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Lot sizing Description Economic lot sizing (ELS) considers production planning
over a given planning horizon. In every period, there is
a given demand for every product that must be satisfied by
the production in this period and by inventory carried over
from previous periods.
A set-up cost is associated with production in a period,
and the total production capacity per period is limited.
The unit production cost per product and time period is
given. There is no inventory or stock-holding cost.
The model implements a configurable cutting plane algorithm
for this problem. Further explanation of this example: 'Xpress teaching material', Section 2.7 'Branch-and-Cut'; Xpress Whitepaper 'Embedding Optimization Algorithms'; 'Mosel User Guide', Section 11.1 'Cut generation'
Source Files By clicking on a file name, a preview is opened at the bottom of this page.
Data Files els_graph.mos (!******************************************************* Mosel Example Problems ====================== file els.mos ```````````` TYPE: Lot sizing problem DIFFICULTY: 4 FEATURES: MIP problem, implementation of Branch-and-Cut and Cut-and-Branch algorithms, definition of Optimizer callbacks, Optimizer and Mosel parameter settings, `case', `procedure', `function', time measurement DESCRIPTION: Economic lot sizing (ELS) considers production planning over a given planning horizon. In every period, there is a given demand for every product that must be satisfied by the production in this period and by inventory carried over from previous periods. A set-up cost is associated with production in a period, and the total production capacity per period is limited. The unit production cost per product and time period is given. There is no inventory or stock-holding cost. The model implements a configurable cutting plane algorithm for this problem. FURTHER INFO: `Applications of optimization with Xpress-MP teaching material', Section 2.7 `Branch-and-Cut'; `Mosel User Guide', Section 11.1 `Cut generation' -- This model cannot be run with a Community Licence for the provided data instance -- (c) 2008 Fair Isaac Corporation author: S. Heipcke, 2001, rev. July 2023 *******************************************************!) model "ELS" uses "mmxprs","mmsystem", "mmsvg" parameters ALG = 6 ! Algorithm choice [default settings: 0] CUTDEPTH = 10 ! Maximum tree depth for cut generation EPS = 1e-6 ! Zero tolerance end-parameters forward function cb_node:boolean forward procedure tree_cut_gen declarations TIMES = 1..15 ! Range of time PRODUCTS = 1..4 ! Set of products DEMAND: array(PRODUCTS,TIMES) of integer ! Demand per period SETUPCOST: array(TIMES) of integer ! Setup cost per period PRODCOST: array(PRODUCTS,TIMES) of integer ! Production cost per period CAP: array(TIMES) of integer ! Production capacity per period D: array(PRODUCTS,TIMES,TIMES) of integer ! Total demand in periods t1 - t2 produce: array(PRODUCTS,TIMES) of mpvar ! Production in period t setup: array(PRODUCTS,TIMES) of mpvar ! Setup in period t solprod: array(PRODUCTS,TIMES) of real ! Sol. values for var.s produce solsetup: array(PRODUCTS,TIMES) of real ! Sol. values for var.s setup starttime: real end-declarations initializations from "els.dat" DEMAND SETUPCOST PRODCOST CAP end-initializations forall(p in PRODUCTS,s,t in TIMES) D(p,s,t):= sum(k in s..t) DEMAND(p,k) ! Objective: minimize total cost MinCost:= sum(t in TIMES) (SETUPCOST(t) * sum(p in PRODUCTS) setup(p,t) + sum(p in PRODUCTS) PRODCOST(p,t) * produce(p,t) ) ! Satisfy the total demand forall(p in PRODUCTS,t in TIMES) Dem(p,t):= sum(s in 1..t) produce(p,s) >= sum (s in 1..t) DEMAND(p,s) ! If there is production during t then there is a setup in t forall(p in PRODUCTS, t in TIMES) ProdSetup(p,t):= produce(p,t) <= D(p,t,getlast(TIMES)) * setup(p,t) ! Capacity limits forall(t in TIMES) Capacity(t):= sum(p in PRODUCTS) produce(p,t) <= CAP(t) ! Variables setup are 0/1 forall(p in PRODUCTS, t in TIMES) setup(p,t) is_binary ! Uncomment to get detailed MIP output setparam("XPRS_VERBOSE", true) ! All cost data are integer, we therefore only need to search for integer ! solutions setparam("XPRS_MIPADDCUTOFF", -0.999) ! Set Mosel comparison tolerance to a sufficiently small value setparam("ZEROTOL", EPS/100) writeln("**************ALG=",ALG,"***************") starttime:=gettime SEVERALROUNDS:=false; TOPONLY:=false case ALG of 1: setparam("XPRS_CUTSTRATEGY", 0) ! No cuts 2: setparam("XPRS_PRESOLVE", 0) ! No presolve 3: tree_cut_gen ! User branch-and-cut + automatic cuts 4: do ! User branch-and-cut (several rounds), tree_cut_gen ! no automatic cuts setparam("XPRS_CUTSTRATEGY", 0) SEVERALROUNDS:=true end-do 5: do ! User cut-and-branch (several rounds) tree_cut_gen ! + automatic cuts SEVERALROUNDS:=true TOPONLY:=true end-do 6: do ! User branch-and-cut (several rounds) tree_cut_gen ! + automatic cuts SEVERALROUNDS:=true end-do end-case minimize(MinCost) ! Solve the problem writeln("Time: ", gettime-starttime, "sec, Nodes: ", getparam("XPRS_NODES"), ", Solution: ", getobjval) write("Period setup ") forall(p in PRODUCTS) write(strfmt(p,-7)) forall(t in TIMES) do write("\n ", strfmt(t,2), strfmt(getsol(sum(p in PRODUCTS) setup(p,t)),8), " ") forall(p in PRODUCTS) write(getsol(produce(p,t)), " (",DEMAND(p,t),") ") end-do writeln ! Draw the solution forall(p in PRODUCTS) do svgaddgroup("P"+p, "Product "+p) svgsetstyle(SVG_FILL, SVG_CURRENT) svgsetstyle(SVG_STROKEWIDTH, 0) end-do forall(t in TIMES) do cum:=0.0 forall(p in PRODUCTS | produce(p,t).sol>0) do svgaddrectangle("P"+p,t, cum, 0.9, setup(p,t).sol) svgsetstyle(svggetlastobj, SVG_OPACITY, 0.5) cum+=setup(p,t).sol svgaddrectangle("P"+p,t, cum, 0.9, produce(p,t).sol) cum+=produce(p,t).sol end-do end-do ! svgsetgraphviewbox(0, 0, TIMES.last+1, max(t in TIMES) CAP(t) +1) svgsetgraphlabels("Time", "Production amounts and setup") svgsave("els.svg") svgrefresh svgwaitclose("Close browser window to terminate model execution.", 1) !************************************************************************* ! Cut generation loop: ! get the solution values ! identify and set up violated constraints ! load the modified problem and load the saved basis !************************************************************************* function cb_node:boolean declarations ncut:integer ! Counter for cuts cut: array(range) of linctr ! Cuts cutid: array(range) of integer ! Cut type identification type: array(range) of integer ! Cut constraint type ds: real end-declarations returned:=false ! OPTNODE: This node is not infeasible depth:=getparam("XPRS_NODEDEPTH") cnt:=getparam("XPRS_CALLBACKCOUNT_OPTNODE") if ((TOPONLY and depth<1) or (not TOPONLY and depth<=CUTDEPTH)) and (SEVERALROUNDS or cnt<=1) then ncut:=0 ! Get the solution values forall(t in TIMES, p in PRODUCTS) do solprod(p,t):=getsol(produce(p,t)) solsetup(p,t):=getsol(setup(p,t)) end-do ! Search for violated constraints forall(p in PRODUCTS,l in TIMES) do ds:=0 forall(t in 1..l) if (solprod(p,t) < D(p,t,l)*solsetup(p,t) + EPS) then ds += solprod(p,t) else ds += D(p,t,l)*solsetup(p,t) end-if ! Generate the violated inequality if ds < D(p,1,l) - EPS then cut(ncut):= sum(t in 1..l) if(solprod(p,t)<(D(p,t,l)*solsetup(p,t))+EPS, produce(p,t), D(p,t,l)*setup(p,t)) - D(p,1,l) cutid(ncut):= 1 type(ncut):= CT_GEQ ncut+=1 end-if end-do ! Add cuts to the problem if ncut>0 then addcuts(cutid, type, cut); if getparam("XPRS_VERBOSE")=true then writeln("Cuts added : ", ncut, " (depth ", depth, ", node ", getparam("XPRS_NODES"), ", obj. ", getparam("XPRS_LPOBJVAL"), ")") end-if end-if end-if end-function ! ****Optimizer settings for using the cut manager**** procedure tree_cut_gen setparam("XPRS_PRESOLVE", 0) ! Switch presolve off setparam("XPRS_ROOTPRESOLVE", 0) ! Switch B&B root presolve off setparam("XPRS_EXTRAROWS", 5000) ! Reserve extra rows in matrix setcallback(XPRS_CB_OPTNODE, ->cb_node) ! Set the optnode callback function end-procedure end-model | |||||||||||
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