A company sells seven types of boxes, ranging in volume from 17 to 33 cubic feet. The demand and size of each box is given in Table 7. The variable cost (in dollars) of producing each box is equal to the box’s volume. A fixed cost of $1,000 is incurred to produce any of a particular box. If the company desires, demand for a box may be satisfied by a box of larger size. Formulate and solve a shortest-path problem whose solution will minimize the cost of meeting the demand for boxes. TABLE 7:

$\begin{matrix} \text{ } & \text{Box}\\ \text{ } & \text{1} & \text{2} & \text{3} & \text{4} & \text{5} & \text{6} & \text{7}\\ \text{Size} & \text{33} & \text{30} & \text{26} & \text{24} & \text{19} & \text{18} & \text{17}\\ \text{Demand} & \text{400} & \text{300} & \text{500} & \text{700} & \text{200} & \text{400} & \text{200}\\ \end{matrix}$

Solution

VerifiedThis can be modeled as a shortest-path problem with 8 nodes, one labeled 0 and the other 7 labeled by the possible box sizes, and with edges $(i,j)$ for all $i<j$ and associated cost given by the formula

$c_{i,j}=j\cdot\left(\sum_{i<k\leq j}d_k\right)+1000$

where $d_k$ is the demand for boxes of size $k$. The associated transshipment problem is given by the tableau

$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline &B17&B18&B19&B24&B26&B30&B33&\\\hline B0&4400&11800&16200&37000&53000&70000&90100&1\\\hline B17&0&8200&12400&32200&47800&64000&83500&1\\\hline B18&M&0&4800&22600&37400&52000&70300&1\\\hline B19&M&M&0&17800&32200&46000&63700&1\\\hline B24&M&M&M&0&14000&25000&40,600&1\\\hline B26&M&M&M&M&0&10000&24100&1\\\hline B30&M&M&M&M&M&0&14200&1\\\hline &1&1&1&1&1&1&1&\\\hline \end{array}$

which has optimal solution

$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline &B17&B18&B19&B24&B26&B30&B33&\\\hline B0&&&1&&&&&1\\\hline B17&1&&&&&&&1\\\hline B18&&1&&&&&&1\\\hline B19&&&&1&&&&1\\\hline B24&&&&&1&&&1\\\hline B26&&&&&&&1&1\\\hline B30&&&&&&1&&1\\\hline &1&1&1&1&1&1&1&\\\hline \end{array}$

which tells us the company should produce boxes of sizes 19, 24, 26 and 33.