False Z= Maximize monthly earnings = 50X + 20Y New version: Z=45X+25Y Objective Function Coefficient for Large rooms : 50-45=5 -decrease by 5 5 is within the allowable decrease of 10 (table "Constraints"). (5/10)*100%=50% Objective Function Coefficient for Small rooms: 25-20=5-increase by 5 5 is within the allowable increase of 5 (table "Constraints"). (5/5)*100%=100% Sum of the percentages: 50%+100%=150% 150%>100% Thus, the answer is False. Please, look at Interpreting Excel's Sensitivity Report -- 100 % Rule. The upper table of the Sensitivity Report can be used to identify changes to one or more objective function coefficients for which the current optimal solution will remain optimal. The table provides for each objective function coefficient an allowable increase and an allowable decrease. If multiple objective function coefficients are changed (each either increasing or decreasing within its allowable range) and if the sum of those percentage changes is ≤ 100%: the current optimal solution will remain optimal; and the resulting optimal value of the objective function can be calculated using the "new" set of coefficients
Assuming all other parameters remain unchanged, if another firm is ready to provide a loan to increase the Advertising Budget of Personal Mini Warehouses at an interest rate of 0.8% per month, should Personal Mini Warehouses accept the offer?
No No because shadow price is 0. If we add some units of resources, it will have no impact on the optimal value of the objective function. Please, note: A non-binding constraint will always have a shadow price of 0 and changing the RHS of a non-binding constraint by any amount within its allowable increase or decrease will have no impact on the optimal solution and no impact on the optimal value of the objective function
A linear programming problem contains a restriction that reads "the quantity of Q must be no larger than the sum of R, S, and T." Formulate this as a linear programming constraint.
Q − R − S − T ≤ 0
Which of the constraints is binding?
Storage Space &Rental Limit RHS=LHS Storage Space 8,000=8,000 Rental Limit 60=60
Management resources that need control include machinery usage, labor volume, money spent, time used, warehouse space used, and material usage.
TRUE
Resource restrictions are called constraints.
TRUE
Sensitivity analysis enables us to look at the effects of changing the coefficients in the objective function, one at a time.
TRUE
The set of solution points that satisfies all of a linear programming problem's constraints simultaneously is defined as the feasible region in graphical linear programming.
TRUE
The solution to a linear programming problem must always lie on a constraint.
TRUE
The term surplus is associated with ≥ constraints.
TRUE
Assuming all other parameters remain unchanged, if the objective function coefficient associated with "Large Rooms" increases by $24, what will be the change in the optimal solution?
The optimal number of Large Rooms remains the same Allowable increase is 10^30 (infinit number). $24 is within the allowable increase. The optimal solution (60,40) will not change. The optimal number of large room will remain the same-60 rooms.
The optimal solution to a linear programming problem lies within the feasible region.
True
If 500 additional units become available for the right-hand side constraint of storage space, and if 11 fewer units for the right-hand side constraint of the rental limit were available, the maximum profit will be $3,890 and the optimal solution will change.
True Interpreting Excel's Sensitivity Report. 100 % Rule The lower table of the Sensitivity Report can be used to assess the impact of changing (within limits) the RHS of one or more constraints. If, making no other changes, multiple constraints have their RHS changed, each increased by some percentage of its allowable increase or decreased by some percentage of its allowable decrease, and if the sum of those percentages is ≤ 100%: The optimal value of the objective function will be changed by the sum of the changes attributable to each individual change in a RHS; and a change in the optimal value of the objective function will accompanied by a change in the optimal solution RHS constraint of the additional storage space: 500 units of the additional storage space=500/1500=33.33% of allowable increase. RHS constraint of the of the rental limit: 11 fewer units of the rental limit=11/20=55% of allowable decrease. Sum of the percentages=33.33%+55%=88.33% ≤100%, the shadow prices are operative. The maximum profit would be 50(60)+20(40)+500(0.4)-11(10)=$3,890. Yes, the optimal solution would change to a more profitable one.
Which of the following is not a property of all linear programming problems?
a computer program
The simultaneous equation method is
an algebraic means for solving the intersection of two or more constraint equations.
A constraint with zero slack or surplus is called a
binding constraint
The mathematical theory behind linear programming states that an optimal solution to any problem will lie at a(n) ________ of the feasible region.
corner point or extreme point
Assuming all other parameters remain unchanged, if the objective function coefficient associated with "Small Rooms" decreases by $8, what will be the change in the objective function?
decreases by $320 The allowable decrease is $20. $8 is within the allowable decrease of $20. Thus the optimal solution (60,40) will remain optimal. Let's compute new maximum of earnings 50X+12Y=50(60)+12(40)=3000+480=3480. 3800-3480=320. Or 8*40=320.
When appropriate, the optimal solution to a maximization linear programming problem can be found by graphing the feasible region and
finding the profit at every corner point of the feasible region to see which one gives the highest value.
What is a limitation of the graphical method over the "excel solver" method?
graphical method is not suitable when the number of decision variables is more than 2
A widely used mathematical programming technique designed to help managers and decision making relative to resource allocation is called
linear programming
A feasible solution to a linear programming problem
must satisfy all of the problem's constraints simultaneously
A feasible solution to a linear programming problem
must satisfy all of the problem's constraints simultaneously.
A constraint with positive slack or surplus is called a
nonbinding constraint
The maximization or minimization of a quantity is the
objective of linear programming
What type of problems use LP to decide how much of each product to make, given a series of resource restrictions?
product mix
The corner point solution method
requires that the profit from all corners of the feasible region be compared.
The difference between the left-hand side and right-hand side of a less-than-or-equal-to constraint is referred to as
slack
Whenever all the constraints in a linear program are expressed as equalities, the linear program is said to be written in
standard form
The difference between the left-hand side and right-hand side of a greater-than-or-equal-to constraint is referred to as
surplus
The coefficients of the variables in the constraint equations that represent the amount of resources needed to produce one unit of the variable are called
technological coefficients
A solution that satisfies all the constraints in an LP minimization problem and gives the lowest value in the objective function is called
the optimal solution
