Robert Gardner Runs a Small, Local-Only Delivery Service Robert Would Like Each Truck Equally Loaded, Both in Terms

Essay

Robert Gardner runs a small, local-only delivery service. His fleet consists of three smaller panel trucks. He recently accepted a contract to deliver 12 shipping boxes of goods for delivery to 12 different customers. The box weights are: 210, 160, 320, 90, 110, 70, 410, 260, 170, 240, 80 and 180 for boxes 1 through 12, respectively. Since each truck differs each truck has different load capacities as given below:
$$\begin{array} { c c c c } \text { Truck } & \text { Weight Capacity } & \text { Box Capacity } & \text { Cast per pound } \\\hline 1 & \text { 800 pounds } & 5 & \$ 0.34 \\2 & 900 \text { pounds } & 6 & \$ 0.42 \\3 & 700 \text { pounds } & 4 & \$ 0.25\end{array}$$ Robert would like each truck equally loaded, both in terms of number of boxes and in terms of total weight, while minimizing his shipping costs. Assume a cost of $50 per item for trucks carrying extra boxes and $0.10 per pound cost for trucks carrying less weight.
Formulate the integer goal programming problem for Robert. (Hint: objective function involves decision and deviation variables.)