Question 1

A firm produces shampoo (S) and conditioner (C). The profit contribution of shampoo is $3 per unit, while conditioner's contribution is $6 per unit. What is the slope, (∆C/∆S) of the firm's objective function?

Accepted Answer

The slope of the firm's objective function, represented as (∆C/∆S), indicates the rate at which the firm can trade off producing conditioner for shampoo while keeping profit constant. Given the profit contributions ($3 for shampoo and $6 for conditioner), the slope is calculated as the negative ratio of the profit contributions: -($3/$6) = -0.5. This means for every unit increase in shampoo production, the firm has to reduce conditioner production by 0.5 units to maintain the same profit level.

Question 2

In a linear programming problem, multiple optimal solutions are possible if:

Accepted Answer

Multiple optimal solutions in a linear programming problem occur when the slope of the objective function is parallel to the slope of a binding constraint. This means that the objective function's contour lines and a constraint line coincide over a range, indicating that any point along this range on the constraint line gives the same optimal value.

Question 3

In a linear programming problem, the objective function:

Accepted Answer

The objective function is the mathematical expression that represents the goal or target of the linear programming problem, and it is formulated in terms of the relevant decision variables. Its purpose is to maximize or minimize a certain value, such as profit or cost, while satisfying the constraints defined by the problem. Therefore, option A is the correct choice.

Question 4

To identify the feasible region, one must graph:

Accepted Answer

The feasible region in a graphical linear programming problem is identified by graphing all binding constraints, which are the constraints that form the edges of the feasible region. Non-binding constraints, objective function contours, the complete set of simultaneous equations, and capacities of all relevant resources do not directly define the feasible region's boundaries.

Question 5

A firm is pondering the introduction of a new good with a profit contribution of $70 per unit. Each unit of the good uses 2 units of input A and 3 units of input B. The shadow price of A is $10, and the shadow price of input B is $15. Which of the following statements is true?

Accepted Answer

The shadow prices represent the opportunity cost of using one more unit of each input. Since the shadow price of input A is $10 and one unit of the new good uses 2 units of input A, the opportunity cost of using 2 units of input A is $20. Similarly, the opportunity cost of using 3 units of input B is $45. Therefore, the total opportunity cost of producing one unit of the new good is $65 ($20+$45), leaving a profit contribution of $70. Thus, producing the new good would be profitable and the firm should introduce it. Option A states this, so it is the correct answer. Option B is incorrect because it calculates the total opportunity cost incorrectly by multiplying the shadow prices by the amounts of inputs used, rather than by the amounts of inputs given in the problem. Option C is incorrect because it does not take into account the profit contribution of the new good. Option D is incorrect because it only states the value of the shadow price of input A, which is not relevant to calculating the profit earned by producing the new good. Option E is incorrect because it assumes that the total cost of producing one unit of the new good is $75, which is incorrect.

Question 6

In an LP problem the inequalities 2X + Y ≤ 800 and X + 2Y ≤ 700, hold as binding constraints. The optimal solution is:

Accepted Answer

The optimal solution can be found by solving the system of equations formed by the given binding constraints: 2X + Y = 800 and X + 2Y = 700. Solving these equations simultaneously gives X = 300 and Y = 200.

Question 7

The combination of decision variables optimizing a linear programming problem, occurs at:

Accepted Answer

The optimal solution of a linear programming problem occurs at a point where the objective function contour touches the feasible region, not necessarily intersecting or being inside the feasible region. This is because the objective function represents the value to be optimized (maximized or minimized), and its optimal value is found on the boundary of the feasible region defined by the constraints.

Question 8

The optimal solution in a linear programming problem:

Accepted Answer

The optimal solution in a linear programming problem occurs at a corner of the feasible region, where at least one binding constraint is active. This is known as the corner-point or vertex solution. It is not necessary for all constraints to be binding at the optimal solution.

Question 9

In an LP problem, the feasible region contains values of the decision variables that:

Accepted Answer

The feasible region in an LP problem contains all the values of the decision variables that satisfy all the constraints, including both binding and nonbinding constraints. Therefore, option D is the best choice as it specifies that the feasible region includes only those values of the decision variables that satisfy all the binding constraints.

Question 10

Linear programming is useful for solving optimization problems involving:

Accepted Answer

Linear programming is specifically designed to solve optimization problems with linear objectives and linear constraints. While linear programming can also handle a large number of decision variables, the other choices are not accurate descriptions of the types of problems that can be solved using linear programming.

Question 11

In a linear programming problem, there are 3 binding constraints. Constraint A has slope -1.5, constraint B has slope -0.5, and constraint C has slope -0.2. The objective function's slope is -1.2. Where would the optimal solution lie?

