# Planning the Production of Bicycles

## Contents

## Problem description

A company produces bicycles for children. The sales forecast in thousand of units for the coming year are given in the following table. The company has a capacity of 30,000 bicycles per month. It is possible to augment the production by up to 50% through overtime working, but this increases the production cost for a bicycle from the usual $ 32 to $ 40.

Sales forecasts for the coming year in thousand units

+---+---+---+---+---+---+---+---+---+---+---+---+ |Jan|Feb|Mar|Apr|May|Jun|Jul|Aug|Sep|Oct|Nov|Dec| +---+---+---+---+---+---+---+---+---+---+---+---+ | 30| 15| 15| 25| 33| 40| 45| 45| 26| 14| 25| 30| +---+---+---+---+---+---+---+---+---+---+---+---+

Currently there are 2,000 bicycles in stock. The storage costs have been calculated as $ 5 per unit held in stock at the end of a month. We assume that the storage capacity at the company is virtually unlimited (in practice this means that the real capacity, that is quite obviously limited, does not impose any limits in our case). We are at the first of January. Which quantities need to be produced and stored in the course of the next twelve months in order to satisfy the forecast demand and minimize the total cost?

## Variables

normcapacity the normal production capacity extracapacity extra capacity normcost normal cost extracost cost per bike if overtime demand bikes wanted per month startstock bikes in store storagecost cost to have a bike in store one 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

normcapacity = 30000; extracapacity = 15000; normcost = 32; extracost = 40; demand = [30;15;15;25;33;40;45;45;26;14;25;30]*1000; startstock = 2000; storagecost = 5; t = length(demand); % Months pnorm = tom('pnorm',t,1,'int'); pover = tom('pover',t,1,'int'); store = tom('store',t,1,'int'); % All slots are integers bnds = {0 <= pnorm <= normcapacity, 0 <= pover <= extracapacity,... 0 <= store}; % Initial demand constraint con1 = {pnorm(1) + pover(1) + startstock == demand(1) + store(1)}; % Monthly demand constraint con2 = {pnorm(2:end) + pover(2:end) + store(1:end-1) == ... demand(2:end) + store(2:end)}; % Objective objective = sum(normcost*pnorm) + sum(extracost*pover) + ... sum(storagecost*store); constraints = {bnds, con1, con2}; options = struct; options.solver = 'cplex'; options.name = 'Planning the Production of Bicycles'; sol = ezsolve(objective,constraints,[],options); PriLev = 1; if PriLev > 0 months = length(demand); temp = [sol.pnorm, sol.pover, sol.store]; for i = 1:months, disp(['Month ' num2str(i) ':']) disp([' produce ' num2str(temp(i,1)) ' regular bikes' ]) if temp(i,2) > 0, disp([' and ' num2str(temp(i,2)) ' extras' ]) end if temp(i,3) > 0, disp([' let ' num2str(temp(i,3)) ' be stored' ]) end end end % MODIFICATION LOG % % 051017 med Created. % 060110 per Added documentation. % 060125 per Moved disp to end % 090308 med Converted to tomSym

Problem type appears to be: mip ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2010-02-05 ===================================================================================== Problem: --- 1: Planning the Production of Bicycles f_k 11247000.000000000000000000 f(x_0) 0.000000000000000000 Solver: CPLEX. EXIT=0. INFORM=101. CPLEX Branch-and-Cut MIP solver Optimal integer solution found FuncEv 18 Month 1: produce 28000 regular bikes Month 2: produce 15000 regular bikes Month 3: produce 15000 regular bikes Month 4: produce 28000 regular bikes let 3000 be stored Month 5: produce 30000 regular bikes Month 6: produce 30000 regular bikes and 10000 extras Month 7: produce 30000 regular bikes and 15000 extras Month 8: produce 30000 regular bikes and 15000 extras Month 9: produce 26000 regular bikes Month 10: produce 14000 regular bikes Month 11: produce 25000 regular bikes Month 12: produce 30000 regular bikes