APPLICATIONS of Linear and Nonlinear programming and Simulation

PART A: USE CASES OF LINEAR AND NONLINEAR PROGRAMMING:

·      USE CASES OF LINEAR and Nonlinear PROGAMMING:

Linear Programming has been used to solve optimization problems in banking, forestry, petroleum, and medical industries. Optimization can be completed with linear and non linear models.

There are three components of optimization model:

A)   Decision variables.

B)   Objective function.

C)   Constraints.

Optimization needs to be combined with descriptive and predictive analytical tools.

Numerous sectors and functional areas can benefit from optimization as a technique. The limiting conditions contained in the maximization/minimization problem are represented by the constraints in optimization problems. The costs of labor, production, and advertising would be the restraints in our hypothetical business case. 

1.     Supply chain optimization:

By using optimization models, companies can optimize their supply chain processes, including to but not limited to inventory management , transportation planning and production scheduling.

2.     Staff scheduling:

Hospitals may use an optimization model to determine an optimal schedule for nurses and doctors, taking into account shift preferences , workload distribution and workload regulations.

3.     Portfolio Optimization:

Financial institutions use optimization models to determine an optimal allocation of funds across different investment options by considering various risk factors such as risk, return, and asset correlation.

4.     Vehicle Routing:

Optimization model may be used to optimize vehicle routing and scheduling, considering factors such as traffic congestion,  delivery windows, vehicle capacity.

5.     Production Planning:

Optimization model may be used to determine optimal production schedule, considering factors such as production capacity, raw material availability, and production costs.

6.     Other examples of application problems:

·      Approaches to build optimization model:

Making Optimal Decisions in Practice:

Solver optimizer in Excel is used to optimize resources among competing products. Moreover, solver can be applied under the following circumstances:

A)   Resource allocation. It determines the best available resource allocation based on the constraints. Allocated resources cannot exceed available resources.

B)   Network optimization. For example, it should be determined how many plants should be operated in different parts of Germany.

With the help of the solver, statistician can check for alternatives and identify the best possible solution based on the dearth of resources.

When using solver as an optimization tool, one should identify relevant constraints. There is an option to make constraints non negative.

In the subject to constraint box, input relevant constraints.

The following are examples of constraints of objective function:

A)   Weight constraint.

B)   Volume constraint.

C)   Demand constraint.

In the Solving Method, the following methods are available:

A)   Simplex LP is applied to linear function. Model is built as a linear model.

B)   GRG Nonlinear allows us to work with linear and non linear models.

In the Options, uncheck integer values to ensure optimized units of production does not include integer values.

Lastly, one can make unconstrained variables non-negative.

If one faces the following error message, it indicates that some of the constraints were not included:

If there are incompatible constraints, the following error message may arise.

PART B: USE CASES OF SIMULATIONS

Decisions can be made in high and low uncertainty settings.

·       Simulation runs can be repeated as many times as necessary to generate the sample distribution of output values.

·       Sample distribution can be analyzed to determine estimates for the expected value and standard deviation.

·       Excel is used to run simulation and analyze the results.

To perform data simulation in Excel, the following steps are followed:

A)   Use Data Analysis toolkit in the data tab.

B)   Purpose of running simulation is to determine risks and rewards associated with certain outcomes.

St deviation is a statistical measure used to measure risks associated with certain outcome. Mean/Average is the measure of reward.

The following steps are followed to generate 10 instances of monthly data usage of what the future may hold:

Number of variables indicated as 1 means a single random variable.

Number of random numbers 10 indicates 10 instances of monthly data usage.

Distribution is indicated as normal.

Random seed 123 generates values from distribution specified in a particular way. It does not matter what seed is selected as long as the seed is high.

C)   Interpreting simulation output:

Sample mean is a reward measure. St deviation is a risk measure. The more values are generated, the closer are estimates to a true value.

Compare results of different simulation runs:

Based on the following assessment, the longer simulation run, the closer is sample mean and st deviation to a true value.

Sample mean and st deviation do not depend on seed values for long simulations. As long as there are many instances, seed values don’t matter much.

D)   Visualization of simulation results:

d.1 Histogram is created to determine frequency of values in data set.