ABC Company operates a $4-$stage supply chain $-$ suppliers, plants, distribution centers and demand regions. The two manufacturing plants are located in Austin and Charlotte, each with annual capacity of $30,000$ units. The supply chain serves $5$ markets as Asia $-$Pacific $($with demand $15,000$ units$),$ Europe $(10,000),$ North America $(8,000),$ Middle East $(4,000),$ South America $(3,000).$ ABC works with $3$ suppliers in China, Australia, and Kazakhstan.

ABC is considering $3$ potential locations for distribution center: Houston $($$$1\mathrm{.}5$ million$)$; New York $($$$2$ million$)$; Los Angeles $($$$1\mathrm{.}8$ million$).$ The capacity will be $40,000$ units.

Transportation costs are given as follows:

Houston New York Los Angeles

Austin $$20$ $$100$ $$70$

Charlotte $$90$ $$30$ $$120$

Asia$-$Pacific $$220$ $$250$ $$200$

Europe $$195$ $$180$ $$210$

North America $$70$ $$75$ $$80$

Middle East $$240$ $$230$ $$235$

South America $$90$ $$115$ $$95$

China Australia Kazakhstan

Austin $$150$ $$175$ $$165$

Charlotte $$160$ $$170$ $$180$

The objective is to minimize the total cost of investment and transportation such that all demands are met.

a$)$ Write the mathematical optimization model for this problem. Define the appropriate notation for decision variables and present the objective function and constraints.

b$)$ Solve your model using an optimization solver and find the optimal facility location and distribution plan.