Case Study: Optimization Of Manufacture Of Personal Computers

1. Introduction

In today's world, computers are ubiquitous in both the home and the workplace. As a result, there is a high demand for computer use, and many types of computers are being developed to fulfill the various needs of individuals. Clearly, there will be a lot of independent variables in the formulation of a real-world optimization problem incorporating computer manufacturing.

Two new computer products are being developed by a personal computer maker. A lot of factors, such as the number of computers sold, the selling price of the computers, the fixed manufacturing expenses, and so on, will affect the manufacturer's total earnings.

The goal is to assist the manufacturer in determining how many computers should be produced and sold in order to maximize annual earnings.

A mathematical model can be built and solved using multivariable calculus techniques if there are no constraints (i.e., unconstrained optimization). This option, on the other hand, assumes that the corporation has the capacity to build an endless number of computers each year.

In a real-world setting, restrictions on available production capacity must be placed, which might be influenced by a variety of circumstances. As a result, restricted optimization can be used to formulate and solve a mathematical model. To account for the introduction of imposed limitations, Lagrange multiplier methods and linear programming are used. The effects of changes in the key parameters can also be considered using sensitivity analysis.

2. Problem Statement

A manufacturer of personal computers is planning the introduction of two new products, a basic model with a retail price of $1250 and an enhanced model with a retail price of 00.

The cost to the company is $700 and $850 for basic and enhanced models, respectively, plus an additional $500,000 in fixed costs. In a competitive market, the number of sales per year will affect the average selling price. It is estimated that for each type of computer, the average selling price will drop by $0.1 for each additional unit sold. Also, sales of the basic model will affect sales of the enhanced model, and vice versa. It is estimated that the average selling price for the basic model will be reduced by an additional $0.03 for each enhanced model sold, and the price for enhanced models will decrease by $0.04 for each basic model sold.

Develop an analytical model to determine how many units of each type should be manufactured.

Now introduce constraints based on the available production capacity. It is estimated that the available production capacity will be sufficient to produce 4,000 computers per year. Due to the restriction of vital electronic parts, the supplier is able to supply parts for 2,000 basic models per year and for 3,000 enhanced models per year. How should the company set production levels? [Suggestion: Develop a mathematical model using Lagrange multiplier methods].

Reconsider the above mathematical model but now make the simplifying assumptions that the company makes a profit of $300 per basic model and $375 per enhanced model.

Find the optimum production levels by solving as a linear programming problem using the computer.

Determine the shadow prices for each constraint and explain what they mean. Which constraints are binding on the optimal solution?

Determine the sensitivity to the objective function coefficients (profit per unit). Consider both profit and optimal production levels.

Draw a graph of the feasible region and include a picture of ∇f at the optimum, where f is the objective function (Ng, 2004).

3. Model Formulation

To create our model, we must first create a list of variables that will be used, naming as shown below such as,

• X1= Basic model computers number of sold per year.
• X2= Enhanced model computers number of sold per year.
• P1= Basic model computers selling price ($).
• P2= Enhanced model computers selling price ($).
• C= Total cost of manufacturing ($year).
• R= Total Revenue from the computers sale ($year).
• P= Total Profit from the computers sale ($year).

Assumptions and arranged in,

P1= 1250 - 0.1 X1 - 0.03X2
P2= 1500 - 0.04 X1 - 0.1X2
R= P1* X1 + P2* X2
C= 500000 + 700 X1 + 850 X2
P= R - C
X1 ≥ 0
X2 ≥ 0

Our goal is to make the most profit P ($year) from computer sales.

The profit from the computer sales is given by,

P= R - C
= P1* X1 + P2* X2 - (500000 + 700 X1 + 850 X2)
= (1250 - 0.1 X1 - 0.03X2) X1 + (1500 - 0.04 X1 - 0.1X2) X2 - (500000 + 700 X1 + 850 X2)

Letting y = P,

y = f (X1, X2)
= (1250 - 0.1 X1 - 0.03X2) X1 + (1500 - 0.04 X1 - 0.1X2) X2 - (500000 + 700 X1 + 850 X2)

4. Mathematical Solution

4.1 Unconstrained Optimization

Proceed to solve the problem using normal multivariable calculus solution methods.

