Linear Programming Chapter 4: Modeling Examples

Linear Programming: Modeling Examples

• This chapter presents more complex model formulation examples, illustrating popular application areas of linear programming and providing guidelines for model formulation and computer solutions.

A Product Mix Example: Quick-Screen

• Quick-Screen is a clothing manufacturing company that produces commemorative shirts for sporting events.
• The company has 72 hours to complete production after the game, with a budget of $25,000 and sufficient blank sweatshirts and T-shirts in stock.
• The company wants to maximize profit by determining the optimal number of dozens of each type of shirt to produce.

Decision Variables

• Four decision variables represent the number of dozens of each type of shirt to produce:
  • x1: sweatshirts, front printing
  • x2: sweatshirts, back and front printing
  • x3: T-shirts, front printing
  • x4: T-shirts, back and front printing

The Objective Function

• The objective is to maximize profit, with the total profit being the sum of individual profits from each type of shirt:
  • maximize Z = $90x1 + 125x2 + 45x3 + 65x4

Model Constraints

• Four constraints:
  • Processing time: total available processing time is 72 hours
  • Shipping capacity: available shipping capacity is 1,200 standard-size boxes
  • Cost budget: total budget available for production is $25,000
  • Availability of blank shirts: limited to 500 dozen sweatshirts and 500 dozen T-shirts

Model Summary

• The linear programming model for Quick-Screen is summarized as:
  • maximize Z = 90x1 + 125x2 + 45x3 + 65x4
  • subject to four constraints: processing time, shipping capacity, cost budget, and availability of blank shirts
  • x1, x2, x3, x4 ≥ 0

Computer Solution with Excel

• The Excel spreadsheet solution for this product mix example is shown in Exhibit 4.1.
• Decision variables are located in cells B14:B17.
• Profit is computed in cell B18.
• Constraint formulas are embedded in cells H7 through H11.
• Formulas for leftover resources are in cells K7 through K11.