Lindo Systems
! Small bucket production sequencing with changeovers (prodseqk);
! Minimize the sum of setup plus inventory costs
for producing multiple products on a single
machine over multiple periods,
Subject to:
At most one product in a time period( the small bucket feature),
Upper limit on amount produced in a period,
No backlogging of demand;
! Keywords: Changeover costs, lot sizing, scheduling, sequencing,
production scheduling, ;
SETS:
prod: iinv, ci, insetup;
timep;
pxt(prod,timep): dem, inv, x, z;
pxp( prod, prod): cs;
pxpxt( pxp, timep): y;
ENDSETS
DATA:
! The products;
prod = A B C D E F G H I J K;
! How many machines producing which product at beginning;
insetup= 1 0 0 0 0 0 0 0 0 0 0;
! Initial inventory;
iinv = 0 2 9 8 6 7 8 3 7 9 8;
! Inventory cost for each product per period;
ci = 3 2 2 1 2 4 2 1 1 2 1;
! The time periods;
timep = 1..20;
! Changeover cost for each product pair;
cs =
! A B C D E F G H I J K;
!A; 0 1 2 1 1 3 1 2 1 1 3
!B; 9 0 1 3 1 1 1 1 1 1 1
!C; 8 6 0 1 1 1 1 3 1 1 2
!D; 9 9 9 0 1 1 1 1 1 1 1
!E; 8 9 9 9 0 1 1 1 3 1 3
!F; 9 7 9 9 9 0 1 1 1 1 1
!G; 9 9 9 9 9 9 0 1 1 1 1
!H; 9 9 9 9 9 9 9 0 1 1 1
!I; 9 9 9 9 9 9 9 9 0 1 1
!J; 9 9 9 9 9 9 9 9 9 0 1
!K; 7 9 6 9 9 7 9 9 8 9 0
;
! Demands for each
product(row)and period(col);
! 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20;
dem =
!A; 0 0 7 0 5 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0
!B; 0 0 2 0 1 0 0 0 0 0 0 2 0 0 0 0 7 0 0 0
!C; 0 0 1 0 1 0 2 0 0 0 0 0 0 0 1 0 0 1 0 0
!D; 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 3 0 0 0 1
!E; 0 0 0 0 2 0 0 3 0 0 1 0 0 0 0 1 1 0 0 1
!F; 0 0 0 0 0 0 0 0 0 0 9 3 0 0 8 3 0 0 0 0
!G; 0 0 1 0 0 3 0 0 1 0 1 2 0 1 8 0 0 0 0 0
!H; 0 0 0 0 0 1 0 2 0 1 1 0 0 0 0 1 0 0 0 0
!I; 3 0 0 0 1 1 0 0 0 0 0 0 0 3 0 0 0 0 0 0
!J; 0 0 2 4 0 2 1 0 0 0 0 0 0 7 0 0 0 0 1 0
!K; 1 0 0 4 1 0 0 0 0 0 0 0 8 7 6 0 0 0 0 0
;
! Production capacity per period. All products measured in
same units;
caplim = 11;
ENDDATA
SUBMODEL SCHED:
! Definitions:
y( i, j, t)= 1 if machine is changed from
i to j at end of time period t.
x( i, t) = amount produced of product i in period t,
z( i, t) = 1 if machine produces (or is capable of)
product i during period t;
! Minimize the sum over all periods of the cost of
changeovers + inventory;
MIN = CCOST + ICOST;
CCOST = @SUM( pxpxt(i,j,t): cs(i,j)*y(i,j,t));
ICOST = @SUM( pxt(i,t): ci(i)*inv(i,t));
! Which product is being produced initially, and if
setup for k, must change to some j at end of period;
@FOR( prod( k):
z( k, 1) = insetup( k);
z( k, 1) = @SUM( prod( j): y( k, j, 1);
);
! If we changed from k to some j at the end of s,
we had to change from some i to k at end of s-1;
@FOR( pxt( k, s)| s #GT# 1:
z( k, s) = @SUM( prod( i): y( i, k, s-1));
z( k, s) = @SUM( prod( j): y( k, j, s));
);
! Setups must be 0 or 1;
@FOR( pxt( i, s):
@BIN( z( i, s))
);
! If we produce product i in period s, the machine
must be in product i mode in period s.
An upper limit on production is total demand of product;
@FOR( pxt( i, s):
x( i, s) <= @SMIN( caplim, @SUM( timep( t) | t #GE# s: dem( i, t)))* z( i, s)
);
);
! Inventory at end of period 1;
@FOR( prod(i):
[IEND1] inv( i, 1) = iinv( i) + x( i, 1) - dem( i, 1);
);
! and subsequent periods;
! Compute ending inventory in each period;
@FOR( pxt( i, s)| s #gt# 1:
[IEND] inv( i, s) = inv( i, s-1) + x( i, s) - dem( i, s);
);
! Production capacity constraints;
@FOR( timep( t):
@SUM( prod( i): x( i, t)) <= caplim;
);
ENDSUBMODEL
CALC:
! Turn off default output;
@SET("TERSEO", 2);
! Set relative optimality tolerance;
@SET("IPTOLR", .015);
! Time in seconds at which to apply rel tolerance;
@SET("TIM2RL", 30);
@SOLVE(SCHED);
! Produce a little report;
@WRITE(@NEWLINE(2),'Inventory cost = ',@FORMAT(ICOST,"9.2f"), @NEWLINE(1));
@WRITE( 'Changeover costs= ',@FORMAT(CCOST,"9.2f"), @NEWLINE(1));
@WRITE( 'Total costs = ',@FORMAT(ICOST+CCOST,"9.2f"), @NEWLINE(1));
@WRITE(@NEWLINE(1),' Period Product Production', @NEWLINE(1));
@FOR(timep(s):
@WRITE(' ',@FORMAT(s,"2.0f"),' ');
@FOR(prod(j) | z(j,s) #GT# .5:
@WRITE(prod(j),' ',@FORMAT(@ABS(x(j,s)),"8.1f"),@NEWLINE(1)));
);
ENDCALC