Partial derivatives of f(X1, X2) with respect to X1 and X2 respectively,

∂y/∂X1= 1250 - 0.2 X1 - 0.03 X2 - 0.04 X2 - 700 = 0
∂y/∂X2= 1500 - 0.04 X1 - 0.2 X2 - 0.03 X2 - 850 = 0

Solving the equation 3,

0.2 X1 + 0.07 X2 = 550
0.07 X1 + 0.2 X2 = 650

Cramer's rule,

One of the most essential approaches for solving a system of equations is Cramer's rule. The values of the system's variables are computed using determinants of matrices in this method. As a result, the determinant approach is often known as Cramer's rule (Cuemath, n.d.).

Using Cramer's rule,

X1= (5500.2 - 6500.07) / (0.20.2 - 0.070.07) = 1838
X2= (0.2650 - 0.07550) / (0.20.2 - 0.070.07) = 2607

Substituting the X1 and X2 in equation 2, maximum value of the objective function,

y = (1250 - 0.11838 - 0.032607) * 1838 + (1500 - 0.041838 - 0.12607) * 2607 - (500000 + 7001838 + 8502607)
y = 852564

The manufacture needs 1838 units of the basic model and 2607 of the enhanced model computers per year to make a maximum profit of $852,564.

4.2 Constrained Optimization

The same objective function,

y = f (X1, X2) = 300 X1 + 375 X2

The constraints,

X1 ≤ 2000
X2 ≤ 3000
X1 * X2 ≤ 4000
X1 ≥ 0
X2 ≥ 0

Using the Lagrange multiplier method,

The Lagrange multiplier method is a method for determining the maximum or minimum of a function F(x, y, z) subject to a constraint of the form G(x, y, z) = 0.

∇F (X, Y) = -λ ∇G (X, Y)

The scalar parameter λ is called a Lagrange multiplier (Salih, 2013).

Applying the Lagrange multiplier equation,

∇f = (550 - 0.2 X1 - 0.07 X2, 650 - 0.07 X1 - 0.2 X2)

Boundary segment on the constraint line,

∇g = g(X1, X2) = X1 * X2 = 4000
∇g = (1, 1)

The Lagrange multiplier equation is,

550 - 0.2 X1 - 0.07 X2 = λ
650 - 0.07 X1 - 0.2 X2 = λ

X1 = 1615
X2 = 2385
λ = 60

Substituting the equation 10 in the objective function, Profit $685,079.

With constrained one manufacture needs 1615 units of the basic model and 2385 of the enhanced model computers per year to make a maximum profit of $839,231.

4.3 Computer Solution (Linear Programming)

Linear Programming problem

The purpose of Linear Programming Problems (LPP) is to find the optimal value for a given linear function. The best value can be either the highest or the lowest. In this scenario, the stated linear function is regarded as an objective function. The objective function might have several variables that are subject to conditions, and it must meet a set of linear inequalities known as linear constraints (BYJUS, n.d.) and (Schulze, 2000).

The objective function as assumed,

y = f (X1, X2) = 300 X1 + 375 X2

The constraints,

X1 ≤ 2000
X2 ≤ 3000
X1 * X2 ≤ 4000
X1 ≥ 0
X2 ≥ 0

By using MATLAB (Code shown in Model validation) to calculate, the manufacture requests 1000 units of the basic model and 3000 of the enhanced model computers per year to make a maximum profit of $1,425,000.

