## "Polite customer" problem and Z3 SMT-solver

This is probably not relevant in the era of credit cards, but...

In a shop or bar, you have to pay something in cash and, as a polite customer, you try to give such a set of banknotes, so that the cashier will not touch small banknotes in his/her cash desk, which are valuable (because of impoliteness of other customers).

For small amount of money, this can be solved in one's mind, but for large...

The problem is surprisingly easy to solve using Z3 SMT-solver:

from z3 import *

# we don't use coins for simplicity, but it's not a problem to add them

# banknotes in cash register
cash_register_1=1
cash_register_3=0
cash_register_5=2
cash_register_10=2
cash_register_25=0
cash_register_50=12
cash_register_100=10

# ... in customer's wallet
customer_wallet_1=2
customer_wallet_3=1
customer_wallet_5=1
customer_wallet_10=1
customer_wallet_25=0
customer_wallet_50=15
customer_wallet_100=20

# what customer have to pay
checkout=2135

from_cash_register_1=Int('from_cash_register_1')
from_cash_register_3=Int('from_cash_register_3')
from_cash_register_5=Int('from_cash_register_5')
from_cash_register_10=Int('from_cash_register_10')
from_cash_register_25=Int('from_cash_register_25')
from_cash_register_50=Int('from_cash_register_50')
from_cash_register_100=Int('from_cash_register_100')

from_customer_wallet_1=Int('from_customer_wallet_1')
from_customer_wallet_3=Int('from_customer_wallet_3')
from_customer_wallet_5=Int('from_customer_wallet_5')
from_customer_wallet_10=Int('from_customer_wallet_10')
from_customer_wallet_25=Int('from_customer_wallet_25')
from_customer_wallet_50=Int('from_customer_wallet_50')
from_customer_wallet_100=Int('from_customer_wallet_100')

s=Optimize()

# banknotes pulled from cash_register are limited by 0 and what is defined:
s.add(And(from_cash_register_1 >= 0, from_cash_register_1 <= cash_register_1))
s.add(And(from_cash_register_3 >= 0, from_cash_register_3 <= cash_register_3))
s.add(And(from_cash_register_5 >= 0, from_cash_register_5 <= cash_register_5))
s.add(And(from_cash_register_10 >= 0, from_cash_register_10 <= cash_register_10))
s.add(And(from_cash_register_25 >= 0, from_cash_register_25 <= cash_register_25))
s.add(And(from_cash_register_50 >= 0, from_cash_register_50 <= cash_register_50))
s.add(And(from_cash_register_100 >= 0, from_cash_register_100 <= cash_register_100))

# same for customer's wallet:
s.add(And(from_customer_wallet_1 >= 0, from_customer_wallet_1 <= customer_wallet_1))
s.add(And(from_customer_wallet_3 >= 0, from_customer_wallet_3 <= customer_wallet_3))
s.add(And(from_customer_wallet_5 >= 0, from_customer_wallet_5 <= customer_wallet_5))
s.add(And(from_customer_wallet_10 >= 0, from_customer_wallet_10 <= customer_wallet_10))
s.add(And(from_customer_wallet_25 >= 0, from_customer_wallet_25 <= customer_wallet_25))
s.add(And(from_customer_wallet_50 >= 0, from_customer_wallet_50 <= customer_wallet_50))
s.add(And(from_customer_wallet_100 >= 0, from_customer_wallet_100 <= customer_wallet_100))

from_cash_register=Int('from_cash_register')

# sum:
from_cash_register_1*1 +
from_cash_register_3*3 +
from_cash_register_5*5 +
from_cash_register_10*10 +
from_cash_register_25*25 +
from_cash_register_50*50 +
from_cash_register_100*100)

from_customer=Int('from_customer')

from_customer_wallet_1*1 +
from_customer_wallet_3*3 +
from_customer_wallet_5*5 +
from_customer_wallet_10*10 +
from_customer_wallet_25*25 +
from_customer_wallet_50*50 +
from_customer_wallet_100*100)

# the main constraint:
s.add(from_customer - checkout == from_cash_register)

value_of_banknotes_from_cash_register=Int('value_of_banknotes_from_cash_register')

# cashiers value small banknotes more:
from_cash_register_1*100 +
from_cash_register_3*50 +
from_cash_register_5*20 +
from_cash_register_10*10 +
from_cash_register_25*5 +
from_cash_register_50*2 +
from_cash_register_100)

# try to minimize this value during transaction:
s.minimize(value_of_banknotes_from_cash_register)

print s.check()
m=s.model()

print "from_cash_register_1=", m[from_cash_register_1]
print "from_cash_register_3=", m[from_cash_register_3]
print "from_cash_register_5=", m[from_cash_register_5]
print "from_cash_register_10=", m[from_cash_register_10]
print "from_cash_register_25=", m[from_cash_register_25]
print "from_cash_register_50=", m[from_cash_register_50]
print "from_cash_register_100=", m[from_cash_register_100]

print "from_cash_register=", m[from_cash_register]

print ""

print "from_customer_wallet_1=", m[from_customer_wallet_1]
print "from_customer_wallet_3=", m[from_customer_wallet_3]
print "from_customer_wallet_5=", m[from_customer_wallet_5]
print "from_customer_wallet_10=", m[from_customer_wallet_10]
print "from_customer_wallet_25=", m[from_customer_wallet_25]
print "from_customer_wallet_50=", m[from_customer_wallet_50]
print "from_customer_wallet_100=", m[from_customer_wallet_100]

print "from_customer=", m[from_customer]

sat
from_cash_register_1= 0
from_cash_register_3= 0
from_cash_register_5= 0
from_cash_register_10= 2
from_cash_register_25= 0
from_cash_register_50= 0
from_cash_register_100= 0
from_cash_register= 20

from_customer_wallet_1= 2
from_customer_wallet_3= 1
from_customer_wallet_5= 0
from_customer_wallet_10= 0
from_customer_wallet_25= 0
from_customer_wallet_50= 11
from_customer_wallet_100= 16
from_customer= 2155

So that the cashier have to pull only 2 tenners from cash desk.

Now what if we don't care about small banknotes, but want to make transaction as paperless as possible:

...

# minimize number of all banknotes in transaction:
s.minimize(
from_cash_register_1 +
from_cash_register_3 +
from_cash_register_5 +
from_cash_register_10 +
from_cash_register_25 +
from_cash_register_50 +
from_cash_register_100 +
from_customer_wallet_1 +
from_customer_wallet_3 +
from_customer_wallet_5 +
from_customer_wallet_10 +
from_customer_wallet_25 +
from_customer_wallet_50 +
from_customer_wallet_100)

...
sat
from_cash_register_1= 0
from_cash_register_3= 0
from_cash_register_5= 1
from_cash_register_10= 1
from_cash_register_25= 0
from_cash_register_50= 0
from_cash_register_100= 0
from_cash_register= 15

from_customer_wallet_1= 0
from_customer_wallet_3= 0
from_customer_wallet_5= 0
from_customer_wallet_10= 0
from_customer_wallet_25= 0
from_customer_wallet_50= 3
from_customer_wallet_100= 20
from_customer= 2150

Likewise, you can minimize banknotes from cashier only, or from customer only.

Please drop me email about any bug(s) and suggestion(s): dennis(@)yurichev.com.