A trucking company is evaluating two alternative designs for its transportation network (which implies locations for its hubs and terminals). The company is mainly concerned with two attributes:

X₁ = Expected number of deadhead miles driven in a typical year (in thousands).

X₂ = Expected percentage of deliveries made on time during a typical year.

There are other attributes for which the company is concerned (e.g., construction costs, workforce availability, desirability of locations chosen) but each design gives the same values for these other attributes, under each scenario evaluated.

A simulation model has been developed to evaluate each design under each of three scenarios (high demand (HD), medium demand (MD), and low demand (LD)). The following payoff table has been developed through the use of the simulation model. In this table the first number represents the value for X₁ and the second number represents the value for X₂.

HD MD LD

Design 1 (25,75) (22,85) (15,98)

Design 2 (30,82) (27,90) (25,99)

Probability .3 .2 .5

The utility function of the company's president has been assessed. The function has been shown to be multiplicative, with scaling constants of w₁ = .1, w₂ = .5. The individual attribute utility function values for the scaled utility functions are given as:

x₁: 15 22 25 27 30

u₁(x₁): 1 0.8 0.6 0.4 0

x₂: 75 82 85 90 98 99

u₂(x₂): 0 0.4 0.5 0.7 0.9 1

Compute the value for the scaling constant, w, the expected utilities for each

design, and rank the two designs according to their expected utilities.