Question

EZ-Windows, Inc., manufactures replacement windows for the home remodeling business. In January, the company produced 14,000 windows and ended the month with 9000 windows in inventory. EZ- Windows’ management team would like to develop a production schedule for the next three months. A smooth production schedule is obviously desirable because it maintains the current workforce and provides a similar month-to-month operation. However, given the sales forecasts, the production capacities, and the storage capabilities as shown, the management team does not think a smooth production schedule with the same production quantity each month possible.

February March Apri
l Sales forecast 15,000 16,500 20,000
Production capacity 14,000 14,000 18,000
Storage capacity 6,000 6,000 6,000
The company’s cost accounting department estimates that increasing production by one window from one month to the next will increase total costs by $ 1.00 for each unit increase in the production level. In addition, decreasing production by one unit from one month to the next will increase total costs by $ 0.75 for each unit decrease in the production level. a. [10] Ignoring production and inventory carrying costs, formulate and solve a linear programming model that will minimize the cost of changing production levels while still satisfying the monthly sales forecasts. [Hint: If you decide to decrease the production by 2,000 for February, this will change the production to 15,000 (Production in Jan.) – 2,000 = 13,000 which is less than the maximum production capacity in Feb. and the end inventory will be 9,000 (Carried inventory in Jan.) + 13,000 (production in Feb.) – 15,000 (Sales in Feb.) = 7,000, which will violate the maximum storage capacity in Feb.] b. [5] Prepare the sensitivity analysis. If decreasing production by one unit from one month to the next will increase total costs by $ 0.80 for each unit decrease in the production level, what will be your optimal solution and the corresponding total cost.

Answer 1

The primary task is to minimize costs by establishing an optimal production schedule for EZ-Windows, Inc., considering sales forecasts, production and storage capacities, and the cost changes associated with altering production levels. Sensitivity analysis is then conducted to determine the impact of updated cost parameters on the optimal solution.

Step-by-step explanation:

Formulating and solving a linear programming model for EZ-Windows, Inc., involves determining the production levels for February, March, and April that minimize the cost of changing production levels while satisfying sales forecasts and adhering to production and storage capacity constraints. To solve part a, let's denote the production in month i as Pi. The cost variables for increasing and decreasing production levels month over month are given, and the goal is to minimize the total cost function subject to the constraints provided by sales forecasts, production capacities, and storage capacities.

For part b, conducting a sensitivity analysis would involve adjusting the cost of decreasing production to $0.80 per unit and recalculating the optimal production schedule to determine the new minimum cost. This requires resolving the linear programming model with the updated cost parameter.

In the context of economies of scale, these refer to the cost advantages companies experience when production becomes efficient, as production scales up. For example, production plants with larger scales of production (like Plant L and Plant V in the given information) can produce goods at a lower average cost compared to smaller plants (Plant S), but only up to a certain output level.