A company must accelerate the production of a certain good in order to get ahead of the competition. The production process includes four phases that must be executed one after the other. Each phase can be accomplished in normal, fast, or urgent mode. The corresponding times are presented in Table $1($normal mode has been deemed unacceptable for the last three phases$).$ A total budget of $30$ million dollars is available for production, and the cost of each phase according to the chosen mode is presented in Table $2.$ The problem then consists of determining the mode for each phase that minimizes the total production time without exceeding the available budget. Represent this problem as a shortest path problem by defining an appropriate network. Provide an interpretation for the vertices and edges of the network, as well as for the edge$-$costs. Table $1$: Duration in months of the production phases according to the chosen mode. Table $2$: Cost in millions of dollars of the production phases according to the chosen mode.