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Chapter 4: Problem 33

Y o u ~ m a n a g e ~ a n ~ i c e ~ c r e a m ~ f a c t o r y ~ t h a t ~ makes two flavors: Creamy Vanilla and Continental Mocha. Into each quart of Creamy Vanilla go 2 eggs and 3 cups of cream. Into each quart of Continental Mocha go 1 egg and 3 cups of cream. You have in stock 500 eggs and 900 cups of cream. Draw the feasible region showing the number of quarts of vanilla and number of quarts of mocha that can be produced. Find the corner points of the region. HINT [See Example 4.1

Short Answer

Expert verified
The feasible region for the ice cream factory problem can be determined by graphing the system of inequalities with constraints: \(2V + M \le 500\) (eggs), \(3V + 3M \le 900\) (cream), \(V \ge 0\), and \(M \ge 0\). The feasible region is a polygon representing all possible combinations of producing Creamy Vanilla (V) and Continental Mocha (M) based on the available stock. To find the corner points of this region, consider all intersections between boundary lines and solve for coordinates (V, M), keeping only intersections that are within the feasible region. These corner points represent the vertices of the polygon and can be used for further analysis, such as finding the optimal production allocation.

Step by step solution

01

Formulate the inequalities

We need to create inequalities based on the given information: - For every quart of Creamy Vanilla, 2 eggs and 3 cups of cream are required. - For every quart of Continental Mocha, 1 egg and 3 cups of cream are required. - 500 eggs and 900 cups of cream are available in stock. Let V be the number of quarts of Creamy Vanilla, and M be the number of quarts of Continental Mocha. The inequalities representing the constraints on eggs and cream are: 1. \(2V + M \le 500\) (Eggs constraint) 2. \(3V + 3M \le 900\) (Cream constraint) Further, both V and M must be non-negative, as we cannot produce a negative number of quarts: 3. \(V \ge 0\) 4. \(M \ge 0\) Step 2: Graph the inequalities
02

Graph the inequalities

Now we need to graph these inequalities to find the feasible region. The best way to graph these inequalities is to first plot each constraint as if it were an equation (equality) by considering the boundary lines: 1. \(2V + M = 500\) 2. \(3V + 3M = 900\) Graph these two equations, along with the equations \(V = 0\) and \(M = 0\). Then apply the inequalities by shading the appropriate region for each constraint, including regions where V and M are non-negative. Step 3: Find the feasible region
03

Find the feasible region

The feasible region will be where all shaded regions from the inequalities overlap, taking into account the constraints on eggs, cream, and non-negativity of V and M. This feasible region is a polygon and represents all possible combinations of producing Creamy Vanilla and Continental Mocha based on the available stock. Step 4: Identify corner points
04

Identify corner points

The corner points are the vertices of the polygon that forms the feasible region. These points can be found by solving the system of inequalities and finding intersection points between constraints. Since there are four inequalities in our system, we will have to consider all four intersections (where the boundary lines meet) to determine the corner points. Remember, we are only considering intersections that are within the feasible region. The corner points are the coordinates (V, M). Following these steps, you will have drawn the feasible region for the ice cream factory problem, and found the corner points of the region.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constraint Optimization
Linear programming involves finding the best solution under given conditions. In an ice cream factory producing Creamy Vanilla and Continental Mocha, this is achieved by maximizing or minimizing something, such as profit or cost, considering the constraints of available resources: eggs and cream.
To start, inequalities representing these constraints are formulated based on resource availability.
This process of forming inequalities helps the decision-maker understand how different combinations of ice cream quarts can be optimized within given boundaries.
  • The first inequality represents eggs availability: \(2V + M \le 500\)
  • The second inequality represents cream availability: \(3V + 3M \le 900\)
  • Non-negativity constraints ensure that production quantities can't be negative: \(V \ge 0\) and \(M \ge 0\)
Feasible Region
The feasible region is the area on a graph where all constraints overlap and align. In linear programming, this region depicts all possible solutions that meet the given constraints.
For our ice cream factory, it represents all points or combinations of Creamy Vanilla and Continental Mocha quarts that satisfy both resource limits.
Creating this region involves graphing the inequality constraints:
  • Graph \(2V + M = 500\) and \(3V + 3M = 900\) as equality lines first.
  • Shade the areas that satisfy the original inequalities.
  • Only the overlapping shaded area, where both constraints are met, is the feasible region.
  • Ensure that this region adheres to the non-negativity constraints \(V \ge 0\) and \(M \ge 0\).
The feasible region is crucial because it logically narrows down the range of options, enabling precise decision-making.
Graphical Method
A graphical method in linear programming provides a visual way to solve constraint optimization problems. In our scenario, this involves visually plotting inequality constraints on a graph.
To use this method, follow these steps:
  • Convert each inequality into an equality line.
  • Plot these lines on a coordinate plane where the axes represent the different ice cream quarts (e.g., V and M).
  • Identify and shade areas that meet each constraint.
  • Highlight the area where all shaded sections overlap鈥攖his is your feasible region.
  • Mark all corners or vertices of this region; these are the possible optimal solutions.
Using the graphical method allows for direct visualization of constraints and potential solutions, making it easier to interpret complex scenarios like the allocation of limited resources.
Inequalities in Two Variables
Inequalities in two variables form the basis of constraint optimization problems like in our ice cream factory scenario. These inequalities define how resources can be distributed across different options.
Each inequality involves two variables, in this case, the number of quarts of Creamy Vanilla (V) and Continental Mocha (M).
Here's how they work:
  • The coefficients in the inequality represent the number of resources needed per quart and are used to form constraints like \(2V + M \le 500\) and \(3V + 3M \le 900\).
  • The constant in the inequality (500, for eggs, and 900 for cream) represents the total resources available.
  • The inequalities guide the creation of a feasible region, critical for making informed decisions about how much of each product can be produced.
Understanding these inequalities helps clarify how resource constraints translate into actionable limits in production scenarios.

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Most popular questions from this chapter

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The Scottsville Textile Mill produces several different fabrics on eight dobby looms which operate 24 hours per day and are scheduled for 30 days in the coming month. The Scottsville Textile Mill will produce only Fabric 1 and Fabric 2 during the coming month. Each dobby loom can turn out \(4.63\) yards of either fabric per hour. Assume that there is a monthly demand of 16,000 yards of Fabric 1 and 12,000 yards of Fabric 2. Profits are calculated as 33 d per yard for each fabric produced on the dobby looms. a. Will it be possible to satisfy total demand? b. In the event that total demand is not satisfied, the Scottsville Textile Mill will need to purchase the fabrics from another mill to make up the shortfall. Its profits on resold fabrics ordered from another mill amount to \(20 \mathrm{~d}\) per yard for Fabric 1 and \(16 \mathrm{e}\) per yard for Fabric \(2 .\) How many yards of each fabric should it produce to maximize profits?
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