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Maximise discount at a bookstore Description A bookstore has the following discount policy:
For each $1 you spend you get 0.1% discount on your next purchase.Example: If you have to buy three books that cost $10, $20 and $30, you could buy the $30 book today, the $10 book tomorrow (on which you'll get a 3% discount), and the $20 book the following day (on which you'll get a 1% discount). Or you could buy the $30 book and the $20 book today, and the $10 book tomorrow (with a 5% discount). What is the cheapest way to buy N books (for given prices) ?
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bookdisc.mos (!********************************************************************* Mosel NL examples ================= file bookdisc.mos ````````````````` A bookstore has the following discount policy: For each $1 you spend you get 0.1% discount on your next purchase. Example: If you have to buy three books that cost $10, $20 and $30, you could buy the $30 book today, the $10 book tomorrow (on which you'll get a 3% discount), and the $20 book the following day (on which you'll get a 1% discount). Or you could buy the $30 book and the $20 book today, and the $10 book tomorrow (with a 5% discount). What is the cheapest way to buy N books (for given prices) ? Based on AMPL model bookdisc.mos by M.J.Chlond Reference: J. & L. Poniachik, Hard-to-Solve Brainteasers (p16), Sterling (c) 2008 Fair Issac Corporation author: S. Heipcke, Jul. 2003, rev. Mar 2013 *********************************************************************!) model "bookdisc" uses "mmxnlp" parameters N = 8 ! Number of books D = 4 ! Number of days DISC = 0.001 ! Discount factor end-parameters declarations BOOKS = 1..N ! Set of books DAYS = 1..D ! Set of days COST: array(BOOKS) of real ! Cost of books ifbuy: array(BOOKS,DAYS) of mpvar ! ifbuy[i,j]=1 if book i bought on day j, 0 otherwise total: array(DAYS) of mpvar ! Total cost of books on day j totcost: mpvar ! Total cost of all books end-declarations forall(i in BOOKS,j in DAYS) ifbuy(i,j) is_binary ! forall(j in DAYS) total(j) is_integer ! Generate random book prices setrandseed(3) forall(i in BOOKS) COST(i):= 10+round(50*random) ! Objective: total cost totcost = sum(i in BOOKS) COST(i) - sum(k in DAYS | k>1) DISC*total(k-1)*total(k) ! Total cost per day forall(j in DAYS) total(j) = sum(i in BOOKS) COST(i)*ifbuy(i,j) ! Every book is bought on one day forall(i in BOOKS) sum(j in DAYS) ifbuy(i,j) = 1 ! Solve the problem setparam("xnlp_verbose",true) setparam("xnlp_solver",0) minimize(totcost) ! Solution printing TOTCOST:= sum(i in BOOKS) COST(i) writeln("Total cost: ", strfmt(getsol(totcost),5,2), " (orig: ", TOTCOST, ") discount:", strfmt((TOTCOST-totcost.sol)/TOTCOST * 100,5,2), "%" ) forall(i in BOOKS) BDay(i):=round(getsol(sum(j in DAYS) j*ifbuy(i,j))) forall(i in BOOKS) writeln("Book ", i, " bought on day ", BDay(i), " at ", strfmt(COST(i)*(1-if(BDay(i)>1, DISC*total(BDay(i)-1).sol,0)),5,2), " (orig: ", strfmt(COST(i),5,2), ")" ) end-model | |||||||||||

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