Planning the Production of Electronic Components
Contents
Problem description
To augment its competitiveness a small business wishes to improve the production of its best selling products. One of its main activities is the production of cards with microchips and electronic badges. The company also produces the components for these cards and badges. Good planning of the production of these components therefore constitutes a decisive success factor for the company. The demand for components is internal in this case and hence easy to anticipate.
For the next six months the production of four products with references X43-M1, X43-M2, Y54-N1, Y54-N2 is to be planned. The production of these components is sensitive to variations of the level of production, and every change leads to a non-negligible cost through controls and readjustments. The company therefore wishes to minimize the cost associated with these changes whilst also taking into account the production and storage costs.
The demand data per time period, the production and storage costs, and the initial and desired final stock levels for every product are listed in the following table. When the production level changes, readjustments of the machines and controls have to be carried out for the current month. The cost incurred is proportional to the increase or reduction of the production compared to the preceding month. The cost for an increase of the production is $ 1 per unit but only $ 0.50 for a decrease of the production level.
Data for the four products
+------------------------------------+------------------+-------------+ | Demands | Cost | Stock | +------+----+----+----+----+----+----+----------+-------+-------+-----+ | Month| 1 | 2 | 3 | 4 | 5 | 6 |Production|Storage|Initial|Final| +------+----+----+----+----+----+----+----------+-------+-------+-----+ |X43-M1|1500|3000|2000|4000|2000|2500| 20 | 0.4 | 10 | 50 | |X43-M2|1300| 800| 800|1000|1100| 900| 25 | 0.5 | 0 | 10 | |Y54-N1|2200|1500|2900|1800|1200|2100| 10 | 0.3 | 0 | 10 | |Y54-N2|1400|1600|1500|1000|1100|1200| 15 | 0.3 | 0 | 10 | +------+----+----+----+----+----+----+----------+-------+-------+-----+
What is the production plan that minimizes the sum of costs incurred through changes of the production level, production and storage costs?
Variables
demand the demand for each component and month
prodcost Cost to produce a component
storagecost Cost to store a component
initialstock Initial stock
finalstock Final stock
increasecost, decreasecost Increase or decrease in cost
when producing more or less this month
than last month.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
demand = [1500 3000 2000 4000 2000 2500;... 1300 800 800 1000 1100 900;... 2200 1500 2900 1800 1200 2100;... 1400 1600 1500 1000 1100 1200]; prodcost = [20;25;10;15]; storagecost = [.4;.5;.3;.3]; initialstock = [10;0;50;0]; finalstock = [50;10;30;10]; increasecost = 1; decreasecost = 0.5; t = size(demand,2); p = size(demand,1); produce = tom('produce',p,t,'int'); store = tom('store',p,t,'int'); add = tom('add',t,1,'int'); reduce = tom('reduce',t,1,'int'); % All slots are integers bnds = {produce >= 0, store >= 0, add >= 0, reduce >= 0}; bnds1 = {store(:,end) >= finalstock}; % Production equilibrium constraint at start con1 = {store(:,1) == initialstock + produce(:,1) - demand(:,1)}; % Production equilibrium in process con2 = {store(:,2:end) == store(:,1:end-1) + produce(:,2:end) - demand(:,2:end)}; % Add/reduction constraint con3 = {sum(produce(:,2:end),1)' - sum(produce(:,1:end-1),1)' == ... add(2:end) - reduce(2:end)}; % Objective objective = sum(prodcost'*produce + storagecost'*store) + ... sum(increasecost*add(2:end)' + decreasecost*reduce(2:end)'); constraints = {bnds, bnds1, con1, con2, con3}; options = struct; options.solver = 'cplex'; options.name = 'Production of Electronic Components'; sol = ezsolve(objective,constraints,[],options); PriLev = 1; if PriLev > 0 for month = 1:t, disp(['Solution for month ' num2str(month)]) disp(['produce ' num2str(sol.produce(:,month)')]) disp(['stock ' num2str(sol.store(:,month)')]) disp(['increase ' num2str(sol.add(month))]) disp(['decrease ' num2str(sol.reduce(month))]) disp(' ') end end % MODIFICATION LOG % % 051018 med Created. % 060110 per Added documentation. % 060125 per Moved disp to end % 060203 med Removed printing of temp % 090407 med Converted to tomSym
Problem type appears to be: mip
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TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2010-02-05
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Problem: --- 1: Production of Electronic Components f_k 683929.000000000000000000
sum(|constr|) 0.000000000000056843
f(x_k) + sum(|constr|) 683929.000000000000000000
f(x_0) 0.000000000000000000
Solver: CPLEX. EXIT=0. INFORM=101.
CPLEX Branch-and-Cut MIP solver
Optimal integer solution found
FuncEv 37
CPU time: 0.015625 sec. Elapsed time: 0.016000 sec.
Solution for month 1
produce 1490 1300 2870 1400
stock 0 0 720 0
increase 0
decrease 0
Solution for month 2
produce 3000 800 780 2480
stock 0 0 0 880
increase 0
decrease 0
Solution for month 3
produce 2000 800 2900 1360
stock 0 0 0 740
increase 0
decrease 0
Solution for month 4
produce 4000 1000 1800 260
stock 0 0 0 0
increase 0
decrease 0
Solution for month 5
produce 2000 1100 1900 1100
stock 0 0 700 0
increase 0
decrease 960
Solution for month 6
produce 2550 910 1430 1210
stock 50 10 30 10
increase 0
decrease 0