Lecture 12 Nonlinear Optimization Models - Copy (2) (1)

Page 1: Title Slide

Complex optimization problems may have nonlinear objective or constraint functions.
Such problems are categorized as Nonlinear Programming (NLP) problems.
Reasons for nonlinearity in models include:
- Nonconstant returns to scale, where the impact of an input on an output is nonlinear.
- Revenue models involving price as a nonlinear function of quantity sold.
- Data fitting may introduce nonlinearity through operations like squaring.
While nonlinear models are often more realistic, they present more complex solutions compared to linear models.

Local Optimum: A point that is the best among nearby points.
Global Optimum: The best point in the entire feasible region.
NLP problems may result in solving algorithms getting stuck at a local optimum, missing the global optimum.

Solver guarantees finding the global maximum (or minimum) if:
- The objective function is concave (if maximizing) or convex (if minimizing).
- Constraints are linear.
If assumptions do not hold:
- Try multiple starting values for changing cells.
- Run Solver from each starting point and use the best solution found.

Nonlinear objective functions can complicate optimization due to the presence of local optima.
Every feasible region point can potentially be optimal, complicating the search for the global optimum.
Distinguishing between local and global optima in nonlinear problems is challenging.

Nonlinear models may have multiple local and global optima.
The multi-start option allows for automatic selection of various starting solutions, optimizing the search process via the GRG nonlinear algorithm.

Population Size: Minimum of 10, recommended 100 starting solutions.
Random Seed: Determines consistency in selected starting solutions. Positive values yield the same selections; zero allows random selections.
Require Bounds on Variables: Enforcing explicit lower and upper bounds on changing cells.

Key questions for model building:
- What must be decided?
- What measure will compare alternative decisions?
- What restrictions limit choices?

The algorithm used is the Large-Scale Generalized Reduced Gradient (LSGRG).
Begins with initial values of decision variables, testing potential directions for improvement.
The algorithm iteratively moves towards a local optimum.
It does not guarantee finding the global optimum unless specific conditions are met.

The initial solution is derived from worksheet values at the start of the Solver’s search.
An identical solution from various initial conditions can indicate confidence in reaching a global optimum.
There is no standard for how many initial solutions to try.

Kilroy Paper Company intends to consolidate warehouses into one national distribution center (DC).
Distribution manager maps stores on a grid, associating coordinates with each store location.

For a potential DC at (x, y), the distance to each store can be calculated and summed.
Minimizing total distances equates to minimizing distribution costs, guiding Kilroy to the optimal location.

Madison Company aims to determine optimal pricing for maximizing product profit.
Unit cost for production is $50, and pricing must be at least this amount to generate profit.

Unit price influences demand, impacting revenue and costs.
The first step involves estimating how demand varies with price, forming the demand function.
Demand elasticity measures the sensitivity of demand to price changes.
Two types of demand functions are considered: linear and constant elasticity.

Elasticity is calculated as the percentage change in demand from a 1% price increase.
High elasticity means demand is sensitive to price changes.
Parameter estimates for demand functions are critical for price optimization, achieved through Excel trend lines based on observed data.

Nonlinear pricing involves balancing unit sales against price and profits.
Trade-offs exist between quantity sold at lower prices versus fewer units at higher prices.

CTC analyzes optimal pricing for daytime and evening calling rates, using demand estimation models to maximize revenue.
Formulation of daytime and evening lines demanded based on price variables is included.

EOQ addresses the balance between ordering and carrying costs for minimizing total costs.
Annual demand and quantity ordered interplay with fixed ordering costs (K).

ATC combines ordering costs and inventory costs in the formula ATC = ordering cost + carrying cost.

Woodstock has varying annual demands and needs to optimize order quantities to minimize costs within storage limits.

The company’s warehouse space of 12,000 square feet must be allocated efficiently among four products while minimizing costs.