A Company Wants to Purchase Large and Small Delivery Trucks

question 31

Essay

A company wants to purchase large and small delivery trucks. The company wants to purchase about 10 large and 15 small trucks. Each large truck costs $30,000 and has a 10 ton capacity. Each small truck costs $20,000 and has a 7 ton capacity. The company wants to have about 200 tons of capacity and spend about $600,000.
Based on the following goal programming formulation, associated solution, and spreadsheet model, what formulas should go in cells D6:E6, B9:E9, and B16 of the spreadsheet?
$\begin{array}{|c|l|c|c|c|c|}\hline &{\text { A }} & \text { B } & \text { C } & \text { D } & \text { E } \\\hline 1 & \text { Problem Data } & \text { Large } & \text { Small } & & \\\hline 2 & \text { Cost } & 10 & 7 & & \\\hline 3 & \text { Capacity } & 30 & 20 & & \\\hline 4 & & & & & \\\hline 5 & \text { Goal Constraints } & \text { Large } & \text { Small } & \text { Cost } & \text { Capacity } \\\hline 6 & \text { Actual Amount } & 10 & 15 & 205 & 600 \\\hline 7 & \text { + Under } & 0 & 0 & 0 & 0 \\\hline 8 & \text { - Over } & 0 & 0 & 5 & 0 \\\hline 9 & =\text { Goal } & 10 & 15 & 200 & 600 \\\hline 10 & \text { Target Value } & 10 & 15 & 200 & 600 \\\hline 11 & & & & & \\\hline 12 & \text { Weights } & & & & \\\hline 13 & \text { Under } & 1 & 1 & 1 & 1 \\\hline 14 & \text { Over } & 1 & 1 & 1 & 1 \\\hline 15 & & & & & \\\hline 16 & \text { Objective } & 5 & & & \\\hline\end{array}$

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Definitions:

Increasing Returns to Scale

A situation in which a proportional increase in all inputs results in a more than proportional increase in output, demonstrating economies of scale.

Law of Diminishing Returns

An economic principle stating that, holding all else constant, as one input variable is increased, there is a point at which the marginal per unit output starts decreasing.

Diseconomies of Scale

The phenomenon where an increase in production leads to higher per-unit costs, often due to inefficiencies associated with scale.