## Work-Shift Scheduling: Finding Optimal Assignment

```/*********************************************************************/
/*                                                                   */
/*          S A S   S A M P L E   L I B R A R Y                      */
/*                                                                   */
/*    NAME: clpe03                                                   */
/*   TITLE: Work-Shift Scheduling: Finding Optimal Assignment        */
/*                                                          (clpe03) */
/* PRODUCT: OR                                                       */
/*  SYSTEM: ALL                                                      */
/*    KEYS: OR                                                       */
/*   PROCS: CLP, GANTT                                               */
/*    DATA:                                                          */
/*                                                                   */
/* SUPPORT:                             UPDATE:                      */
/*     REF:                                                          */
/*    MISC: Example 3 from the CLP Procedure chapter of the          */
/*          Constraint Programming book.                             */
/*                                                                   */
/*********************************************************************/

proc clp out=clpout;
/*  Six workers (Alan, Bob, John, Mike, Scott and Ted)
are to be assigned to 3 working shifts.            */
var (W1-W6) = [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. */
gcc (W1-W6) = ( ( 1, 1, 4) ( 2, 2, 3) ( 3, 2, 2) );

/* Alan doesn't work on the first shift. */
lincon W1 <> 1;

/* Bob works only on the third shift. */
lincon W2 = 3;
run;

/* 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='John';
else if _Name_='W4' then Name='Mike';
else if _Name_='W5' then Name='Scott';
else if _Name_='W6' then Name='Ted';
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;

%macro callclp(obj);
%put The objective value is: &obj..;
proc clp out=clpout;
/* Six workers (Alan, Bob, John, Mike, Scott and Ted)
are to be assigned to 3 working shifts.  */
var (W1-W6) = [1,3];
var (C1-C6) = [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. */
gcc (W1-W6) = ( ( 1, 1, 4) ( 2, 2, 3) ( 3, 2, 2) );

/* Alan doesn't work on the first shift. */
lincon W1 <> 1;

/* Bob works only on the third shift. */
lincon W2 = 3;

/* Specify the costs of assigning the workers to the shifts.
Use 100 (a large number) to indicate an assignment
that is not possible.*/
element (W1, (100,  12, 10), C1);
element (W2, (100, 100,  6), C2);
element (W3, ( 16,   8, 12), C3);
element (W4, ( 10,   6,  8), C4);
element (W5, (  6,   6,  8), C5);
element (W6, ( 12,   4,  4), C6);

/* The total cost should be no more than the given objective value. */
lincon C1 + C2 + C3 + C4 + C5 + C6 <= &obj;
run;

/* when a solution is found,                      */
/* &_ORCLP_ contains the string SOLUTIONS_FOUND=1   */
%if %index(&_ORCLP_,  SOLUTIONS_FOUND=1) %then %let clpreturn=SUCCESSFUL;
%mend;

/* Bisection search method to determine the optimal objective value  */
%macro mincost(lb, ub);
%do %while (&lb<&ub-1);
%put Currently lb=&lb, ub=&ub..;
%let newobj=%eval((&lb+&ub)/2);
%let clpreturn=NOTFOUND;
%callclp(&newobj);
%if &clpreturn=SUCCESSFUL %then %let ub=&newobj;
%else %let lb=&newobj;
%end;

%callclp(&ub);

%put Minimum possible objective value within given range is &ub.;
%put Any value less than &ub makes the problem infeasible. ;

proc print;
run;
%mend;

/* Find the minimum objective value between 1 and 100. */
%mincost(lb=1, ub=100);

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='John';
else if _Name_='W4' then Name='Mike';
else if _Name_='W5' then Name='Scott';
else if _Name_='W6' then Name='Ted';
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;

```