Question $2$: 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 depotrecycling center route:

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

$-$ Depot D$2$ delivers to C$1$ by rail $(\backslash (12/$ t $\backslash ))$ or road $(\backslash (14/$ t $\backslash )).$

$-$ Depot D$3$ delivers to C$2$ by road $(\backslash (9/\backslash$mathrm$\{$t$\}\backslash )),$ and to C$3$ by rail $(\backslash (10/\backslash$mathrm$\{$t$\}\backslash ))$ or road $(\backslash (5/\backslash$mathrm$\{$t$\}\backslash )).$

$-$ Depot D$4$ delivers to C$2$ by rail $(\backslash (11/\backslash$mathrm$\{$t$\}\backslash ))$ or road $(\backslash (14/\backslash$mathrm$\{$t$\}\backslash )),$ and to C$3$ by rail $(\backslash (10/\backslash$mathrm$\{$t$\}\backslash ))$ 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 shipment requirements.

c$.$ Once the model $($mathematical formulation$)$ is formulated, you will implement it in OPL $($Optimization Programming Language$),$ solve the problem, and present your results, including the total transportation cost and the transport plan for each route.

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.

Give me the mathematical formulation for the problem and also give me the OPL code using IBM CPLEX