Question $7$: Transport Optimization for Chemical Products $$ Minimum Cost Network

Flow Optimization Problem

A chemical manufacturing company is tasked with transporting $190$ tonnes of chemical products from four storage depots $($D$1,$ D$2,$ D$3,$ and D$4)$ to three recycling centers $($C$1,$ C$2$ and C$3).$

$$ Depot D$1$ holds $50$ tonnes,

$$ Depot D$2$ holds $40$ tonnes,

$$ Depot D$3$ holds $35$ tonnes, and

$$ Depot D$4$ holds $65$ tonnes.

We assume that recycling centers do not have capacity restrictions. The company aims to transport these products using two available modes of transport: road and rail. The transportation cost per tonne varies depending on the mode of transport and the depot$-$recycling center route:

$$ Depot D$1$ delivers to C$1$ and C$2$ by road at a cost of $$12/$t and $$11/$t$,$ respectively.

$$ Depot D$2$ delivers to C$2$ by rail $($$$12/$t$)$ or road $($$$14/$t$).$

$$ Depot D$3$ delivers to C$2$ by road $($$$9/$t$),$ and to C$3$ by rail $($$$10/$t$)$ or road $($$$5/$t$).$

$$ Depot D$4$ delivers to C$2$ by rail $($$$11/$t$)$ or road $($$$14/$t$),$ and to C$3$ by rail $($$$10/$t$)$ or road

$($$$14/$t$).$

In addition, the contract with the railway company requires that each rail shipment must carry at least $10$ tonnes but not exceed $50$ tonnes. There are no such constraints for road transport. Your task as an Operational Research Analyst for the company is to formulate an optimization model that will minimize the total transportation cost while ensuring all $190$ tonnes are

transported. You should:

a$.$ Draw a network diagram that represents the problem.

b$.$ Write a mathematical formulation for the problem:

i$)$ Define the decision variables clearly.

ii$)$ State the objective function and explain the company's goal.

iii$)$ Identify the constraints including transport mode capacities, depot storage, and rail