A coffee company grows coffee in two farms on the island of Kaua'i and supplies three of the island's restaurants. The coffee is extremely popular, and all three restaurants wants as much of it as they can get. Farm $i$ can produce up to $s_i$ kilograms per week. The management of restaurant $j$ is happy if it receieves at least $d_j$ kilograms per week. The coffee company makes $p_{i,j}$ dollars for each kilogram it produces at farm $i$ and ships to restaurant $j$.
a. Consider the following linear program representing the problem.
$ \begin{array}{l*{12}{r}lr} \mbox{Maximize} & p_{1,1}x_{1,1} & + & p_{1,2}x_{1,2} & + & p_{1,3}x_{1,3} & + & p_{2,1}x_{2,1} & + & p_{2,2}x_{2,2} & + & p_{2,3}x_{2,3} \\ \mbox{Subject to} & x_{1,1} & + & x_{1,2} & + & x_{1,3} & ~ & ~ & ~ & ~ & ~ & ~ &\leq & s_1 \\ ~ & ~ & ~ & ~ & ~ & ~ & ~ & x_{2,1} & + & x_{2,2} & + & x_{2,3} & \leq & s_2 \\ ~ & x_{1,1} & ~ & ~ & ~ & ~ & + & x_{2,1} & ~ & ~ & ~ & ~ & \geq & d_1 \\ ~ & ~ & ~ & x_{1,2} & ~ & ~ & ~ & ~ & + & x_{2,2} & ~ & ~ & \geq & d_2 \\ ~ & ~ & ~ & ~ & ~ & x_{1,3} & ~ & ~ & ~ & ~ & + & x_{2,3} & \geq & d_3 \\ &&&&&&&&&&&x_{i,j} & \geq & 0 \end{array}$
Here we can think of each $x_{i,j}$ as the amount (in kg) that is shipped from farm $i$ to restaurant $j$. Given that $p_{i,j}$ represents the revenue from each kilogram produced at farm $i$ and shipped to restaurant $j$. Naturally, our objective function to maximize the overall profits.\~\ The first two constraints tell us that the total amount of coffee shipped to all three restaurants that comes from farm $i$ must be less than the production maximum for farm $i$, $s_i$. The last three constraints tell us that the amount of coffee coming from both farms to restaurant $j$ must be at least $d_j$, the minimum amount of coffee each restaurant needs. Of course, we have non-negativity constraints on production.
b. Using dual variables $u_1,u_2$ for the first two constraints in the primal, and $v_1,v_2,v_2$ for the last three, we form the dual to this LP.
$ \begin{array}{l*{12}{r}lr} \mbox{Minimize}\hspace{20pt}\ & s_1u_1 & + & s_2u_2 & - & d_1v_1 & - & d_2v_2 & - & d_3v_3 \\ \mbox{Subject to}\hspace{20pt}\ & u_1 & ~ & ~ & - & v_1 & ~ & ~ & ~ & ~ ~ &\geq & p_{1,1} \\ ~ & u_1 & ~ & ~ & ~ & ~ & - & v_2 & ~ & ~ &\geq & p_{1,2} \\ ~ & u_1 & ~ & ~ & ~ & ~ & ~ & ~ & - & v_3 &\geq & p_{1,3} \\ ~ & ~ & ~ & u_2 & - & v_1 & ~ & ~ & ~ & ~ &\geq & p_{2,1} \\ ~ & ~ & ~ & u_2 & ~ & ~ & - & v_2 & ~ & ~ &\geq & p_{2,2} \\ ~ & ~ & ~ & u_2 & ~ & ~ & ~ & ~ & - & v_3 &\geq & p_{2,3} \\ &&&&&&&&&u_{i} & \geq & 0 \\ &&&&&&&&&v_{j} & \geq & 0 \end{array}$
This is where I am at. The only thing left to do is give an economic interpretation of the dual and the dual variables. I cannot seem to wrap my head around these new variables. I believe it has something to with minimizing production costs. Any ideas?
