Applications of Integer Linear Programming: Fixed Charge, Capital Budgeting & Distribution System Design Problems

Instructor: Claude Bourbonniere

Claude has worked as a Company Manager for Cirque du Soleil for the past 20 years and completed an MBA in Global Management at the University of Phoenix in 2014

Integer linear programming can be used to analyze business cost decisions. This lesson will demonstrate how we can use this programming technique to help make the best decisions about business costs.

What Is Integer Linear Programming?

Integer linear programming combines fixed charges and variable costs in an equation. The fixed costs portion is the start of the equation; it's followed by the variable costs, which vary depending on the volume of the production being described by the equation. This equation is useful in business decisions to determine the break even cost of a product.

Description of the Components of the Equation

  • Volume (X): Number of units that will be produced.
  • Fixed costs (F): Costs that will be incurred at any level of production.
  • Variable costs (V): Costs that vary by unit of production.
  • Capital budgeting (C): Fixed cost invested in equipment that will last for more than one year.
  • Years (Y): Length of the useful life of the equipment, in years.
  • Revenue (R): Money gain by the sales of the products.

Formula Used for Integer Linear Programming

The following formula estimate the cost of the products. See above for the definitions of each variable.

Cost of the products = F + (C/Y) + (V * X)

Application of the Formula of Integer Linear Programming

Mr. Joe has a small business that produces customized furniture. He wants to build a new custom chair. He asks you the volume of chairs that he will need to produce and sell to have a return on investment (ROI) of 10%. Let's use the integer linear programming formula to answer his question.

Mr. Joe will need to buy new equipment to build this chair. The cost of the equipment is $60,000 (C), and the equipment has a useful life of 10 years (Y).

His furniture shop has $25,000 in monthly fixed costs. These fixed costs include the utility costs of the shop, equipment maintenance, administration costs, and general supplies. The new chair will use 10% of the time of the furniture shop. This represents $25,000 * 10% = $2,500 (F) in monthly fixed costs to be allocated to this new chair.

Mr. Joe estimates at $1,000 the specific costs to produce a new chair (V). This includes the salary of the employees working on the chair and the specific materials to build the chair.

Finally, Mr. Joe estimates that he should be able to sell 20 chairs per month at $1,500 per chair.

The monthly cost of the new chair, based on a volume of 20 chairs per month, would be: $2,500 + ($60,000 / 10 years / 12 months) + (20 chairs * $1,000) = $23,000.

The monthly revenue of Mr. Joe would be: 20 chairs * $1,500 = $30,000.

Mr. Joe's expected profit would thus be $7,000 ($30,000 in revenue minus $23,000 in costs) on an investment of $60,000. This means that the ROI will be 11.7% ($7,000 / $60,000). This ROI is above Mr. Joe goal's of a 10% ROI.

Break Even

The break even cost is the volume of sales necessary for Mr. Joe to cover his fixed costs.

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