## Optimal Work-Shift Scheduling Assignment (oclpe03)

```/*********************************************************************/
/*                                                                   */
/*          S A S   S A M P L E   L I B R A R Y                      */
/*                                                                   */
/*    NAME: oclpe03                                                  */
/*   TITLE: Optimal Work-Shift Scheduling Assignment (oclpe03)       */
/* PRODUCT: OR                                                       */
/*  SYSTEM: ALL                                                      */
/*    KEYS: OR                                                       */
/*   PROCS: CLP, GANTT                                               */
/*    DATA:                                                          */
/*                                                                   */
/* SUPPORT:                             UPDATE:                      */
/*     REF:                                                          */
/*    MISC: Example from the CLP Solver chapter of the               */
/*          Mathematical Programming book.                           */
/*                                                                   */
/*********************************************************************/

proc optmodel;
/*  Six workers (Alan, Bob, Juanita, Mike, Ravi and Aisha)
are to be assigned to 3 working shifts.            */
set WORKERS = 1..6;
var W {WORKERS} integer >= 1 <= 3;

/* The first shift needs at least 1 and at most 4 people;
the second shift needs at least 2 and at most 3 people;
and the third shift needs exactly 2 people. */
con ShiftNeeds:
gcc(W, /<1,1,4>,<2,2,3>,<3,2,2>/);

/* Alan doesn't work on the first shift. */
con Alan:
W[1] ne 1;

/* Bob works only on the third shift. */
fix W[2] = 3;

solve;
print W;
create data clpout from {j in WORKERS} <col('W'||j)=W[j]>;
quit;

/* print solution */
proc print;
title 'Solution to Work-Shift Scheduling Problem';
run;

/* Produce Gantt Chart */
proc transpose data=clpout out=tmp1;
run;

data tmp2 (drop=_NAME_);
format Name \$8.;
set tmp1;
reName col1=e_start;
e_finish=col1+1;
duration=1;
if _Name_='W1' then Name='Alan';
else if _Name_='W2' then Name='Bob';
else if _Name_='W3' then Name='Juanita';
else if _Name_='W4' then Name='Mike';
else if _Name_='W5' then Name='Ravi';
else if _Name_='W6' then Name='Aisha';
else delete;
run;

proc format;
value shift
1 = '   Shift 1'
2 = '   Shift 2'
3 = '   Shift 3'
4 = ' '
;
run;

goptions htext=1.5;
pattern1 c=ltgray;
title j=c h=6pct 'Example 3: Feasible Work-Shift Assignment';

proc gantt data=tmp2;
format e_start shift.;
chart /
pcompress
skip=2
dur=duration scale=16
chartwidth=84
ref=2 3 lref=20
useformat
nolegend;
id Name;
run;

proc optmodel;
/* Six workers (Alan, Bob, Juanita, Mike, Ravi and Aisha)
are to be assigned to 3 working shifts.  */
set WORKERS = 1..6;
set SHIFTS  = 1..3;
var W {WORKERS} integer >= 1 <= 3;
var C {WORKERS} integer >= 1 <= 100;

/* The first shift needs at least 1 and at most 4 people;
the second shift needs at least 2 and at most 3 people;
and the third shift needs exactly 2 people. */
con GccCon:
gcc(W, /<1,1,4>,<2,2,3>,<3,2,2>/);

/* Alan doesn't work on the first shift. */
con Alan:
W[1] ne 1;

/* Bob works only on the third shift. */
fix W[2] = 3;

/* Specify the costs of assigning the workers to the shifts.
Use 100 (a large number) to indicate an assignment
that is not possible.*/
num a {WORKERS, SHIFTS} = [
100,  12, 10,
100, 100,  6,
16,   8, 12
10,   6,  8
6,   6,  8
12,   4,  4
];
con ElementCon {j in WORKERS}:
element(W[j], {k in SHIFTS} a[j,k], C[j]);

/* explicit objective instead of binary search */
min TotalCost = sum {j in WORKERS} C[j];
con TotalCost_bounds:
1 <= TotalCost <= 100;

solve;
print W;
create data clpout from
{j in WORKERS} <col('W'||j)=W[j]> {j in WORKERS} <col('C'||j)=C[j]>;
quit;

proc transpose data=clpout out=tmp1;
run;

data tmp2 (drop=_NAME_);
format Name \$8.;
set tmp1;
reName col1=e_start;
e_finish=col1+1;
duration=1;
if _Name_='W1' then Name='Alan';
else if _Name_='W2' then Name='Bob';
else if _Name_='W3' then Name='Juanita';
else if _Name_='W4' then Name='Mike';
else if _Name_='W5' then Name='Ravi';
else if _Name_='W6' then Name='Aisha';
else delete;
run;

data tmp4;
set tmp1;
if _n_<=6 then delete;
run;

proc format;
value shift
1 = '   Shift 1'
2 = '   Shift 2'
3 = '   Shift 3'
4 = ' '
5 = ' ';
run;

data tmp5(drop=_Name_);
format e_start shift.;
format cost1 \$8.;
set tmp2;
set tmp4;
rename col1=cost;
cost1="\$"||strip(col1);
run;

data labels;
_y = -1; _xvar="e_start"; _xoffset=0.25; _yoffset=-.15;
_clabel='blue'; _hlabel=1.0; _lvar="cost1";
run;

proc sort data=tmp5 out=tmp6;
by e_start;
run;

data tmp6;
retain total 0;
set tmp6;
total=total+cost;
run;

data tmp7(keep=x y drop=max_y cscale);
set tmp6 end=last;
y=lag(total);
if ( dif(e_start) ) then do;
x=e_start;
output;
end;
if (last ) then do;
y=total;
x=e_start+1;
output;
max_y = y + 5;
cscale = 6 /max_y;
stry=y;
call symput('max_x', put(x+0.15,best.));
call symput('mincost', strip(put(stry,best.)));
call symput('max_y', strip(put(max_y,best.)));
call symput('cscale', put(cscale,best.));
end;
run;

%put &max_y;
%put &cscale;
%put &mincost;

%annomac;
data anno; /* define cost curve. */
%dclanno;
%system(2,2,4);
*length lab \$16;
set tmp7;
when='a';
if _n_ = 1 then do;
do i = 0 to &max_y by 5;
lab=put( i, dollar7.);
%label(&max_x + 0.8, (&max_y -i) * &cscale,lab,black,0,0,1.0, ,4);
output;
%label(&max_x + 0.9, (&max_y -i) * &cscale,'-',black,0,0,1.0, ,4);
end;
%move(1, &max_y * &cscale); /* initial point to start the cost curve.*/
end;
else do;
%draw( x, (&max_y - y) * &cscale, blue,2,2);
end;
run;

goptions htext=1.5;
pattern1 c=ltgray;
title1 j=c h=6pct 'Example 3: Optimal Work-Shift Assignment';
title2 j=c h=5pct "Minimum-Cost Schedule: \$&mincost";

proc gantt data=tmp5 labdata=labels annotate=anno;
chart /
pcompress
skip=2
chartwidth=84
labsplit='/' scale=16
mindate=1 maxdate=5
ref=2 3 4
useformat nolegend
;
id Name;
run;

```