Problem 9 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Acoustical Company manufactures a CD storage cabinet that can be bought fully assembled or as a kit. Each cabinet is processed in the fabrications department and the assembly department. If the fabrication department only manufactures fully assembled cabinets, then it can produce 200 units/day; and if it only manufactures kits, it can produce 200 units/day. If the assembly department only produces fully assembled cabinets, then it can produce 100 units/day; but if it only produces kits, then it can produce 300 units/day. Each fully assembled cabinet contributes $$\$ 50$$ to the profits of the company whereas each kit contributes $$\$ 40$$ to its profits. How many fully assembled units and how many kits should the company produce per day in order to maximize its profits?

Step-by-Step Solution

Answer

Acoustical Company should produce 100 fully assembled cabinets and 200 kits daily to maximize its profits, resulting in a total profit of $$\$ 13,000$$.

Step 1: Define the decision variables

Let's denote x as the number of fully assembled cabinets and y as the number of kits to be produced daily.

Step 2: Create the objective function

The objective is to maximize the profit. Each fully assembled cabinet contributes \(50 to the profits and each kit contributes \)40 to the profits. Thus, the total profit can be represented as the objective function: \[ P(x,y) = 50x + 40y \]

Step 3: Write the constraints based on production capacities

Based on the given capacities of the fabrication and assembly departments, we can write the following constraints: 1. Fabrication department for fully assembled cabinets: \[ x \leq 200 \] 2. Fabrication department for kits: \[ y \leq 200 \] 3. Assembly department for fully assembled cabinets: \[ x \leq 100 \] 4. Assembly department for kits: \[ y \leq 300 \]

Step 4: Set up the linear programming problem

We can now formulate our linear programming problem. Maximize the profit function subject to the constraints: Objective function: \[ Maximize \ P(x,y) = 50x + 40y \] Constraints: \[ x \leq 200 \] \[ y \leq 200 \] \[ x \leq 100 \] \[ y \leq 300 \] \[ x, y \geq 0 \]

Step 5: Solve the linear programming problem

To solve this problem, we can use the graphical method or any linear programming solver software. For this example, let's use the graphical method. Plot the constraints and find the feasible region. The feasible region is a rectangle defined by the intersection of the constraints, with vertices at (0,0), (0,200), (100,0), and (100,200).

Step 6: Evaluate the objective function at the vertices of the feasible region

We will now evaluate the profit function at each vertex: 1. Point (0,0): \[ P(0,0) = 50(0) + 40(0) = 0 \] 2. Point (0,200): \[ P(0,200) = 50(0) + 40(200) = 8000 \] 3. Point (100,0): \[ P(100,0) = 50(100) + 40(0) = 5000 \] 4. Point (100,200): \[ P(100,200) = 50(100) + 40(200) = 13000 \]

Step 7: Determine the optimal solution

The maximum profit is $$\$ 13,000$$, which occurs when 100 fully assembled units (x=100) and 200 kits (y=200) are produced per day. Therefore, in order to maximize its profits, the company should produce 100 fully assembled cabinets and 200 kits daily.

Key Concepts

Objective Function

The objective function is a core concept in linear programming. It represents the goal of our problem, which in many cases is to maximize or minimize a certain quantity. In the context of the Acoustical Company's problem, the aim is to maximize profit. Each fully assembled cabinet contributes \(50 to profits, while each kit contributes \)40. Thus, the objective function is defined as: \[ P(x,y) = 50x + 40y \] Here, \(P(x,y)\) denotes the total profit, and \(x\) and \(y\) represent the number of fully assembled cabinets and kits, respectively. To make full sense of the function, it's important to note that every component of the expression correlates directly with the decisions being made. Decisions here involve the quantities of cabinets and kits produced per day.

Constraint Analysis

In linear programming, constraints refer to the limitations or restrictions placed on the decision variables within the problem. These constraints often stem from the available resources or abilities of the system being analyzed. In this example, the Acoustical Company faces constraints in both the fabrication and assembly departments. Here they are in detail: - The fabrication department can produce no more than 200 units per day, whether they are fully assembled cabinets or kits. Hence, we have: \[ x \leq 200 \] \[ y \leq 200 \] - The assembly department constraints allow only up to 100 fully assembled cabinets or up to 300 kits per day: \[ x \leq 100 \] \[ y \leq 300 \] The above constraints ensure that the solution respects the limits of production capacity, maintaining an efficient use of the company's resources without exceeding capabilities.

Feasibility Region

The feasibility region in linear programming is the graphical representation of all possible values that satisfy a given set of constraints. It is often illustrated as a shape or area on the graph where all the conditions are true, showing possible solutions that work under the problem's limitations. In this problem, the feasibility region is a rectangle formed by the intersecting constraints: - Features key points at (0,0), (0,200), (100,0), and (100,200). - Each vertex represents a potential solution for the production of fully assembled units and kits. Visualizing this area assists in comprehending where feasible solutions fall within the constraints, making sure we're maximizing the objective (profit) while abiding by the company's production limits.

Graphical Method

The graphical method is a handy technique used in linear programming, particularly when dealing with a small number of variables. For two-variable problems, like in our exercise, this method provides a clear and visual approach to finding the optimal solution. Here's how it works: - Plot each of the constraints on a graph, creating lines that form the boundaries. - Identify the feasibility region, which is bounded by these constraint lines. - Evaluate the objective function at each vertex of the feasibility region. These points are essentially where the boundaries of the feasible set intersect. - Determine which vertex provides the highest or lowest value of the objective function, based on the aim of the problem (maximization in this case). Applying this method allowed the Acoustical Company to efficiently pinpoint the optimal daily production schedule of 100 fully assembled units and 200 kits, maximizing profits to $13,000.