Mbot at 13:34, 8 November 2011

2011-11-08T13:34:55Z

← Older revision		Revision as of 08:34, 8 November 2011
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	<pre>		<pre>
	Problem type appears to be: mip		Problem type appears to be: mip
	Time for symbolic processing: 0.~~05305~~ seconds		Time for symbolic processing: 0.053719 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
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	FuncEv 12		FuncEv 12
	CPU time: 0.~~280802~~ sec. Elapsed time: 0.~~089000~~ sec.		CPU time: 0.046800 sec. Elapsed time: 0.012000 sec.
	Problem type appears to be: mip		Problem type appears to be: mip
	Time for symbolic processing: 0.~~090135~~ seconds		Time for symbolic processing: 0.089912 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *

Mbot at 21:25, 2 November 2011

2011-11-02T21:25:52Z

← Older revision		Revision as of 16:25, 2 November 2011
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	Personnel requirement per time period		Personnel requirement per time period

			<pre>
	+------+-------------+------------------------+		+------+-------------+------------------------+
	\|Number\|Time interval\|Minimum number of nurses\|		\|Number\|Time interval\|Minimum number of nurses\|
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	\| 11 \| 10pm - 12am \| 20 \|		\| 11 \| 10pm - 12am \| 20 \|
	+------+-------------+------------------------+		+------+-------------+------------------------+
			</pre>

	==Variables==		==Variables==

			<pre>
	demand nurses needed per shift		demand nurses needed per shift
	hoursperday hours of work per day		hoursperday hours of work per day
Line 36:		Line 39:
	intervallength length of an interval		intervallength length of an interval
	maxdemand max number of nurses in Q 2		maxdemand max number of nurses in Q 2
			</pre>

	==Reference==		==Reference==

Mbot: Created page with "{{Part Of Manual|title=the TomSym Manual|link=TomSym Manual}} ==Problem description== Mr. Schedule has been asked to organize the schedule of nurses for the Cardiolo..."

2011-11-02T21:02:59Z

Created page with "{{Part Of Manual|title=the TomSym Manual|link=TomSym Manual}} ==Problem description== Mr. Schedule has been asked to organize the schedule of nurses for the Cardiolo..."

New page

{{Part Of Manual|title=the TomSym Manual|link=[[TomSym|TomSym Manual]]}}

==Problem description==

Mr. Schedule has been asked to organize the schedule of nurses for the Cardiology service at St. Joseph’s hospital. A working day in this service is subdivided into twelve periods of two hours. The personnel requirement changes from period to period: for instance, only a few nurses are required during the night, but the total number must be relatively high during the morning to provide specific care to the patients. The following table lists the personnel requirement for every time period.

Question 1: Determine the minimum number of nurses required to cover all the requirements, knowing that a nurse works eight hours per day and that she is entitled to a break of two hours after she has worked for four hours.

Question 2: The service only has 80 nurses, which is not sufficient with the given requirements. Mr. Schedule therefore proposes that part of the personnel works two additional hours per day. These two additional hours follow immediately after the last four hours, without any break. Determine the schedule of the nurses in this service that minimizes the number of nurses working overtime.

Personnel requirement per time period

+------+-------------+------------------------+
|Number|Time interval|Minimum number of nurses|
+------+-------------+------------------------+
| 0 | 00am - 02am | 15 |
| 1 | 02am - 04am | 15 |
| 2 | 04am - 06am | 15 |
| 3 | 06am - 08am | 35 |
| 4 | 08am - 10am | 40 |
| 5 | 10am - 12pm | 40 |
| 6 | 12pm - 02pm | 40 |
| 7 | 02pm - 04pm | 30 |
| 8 | 04pm - 06pm | 31 |
| 9 | 06pm - 08pm | 35 |
| 10 | 08pm - 10pm | 30 |
| 11 | 10pm - 12am | 20 |
+------+-------------+------------------------+

==Variables==

demand nurses needed per shift
hoursperday hours of work per day
breakinterval work this long before break
breaklength length of break
intervallength length of an interval
maxdemand max number of nurses in Q 2

==Reference==

Applications of optimization... Gueret, Prins, Seveaux

<source lang="matlab">
% Marcus Edvall, Tomlab Optimization Inc, E-mail: tomlab@tomopt.com
% Copyright (c) 2005-2009 by Tomlab Optimization Inc., $Release: 7.2.0$
% Written Oct 7, 2005. Last modified Apr 8, 2009.
</source>

==Problem setup==

<source lang="matlab">
demand = [15 15 15 35 40 40 40 30 31 35 30 20]';
hoursperday = 8;
breakinterval = 4;
breaklength = 2;
intervallength = 2;
maxdemand = 80;

n = length(demand);

start = tom('start',n,1,'int');
% All variables are integer
bnds = {0 <= start};

% Requirement constraints
idxvec = [1:12, 1:4];
con = cell(n,1);
for i=1:n
idx = idxvec(i:i+4);
idx(3) = [];
con{i} = {sum(start(idx)) >= demand(i)};
end

% Objective
objective = sum(start);

constraints = {bnds, con};
options = struct;
options.solver = 'cplex';
options.name = 'Scheduling Nurses';
sol1 = ezsolve(objective,constraints,[],options);
f_k1 = subs(objective,sol1);

overtime = tom('overtime',n,1,'int');
bnds = {bnds, overtime >= 0};

