Production Planning with Personnel Assignment

Contents

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 tomSym
Problem type appears to be: lp
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
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 
Problem type appears to be: lp
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Prod Planning with Personnel Assign  f_k  -23431.085043988271000000
                                             sum(|constr|)      0.000000000000909495
                                    f(x_k) + sum(|constr|) -23431.085043988271000000
                                                    f(x_0)      0.000000000000000000

Solver: CPLEX.  EXIT=0.  INFORM=1.
CPLEX Dual Simplex LP solver
Optimal solution found

FuncEv   10 Iter   10 
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