A Beverage Producer Produces Cola at Two Bottling Plants and Distributes

A beverage producer produces cola at two bottling plants and distributes it to 4 major cities in the Southeast. The table below shows the bottling plant capacities, the number of cases to be shipped to each city, and the transport costs ($ per case) between plants and cities. The producer's objective is to meet its delivery requirements while minimizing its total transport costs.
$\begin{array} { c c c } & \text { Plart X } ( 3,000 ) & \text { Plart } Y ( 3,500 ) \\\text { City A } ( 1,000 ) & 0.37 & 0.10 \\\text { City B } ( 1,400 ) & 0.30 & 0.20 \\\text { City C } ( 2,100 ) & 0.25 & 0.35 \\\text { City D } ( 1,800 ) & 0.10 & 0.40\end{array}$ (a) Formulate the firm's linear programming problem.
(b) The optimal solution is: XC = 1,200, XD = 1,800, YA = 1,000, YB = 1,400, and YC = 900. What is the shadow price associated with Plant Y capacity? Show that the increase in total cost in delivering an additional case to City D is $.20.