Online Bundle Purchasing Business Rules and Web Services Group April 4, 2002
The Goal • Need to purchase a “bundle” of items • May have choice of several satisfactory bundles e. g. AB AC DE FG • Must decide which bundle is the best one to purchase
Problems • Some items are more expensive than others • Some bundles are more preferable to the buyer than others • Some items might not be currently available but may become available in the future. In these cases, the buyer may have probability measures on their prices
Problems (continued) • Items that are currently available may become unavailable in the future • Some items in a bundle might not be available at the same time • Some bundles currently unavailable may be available at the same time in the future
Other Things to Consider • The buyer’s attitude toward risk • The buyer’s attitude toward suppliers of certain items • By when certain items are needed • Until when certain items are unwanted
Bundle Purchase Utility • Set of bundles B • Each bundle b B has cost c(b) • Two utility functions: – For bundles ub : B R – For monetary expenditures uz : Z R
Bundle Purchase Utility • For any b 1, b 2 B, z 1, z 2 Z, b 1 is preferred to b 2 iff ub(b 1) > ub(b 2) z 1 is preferred to z 2 iff uz(z 1) > uz(z 2) • A decision-maker is indifferent between two alternatives iff their utilities are equal
Bundle Purchase Utility • The utility of a bundle purchase can be computed as an additive function of the buyer’s utilities for the bundle and its cost: u(b, c(b)) = kbub(b) + kzuz(c(b)) where kb and kz are scaling constants
Example ub({AB}) =. 7 ub({DE}) = 1 ub({AC}) =. 4 ub({FG}) = 0 uz(z) = 1 - z/20 (20 is max bundle cost) additive utility function: u(b, c(b)) =. 2 ub(b) +. 8 uz(c(b)) The utility of purchasing bundle AC at a cost of 14: u({AC}, 14) =. 2 ub({AC}) +. 8 uz(14) =. 2(. 4) +. 8(. 3) =. 32
Expected Utility • What if: – A is currently available at a cost of 5 – B is currently unavailable but is believed to be coming available in the future at a cost of either 6 or 7 (50 -50 chance) • Need to calculate the expected utility, Eu({AB}), of purchasing AB
Expected Utility of AB • Recall ub({AB}) =. 7 and uz(z) = 1 - z/20 • Need to sum the products of the utility of each possible outcome with its associated probability of occurring • Eu({AB}) =. 5 u({AB}, 11) +. 5 u({AB}, 12) =. 5(. 2 ub({AB}) +. 8 uz(11)) +. 5(. 2 ub({AB}) +. 8 uz(12)) =. 5(. 2(. 7) +. 8(. 45)) +. 5(. 2(. 7) +. 8(. 4)) =. 5(. 5) +. 5(. 46) =. 48
The Online Problem • A second dimension to the problem of finding the best bundle purchase • Not only need to be able to decide which bundle purchase is the best to pursue, but also must decide when to decide which is the best bundle purchase to pursue
Quote Intervals • Consider a framework where each bundle has a quote interval and a pre-quote interval Quote interval - an interval of time during which the cost of the bundle is known and the bundle is available for purchase Pre-quote interval - an interval of time before the quote interval during which the quote interval for the bundle and also a probability measure on the price of it are known
Visual Timeline Example AB Quote Pre-quote AC DE FG
Visual Timeline Example AB Quote Pre-quote AC DE FG t
Competition Intervals • Time intervals between dotted vertical lines • These are the only purchasing intervals under consideration • Need to predict the expected utility that one would achieve if a purchase was made during such an interval. This is equal to the expected highest utility of all bundles available during the interval
Computing Expected Utility of a Competition Interval Recall ub({DE}) = 1 and ub({FG}) = 0 50 -50 chance DE costs 15 or 16 50 -50 chance FG costs 10 or 11 Utility of DE: . 5 chance of. 4, . 5 chance of. 36 Utility of FG: . 5 chance of. 4, . 5 chance of. 36
Expected Highest Utility of DE and FG Probability. 25. 25 Outcome DE FG. 4. 4. 4. 36. 36 Expected highest utility =. 25(. 4) + 25(. 36) =. 39
The Purchase Decision • At time t, a decision must be made on whether or not to purchase bundle AB. • In order to make this decision, the following must be examined: 1) The utility of purchasing the bundle about to expire at t (AB) 2) The utility for each other current bundle purchases 3) The expected utility for each future competition interval
The Purchase Decision • If there is a utility calculated in either 2) or 3) that is higher than the utility calculated in 1), do not buy at time t, • Else buy at t
Other Things to Consider • Calculating expected highest utility of dependent bundles • Partial bundle purchases • How to place restrictions on when certain items are needed or unwanted • User interface
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