Ronald Thump is interested in expanding his firm. After careful consideration, he has determined three areas in which he might invest additional funds: (1) product research and development, (2) manufacturing operations improvements, and (3) advertising and sales promotion. He has $500,000 available for investment in the firm. He can invest in its advertising and sales promotion program every year, and each dollar invested in this manner is expected to yield a return of the amount invested plus 20% yearly. He can invest in manufacturing operations improvements every 2 years, with an expected return of the investment plus 30% (at the end of each 2-year period). An investment in product research and development would be for a 3-year period, with an expected return of the investment plus 50% (at the end of the 3-year period). To diversify the total initial investment, he wishes to include the requirement that at least $30,000 must be invested in the advertising and sales promotion program, at least $40,000 in manufacturing operations improvements, and at least $50,000 in product research and development initially (at the beginning of the first year). Ronald wants to know how much should be invested in each of the three alternatives, during each year of a 4-year period, to maximize the total ending cash value of the initial $500,000 investment.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
a. Formulation of the linear programming model:
Let:
x1 = amount invested in advertising and sales promotion every year (in $)
x2 = amount invested in manufacturing operations improvements every 2 years (in $)
x3 = amount invested in product research and development every 3 years (in $)
The objective is to maximize the total ending cash value of the initial $500,000 investment.
The constraints are:
1. The total investment cannot exceed the available funds of $500,000: x1 + x2/2 + x3/3 ≤ 500,000
2. At least $30,000 must be invested in advertising and sales promotion every year: x1 ≥ 30,000
3. At least $40,000 must be invested in manufacturing operations improvements every 2 years: x2 ≥ 40,000
4. At least $50,000 must be invested in product research and development every 3 years: x3 ≥ 50,000
The return on investment per year is:
Advertising and sales promotion: return = investment + 0.2 * investment = 1.2 * investment
Manufacturing operations improvements: return = investment + 0.3 * investment = 1.3 * investment
Product research and development: return = investment + 0.5 * investment = 1.5 * investment
The linear programming model can be written as follows:
Maximize:
Total cash value = (1.2*x1) * 4 + (1.3*x2) * 2 + (1.5*x3) * (4/3)
Subject to:
x1 + x2/2 + x3/3 ≤ 500,000
x1 ≥ 30,000
x2 ≥ 40,000
x3 ≥ 50,000
b. To solve the model using a computer, you can use any linear programming software or mathematical modeling software that supports linear programming. Some popular options include:
1. GNU Linear Programming Kit (GLPK)
2. IBM ILOG CPLEX Optimization Studio
3. Microsoft Excel Solver
4. MATLAB Optimization Toolbox
You would need to input the objective function, constraints, and parameters into the software and run the optimization to obtain the optimal solution, which will provide the amounts to be invested in each alternative during each year of the 4-year period.
a. To formulate a linear programming model for this problem, we need to define the decision variables, the objective function, and the constraints.
Decision variables:
Let X1 represent the amount invested in advertising and sales promotion program during each year.
Let X2 represent the amount invested in manufacturing operations improvements during each 2-year period.
Let X3 represent the amount invested in product research and development during each 3-year period.
Objective function:
We want to maximize the total ending cash value of the initial $500,000 investment. The ending cash value is the sum of the returns from each investment.
Maximize: X1 + 1.2X1 + X2 + 1.3X2 + X3 + 1.5X3
Constraints:
1. The total amount invested should not exceed $500,000:
X1 + X2 + X3 <= 500,000
2. Minimum investment requirements:
X1 >= 30,000
X2 >= 40,000
X3 >= 50,000
3. Yearly investment limitations:
X1 <= 500,000 (There are no limitations on the yearly investment in advertising and sales promotion program)
4. 2-year investment limitations:
X2 <= 250,000 (Investment in manufacturing operations improvements can only be made every 2 years)
5. 3-year investment limitations:
X3 <= 150,000 (Investment in product research and development can only be made every 3 years)
b. To solve the model using a computer, you would need to use a linear programming solver or software. There are several options available, such as Microsoft Excel's Solver add-in, Gurobi, CPLEX, or open-source solvers like GLPK and PuLP. The specific steps to solve the model would depend on the software or solver you choose to use.