TomSym Production Planning with Personnel Assignment
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Problem description
The company Line Production decides to plan the production of four of its products (P1, P2, P3, P4) on its five production lines (L1 to L5). The company gains net profits of $ 7 for the products P1 and P4, $ 8 for P2, and $ 9 for P3. The maximum times during which the five production lines may operate are different during the planning period. The maximum capacities for L1 to L5 are 4500 hours, 5000 hours, 4500 hours, 1500 hours, and 2500 hours respectively. The table below lists the processing time in hours necessary for the production of one unit of every product on every production line. Which quantities of P1 to P4 should be produced to maximize the total profit? If subsequently a transfer of personnel (and hence of working hours) is authorized between production lines during the planning period as shown in the second table below, which is the maximum profit? How many hours are transfered and under what conditions?
Unitary processing times
+--------+-------------------+ | | Lines | +--------+---+---+---+---+---+ |Products| L1| L2| L3| L4| L5| +--------+---+---+---+---+---+ | P1 |1.3|0.9|2.0|0.3|0.9| | P2 |1.8|1.7|1.4|0.6|1.1| | P3 |1.3|1.2|1.3|1.0|1.4| | P4 |0.9|1.1|1.0|0.9|1.0| +--------+---+---+---+---+---+
Possible transfers of personnel
+------+-------------------+------------------+ | | Destination | | +------+---+---+---+---+---+Maximum number of | |Origin| L1| L2| L3| L4| L5|transferable hours| +------+---+---+---+---+---+------------------+ | L1 | -|yes|yes|yes| no| 400 | | L2 | no| -|yes| no|yes| 800 | | L3 |yes|yes| -|yes| no| 500 | | L4 | no| no| no| -|yes| 200 | | L5 |yes|yes|yes| no| -| 300 | +------+---+---+---+---+---+------------------+
Variables
profit Profit per product capacity Hours at each site timemat Time to produce products transfermat Transfer matrix maxtransfer Maximum hours to transfer
Reference
Applications of optimization... Gueret, Prins, Seveaux
% 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.Problem setup
profit = [7 8 9 7]';
capacity = [4500 5000 4500 1500 2500]';
timemat = [ 1.3 0.9 2.0 0.3 0.9;...
1.8 1.7 1.4 0.6 1.1;...
1.3 1.2 1.3 1.0 1.4;...
0.9 1.1 1.0 0.9 1.0];
transfermat = [0 1 1 1 0;...
0 0 1 0 1;...
1 1 0 1 0;...
0 0 0 0 1;...
1 1 1 0 0];
maxtransfer = [400 800 200 500 300]';
p = length(profit); %products
produce = tom('produce',p,1);
% No variables are binary
bnds = {produce >= 0};
% Capacity constraint
con = {timemat'*produce <= capacity};
% Objective
objective = -profit'*produce;
constraints = {bnds, con};
options = struct;
options.solver = 'cplex';
options.name = 'Prod Planning with Personnel Assign';
sol1 = ezsolve(objective,constraints,[],options);
l = length(capacity);
hours = tom('hours',l,1);
transfer = tom('transfer',l,l);
% No variables are binary
bnds = {produce >= 0, hours >= 0, transfer >= 0};
% Line constraint
con = {timemat'*produce <= hours};
% Line transfer constraint
con2 = {hours == capacity + sum(transfermat.*transfer,1)' - ...
sum(transfermat.*transfer,2)};
% Line max transfer constraint
con3 = {sum(transfermat.*transfer,2) <= maxtransfer};
% Objective
objective = -profit'*produce;
constraints = {bnds, con, con2, con3};
options = struct;
options.solver = 'cplex';
options.name = 'Prod Planning with Personnel Assign';
sol2 = ezsolve(objective,constraints,[],options);
PriLev = 1;
if PriLev > 0
disp('without transfer')
x1 = sol1.produce;
for i = 1:length(x1)
if x1(i) > 0
disp([' produce ' num2str(x1(i)) ' units of P' num2str(i) ])
end
end
disp('with transfer')
x2 = sol2.produce;
for i = 1:length(x2),
if x2(i) > 0,
disp([' produce ' num2str(x2(i)) ' units of P' num2str(i) ])
end
end
transfer = sol2.transfer;
for i = 1:size(transfer,1),
idx = find(transfer(i,:));
for j = 1: length(idx),
disp([' transfer ' num2str(transfer(i,idx(j))) ...
' hours from ' num2str(i) ' to ' num2str(idx(j)) ])
end
end
hours = sol2.hours;
for j = 1:length(hours),
disp([' work ' num2str(hours(j)) ' hours at L' num2str(j) ])
end
end
% MODIFICATION LOG
%
% 051205 med Created.
% 060117 per Added documentation.
% 060126 per Moved disp to end
% 060131 per Really moved disp to end
% 090325 med Converted to tomSymProblem type appears to be: lp
Time for symbolic processing: 0.0080349 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Prod Planning with Personnel Assign f_k -18882.978723404256000000
f(x_0) 0.000000000000000000
Solver: CPLEX. EXIT=0. INFORM=1.
CPLEX Dual Simplex LP solver
Optimal solution found
FuncEv 6 Iter 6
Elapsed time: 0.002000 sec.
Problem type appears to be: lp
Time for symbolic processing: 0.029608 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Prod Planning with Personnel Assign f_k -23431.085043988263000000
sum(|constr|) 0.000000000000682121
f(x_k) + sum(|constr|) -23431.085043988263000000
f(x_0) 0.000000000000000000
Solver: CPLEX. EXIT=0. INFORM=1.
CPLEX Dual Simplex LP solver
Optimal solution found
FuncEv 10 Iter 10
Elapsed time: 0.002000 sec.
without transfer
produce 1542.5532 units of P1
produce 1010.6383 units of P2
with transfer
produce 702.346 units of P1
produce 942.8152 units of P2
produce 554.2522 units of P3
produce 854.8387 units of P4
transfer 400 hours from 1 to 4
transfer 800 hours from 2 to 5
transfer 200 hours from 3 to 4
work 4100 hours at L1
work 4200 hours at L2
work 4300 hours at L3
work 2100 hours at L4
work 3300 hours at L5
