Problem razmene denarja : delo diplomskega seminarja

Šinkovec, Luka (author), Vavpetič, Aleš (mentor)

Posebni problem razmene denarja je le ena od variacij klasičnega problema razmene denarja, ki pa je prav tako le ena od variacij najširšega optimizacijskega problema v tej zgodbi – preprostega problema nahrbtnika. Najožja izmed teh variacij, posebni problem razmene denarja, sprašuje po vrednostih kovancev $a_1, a_2, \ldots , a_t$, za katere obstaja natanko ena razmena z najmanjšim možnim skupnim številom kovancev teh vrednosti, ki jih uporabimo za razmeno, za vsako vsoto denarja, ki se jo s kovanci teh vrednosti da razmenjati. Predstavili bomo rešitev problema v primeru dveh kovancev različnih vrednosti ter nekaj metod za iskanje (oz. preverjanje ustreznosti) rešitev v primeru treh kovancev različnih vrednosti, ki ustrezajo določenim pogojem.

• Language: Slovenian
• Keywords: preprosti problem nahrbtnika, problem poštnih znamk, klasični problem razmene denarja, problem kovancev, Frobeniusova formula, totalna razmena, optimalna razmena, posebni problem razmene denarja
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2021
• PID: 20.500.12556/RUL-131150
• UDC: 511
• COBISS.SI-ID: 78578179
• Publication date in RUL: 23.09.2021

Secondary language

• Language: English
• Title: Money-changing problem

Abstract

The special money-changing problem is only one of the many variations of the classical money-changing problem, which is also only one of the many variations of the widest optimisation problem in this story – the knapsack problem. The narrowest of these variations, the special money-changing problem, asks for what denominations of money $a_1, a_2, \ldots , a_t$, is there exactly one way to make change, using the fewest number of coins possible, for every amount for which change can be made using only coins of these denominations. We will provide a solution to this problem in the case with coins of two different denominations and a few methods for finding (or testing the correctness of) solutions in the case with coins of three different denominations, which all meet certain conditions.

• Keywords: knapsack problem, postage stamp problem, classical money-changing problem, coin problem, Frobenious formula, denumerant (total change), optimal change, special money-changing problem