4.3.1 Shadow Prices and Sensitivity Analysis

The dual prices are $0 for the basic model, $75 for the improved model, and $300 for each unit of production capacity.

y = C1 X1 + C2 X2
C1 = 300, C2 = 375, constraints with (1000, 3000)
S(X1, C1) = S(X1, C2) = 0
S(X2, C1) = S(X2, C2) = 0

Sensitivity of the function y to C1 and C2,

S(Y, C1) = (1000) * (300 - 1425000) = 0.21
S(Y, C2) = (3000) * (375 - 1425000) = 0.79

Re-optimize with MATLAB, the new objective functions are,

y = 303 X1 + 375 X2 and y = 300 X1 + 378.75 X2

The respective objective values are,

y = 1,428,000 and y = 1,436,250

Optimize production levels X1 and X2,

g (X1, X2) = X1 * X2 = C

5. Discussion

The analysis and optimization of the production levels for the two computer models provided valuable insights into how a manufacturer can maximize its profit while considering various constraints. In the unconstrained optimization scenario, where there were no restrictions on production capacity, the optimal solution was to produce 1,838 units of the basic model and 2,607 units of the enhanced model, resulting in a maximum profit of $852,564.

However, in real-world manufacturing, there are often constraints that must be taken into account. In the constrained optimization scenario, where production capacity limitations were imposed, the optimal solution was to produce 1,615 units of the basic model and 2,385 units of the enhanced model, leading to a maximum profit of $839,231. This scenario demonstrates the importance of considering practical constraints, such as production capacity, in decision-making processes.

Furthermore, the linear programming approach, implemented using MATLAB, provided another perspective on the problem. It suggested that manufacturing 1,000 units of the basic model and 3,000 units of the enhanced model would result in a maximum profit of $1,425,000. This approach allows manufacturers to use mathematical optimization techniques to determine the most profitable production levels while considering both constraints and profit objectives.

The shadow prices obtained from the linear programming solution provide valuable insights into the impact of constraint changes on profit. In this case, the shadow price for production capacity constraints was $300 per unit of capacity. This means that if additional production capacity were available, the company could increase its profit by $300 for each additional unit of capacity.

The sensitivity analysis revealed the sensitivity of the objective function to changes in profit coefficients. It showed that a slight increase in profit per unit of the basic model and enhanced model would result in higher maximum profits. This information is essential for decision-makers as it highlights the potential benefits of improving product profitability.

The graphical representation of the feasible region and the gradient of the objective function (∇f) at the optimum point provide a visual understanding of the problem. This visualization aids in comprehending the relationships between variables and constraints, making it easier to interpret and communicate the results to stakeholders.

Overall, the discussion highlights the significance of mathematical modeling and optimization techniques in real-world decision-making processes. These methods enable manufacturers to make informed choices regarding production levels, considering various factors such as pricing, costs, and production capacity constraints.

6. Conclusion

In conclusion, this lab report addressed the optimization of production levels for two computer models, the basic model and the enhanced model, while considering various constraints and profit objectives. Through mathematical modeling and optimization techniques, we derived valuable insights and solutions for the manufacturer.

In the unconstrained optimization scenario, we found that producing 1,838 units of the basic model and 2,607 units of the enhanced model would result in a maximum profit of $852,564. However, real-world manufacturing often involves constraints, and in the constrained optimization scenario, we determined that producing 1,615 units of the basic model and 2,385 units of the enhanced model would yield a maximum profit of $839,231.

Additionally, the linear programming approach using MATLAB suggested that manufacturing 1,000 units of the basic model and 3,000 units of the enhanced model would lead to a maximum profit of $1,425,000. This approach showcased the power of mathematical optimization in making production decisions.

Furthermore, the shadow prices and sensitivity analysis provided insights into the impact of constraint changes and profit coefficients on profit. The shadow price for production capacity constraints was $300 per unit of capacity, highlighting the potential for increased profit with additional capacity.

Overall, this study demonstrates the importance of mathematical modeling and optimization techniques in assisting manufacturers in making informed decisions to maximize profits while considering various real-world constraints. These tools are invaluable in the competitive business environment, where efficient resource allocation is crucial for success.