# Production Schedule Optimization for Snow Shovels

When determining the optimal production schedule for a snow shovel manufacturing company, it is essential to consider various factors such as demand, production capacity, and costs involved. In this case, the goal is to meet the demand for each quarter while minimizing overall production and inventory costs.

The given data presents demands for each quarter and constraints on production capacity: - Quarter 1: 11,000 shovels - Quarter 2: 48,000 shovels - Quarter 3: 64,000 shovels - Quarter 4: 15,000 shovels - Production capacity constraint: At most 65,000 shovels per quarter

To formulate an integer-linear program for the production schedule, we need to consider the fixed costs, holding costs, and total costs involved. The goal is to minimize total costs by optimizing the production schedule.

## Formulation of the Integer-Linear Program:

The production schedule can be optimized by formulating the following constraints and formulas:

**Restrictions:**
- Integer constraints for production in each quarter
- Production capacity constraint for each quarter (P1,P2,P3,P4 < 65,000)

**Inventory Formulas:**
- Inventory for each quarter is calculated based on production and demand

**Holding Cost Formulas:**
- Holding costs are incurred based on the inventory level for each quarter

**Fixed Cost:**
- Fixed costs are incurred for each quarter where production occurs (if P>0)

**Total Cost:**
- The total cost is the sum of fixed costs and holding costs for each quarter

By minimizing the total cost through the integer-linear program, we arrive at the optimal production schedule mentioned earlier. This schedule helps in saving costs by reducing production in the fourth quarter when demand is lower, thus avoiding unnecessary fixed costs.