For each

problem, you need to:

a$)$ Present the mathematical model $($mathematical formulation, e$.$g$.$ decision variables,

objective function and constraints, and any key assumptions$).$

b$)$ Build an OPL model $($the data must be read from a $.$dat file$).$

c$)$ Run the OPL model and record the optimal objective function value and solution of

each problem instance.

d$)$ Write a concise problem summary. 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. Hint: You need to first draw the network of the problem with all possible connections from

depots to recycling centers considering alternative modes of transportation $($road and rail$).$ In

addition to nodes that represent depots and recycling centers, you may consider creating

intermediate dummy nodes of $$road$$ and $$rail$$ for each recycling center. These dummy nodes

will be connecting to recycling centers and they will be connected from depots based on the

descriptions and limitations presented in the problem. Finally, add a dummy demand node

$($sink node$)$ that each recycling center connects to by artificial arcs. You may also choose to

include a dummy source node that connects to the depots via artificial arcs. Each arc in the

network is defined by three parameters: minimum flow, maximum flow or capacity, and cost.