% Requirement constraints
con = cell(n,1);
idxvec2 = [1:12, 1:5];
for i=1:n
idx = idxvec(i:i+4);
idx(3) = [];
idxo = idxvec2(i+5);
con{i} = {sum(start(idx)) + overtime(idxo) >= demand(i)};
end

% Overtime less than regular (extends these)
con2 = {overtime <= start};
con3 = {sum(start) <= maxdemand};
constraints = {bnds, con, con2, con3};
objective = sum(overtime);
sol2 = ezsolve(objective,constraints,[],options);
f_k2 = subs(objective,sol2);

PriLev = 1;
if PriLev > 0
x1 = sol1.start; % nurses working
intervals = length(x1); % number of intervals
wrk = [x1 circshift(x1,-1) circshift(x1,-3) circshift(x1,-4)];
work_1 = sum(wrk')'; % total nurses
n = sol2.start; % regular shifts
o = sol2.overtime; % overtime
work1 = [n circshift(n,-1) ... % regular -> 4 intervals
circshift(n,-3) circshift(n,-4)];
work2 = [o circshift(o,-1) ... % overtime -> 5 intervals
circshift(o,-3) circshift(o,-4) circshift(o,-5)];
work_2 = sum(work1')' + sum(work2')'; % total work

disp(['Without overtime we need ' num2str(f_k1) ' nurses.'])
disp([' This is the schedule:'])
for i = 1:intervals,
disp([' ' num2str((i-1)*2) '.00: ' num2str(x1(i)) ...
' start and ' num2str(work_1(i)) ' are working.' ...
' ( demand is ' num2str(demand(i)) ')' ])
end

disp(['With overtime we have ' num2str(sum(n)) ...
' nurses, and ' num2str(sum(o)) ...
' of them will work overtime.'])
disp([' This is the schedule:'])
for i = 1:intervals,
disp([' ' num2str((i-1)*2) '.00: ' num2str(n(i)) ...
' start a regular shift, ' num2str(o(i)) ...
' start an overtime shift and ' num2str(work_2(i)) ...
' are working.' ...
' ( demand is ' num2str(demand(i)) ')' ])
end
end

% MODIFICATION LOG
%
% 051202 med Created
% 060117 per Added documentation
% 060126 per Moved disp to end
% 090308 med Converted to tomSym
</source>

<pre>
Problem type appears to be: mip
Time for symbolic processing: 0.05305 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Scheduling Nurses f_k 100.000000000000000000
f(x_0) 0.000000000000000000

Solver: CPLEX. EXIT=0. INFORM=101.
CPLEX Branch-and-Cut MIP solver
Optimal integer solution found

FuncEv 12
CPU time: 0.280802 sec. Elapsed time: 0.089000 sec.
Problem type appears to be: mip
Time for symbolic processing: 0.090135 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Scheduling Nurses f_k 40.000000000000000000
f(x_0) 0.000000000000000000

Solver: CPLEX. EXIT=0. INFORM=101.
CPLEX Branch-and-Cut MIP solver
Optimal integer solution found

FuncEv 19
Elapsed time: 0.003000 sec.
Without overtime we need 100 nurses.
This is the schedule:
0.00: 7 start and 24 are working. ( demand is 15)
2.00: 17 start and 34 are working. ( demand is 15)
4.00: 0 start and 29 are working. ( demand is 15)
6.00: 0 start and 35 are working. ( demand is 35)
8.00: 0 start and 40 are working. ( demand is 40)
10.00: 17 start and 40 are working. ( demand is 40)
12.00: 12 start and 46 are working. ( demand is 40)
14.00: 23 start and 36 are working. ( demand is 30)
16.00: 0 start and 31 are working. ( demand is 31)
18.00: 11 start and 35 are working. ( demand is 35)
20.00: 0 start and 30 are working. ( demand is 30)
22.00: 13 start and 20 are working. ( demand is 20)
With overtime we have 80 nurses, and 40 of them will work overtime.
This is the schedule:
0.00: 7 start a regular shift, 0 start an overtime shift and 17 are working. ( demand is 15)
2.00: 5 start a regular shift, 0 start an overtime shift and 15 are working. ( demand is 15)
4.00: 10 start a regular shift, 0 start an overtime shift and 17 are working. ( demand is 15)
6.00: 3 start a regular shift, 2 start an overtime shift and 37 are working. ( demand is 35)
8.00: 0 start a regular shift, 0 start an overtime shift and 55 are working. ( demand is 40)
10.00: 0 start a regular shift, 0 start an overtime shift and 65 are working. ( demand is 40)
12.00: 2 start a regular shift, 0 start an overtime shift and 63 are working. ( demand is 40)
14.00: 15 start a regular shift, 0 start an overtime shift and 71 are working. ( demand is 30)
16.00: 15 start a regular shift, 15 start an overtime shift and 57 are working. ( demand is 31)
18.00: 10 start a regular shift, 10 start an overtime shift and 58 are working. ( demand is 35)
20.00: 13 start a regular shift, 13 start an overtime shift and 43 are working. ( demand is 30)
22.00: 0 start a regular shift, 0 start an overtime shift and 22 are working. ( demand is 20)

</pre>

TomSym Scheduling Nurses - Revision history

Mbot at 13:34, 8 November 2011

Mbot at 21:25, 2 November 2011

Mbot: Created page with "{{Part Of Manual|title=the TomSym Manual|link=TomSym Manual}} ==Problem description== Mr. Schedule has been asked to organize the schedule of nurses for the Cardiolo..."