Accepted Answer

The optimal solution lies at the intersection of constraints A and B because the objective function's slope (-1.2) is between the slopes of these two constraints (-1.5 and -0.5), indicating that the optimal point is at their intersection.

Question 12

A shadow price measures the:

Accepted Answer

A shadow price is defined as the change in the value of the objective function associated with a unit change in a particular constrained resource. It reflects how much the optimal value of the objective function would change if an additional unit of the resource were available. Therefore, option D is the correct answer.

Question 13

Which of the following statements concerning sensitivity analysis is incorrect?

Accepted Answer

Sensitivity analysis is not used to solve equations, but rather to analyze changes in the objective or constraint functions.

Question 14

If the shadow price of a given resource is $100 and the cost of expanding the capacity of that resource is $80 per unit, then the expansion of the capacity will:

Accepted Answer

The shadow price of $100 indicates that the market is willing to pay up to $100 for each additional unit of the resource. Therefore, expanding the capacity by adding units of the resource at a cost of $80 per unit will allow the firm to sell additional units at a price of $100, resulting in a profit of $20 per unit. Thus, expanding the capacity will earn positive profits.

Question 15

In a linear programming problem, the goal to be maximized (or minimized) is referred to as the:

Accepted Answer

The goal to be maximized or minimized in a linear programming problem is referred to as the objective function. This function represents the quantity that is to be optimized or minimized subject to certain constraints. Therefore, the correct choice is option C, the objective function.

Question 16

A small machine shop produces steel shafts and metal plates. The production process depends on labor, milling machine hours, and lathe machine hours. The firm tries to identify how many metal shafts and plates should be produced per week in order to maximize profit. Which, among the following, are the decision variables?

Accepted Answer

The quantities of shafts and plates that can be produced per week are the decision variables that the shop can control in order to maximize profit. The other options listed are constraints or parameters that the shop must take into consideration when making their decision.

Question 17

In a linear programming problem involving two decision variables, a change in one of their coefficients in the objective function will:

Accepted Answer

A change in the coefficient of one decision variable in the objective function will generally change the slope of the objective function's contour, which can in turn affect the optimal solution. However, the magnitude of this change and its impact on the optimal solution will depend on the specific values of the coefficients and the constraints in the problem. So, a change in one coefficient will only change the optimal solution if the slope of the objective function's contour changes significantly.

Question 18

In an LP problem, the goal is to maximize the objective function S + 3T, subject to the binding constraints S + T ≤ 700 and S + 2T ≤ 1,000. The optimal solution is:

Accepted Answer

The answer of In an LP problem, the goal is...

Question 19

In a linear programming problem, the inequalities, X + 2Y ≤ 12 and 3X + 4Y ≥ 28, hold as binding constraints. The problem's optimal solution would be:

Accepted Answer

When a constraint holds as a binding constraint, it means that the optimal solution lies on the boundary of that constraint. Therefore, we can use the given inequalities to find the intersection point of their boundaries.

For X + 2Y ≤ 12, we can rewrite it as Y ≤ (-1/2)X + 6 and graph it as a straight line with intercepts at (0,6) and (12,0).

For 3X + 4Y  28, we can rewrite it as Y  (-3/4)X + 7 and graph it as a straight line with intercepts at (0,7) and (9.33,0).

The feasible region is the region that satisfies both inequalities, which is bounded by the x-axis, the line Y = (-1/2)X + 6, and the line Y = (-3/4)X + 7.

The optimal solution is the point on the boundary of a feasible region that maximizes the objective function. Since the objective function is not given in the problem, we cannot determine the exact solution. However, we can see that the only point on the boundary that satisfies both binding constraints is the intersection point of the line X + 2Y = 12 and 3X + 4Y = 28, which is X = 4 and Y = 4 (choice C). Therefore, C is the correct answer.

Question 20

In an LP problem, the goal is to maximize the objective function 2S + 3T, subject to the binding constraints S + T ≤ 700 and S + 2T ≤ 1,000. The optimal solution is:

Accepted Answer

To find the optimal solution, we can substitute the values of S and T from each option into the objective function and the constraints. Option A, S = 400 and T = 300, gives the highest value of the objective function (2*400 + 3*300 = 1700) while satisfying both constraints (400 + 300 = 700, which is ≤ 700, and 400 + 2*300 = 1000, which is ≤ 1000).

Quiz 16: Linear Programming

A firm produces shampoo (S) and conditioner (C) . The profit contribution of shampoo is $3 per unit, while conditioner's contribution is $6 per unit. What is the slope, (∆C/∆S) of the firm's objective function?