Question

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:

 Box 1234567Size33302624191817Demand400300500700200400200\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

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This 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)(i,j) for all i<ji<j and associated cost given by the formula

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

where dkd_k is the demand for boxes of size kk. The associated transshipment problem is given by the tableau

B17B18B19B24B26B30B33B044001180016200370005300070000901001B170820012400322004780064000835001B18M04800226003740052000703001B19MM0178003220046000637001B24MMM0140002500040,6001B26MMMM010000241001B30MMMMM01420011111111\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

B17B18B19B24B26B30B33B011B1711B1811B1911B2411B2611B30111111111\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.

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