(!*******************************************************
Mosel Example Problems
======================
file lexgoalprog.mos
````````````````````
An example of lexicographic goal programming using the
Xpress multi-objective functionality
Problem description:
A company produces two electrical products, A and B. Both require
two stages of production: wiring and assembly.
The production plan must meet several goals:
1. A profit of $200
2. A contractual requirement of 40 units of product B
3. To fully utilize the available wiring department hours
4. To avoid overtime in the assembly department
(c) 2022 Fair Isaac Corporation
author: S. Heipcke, June 2022
*******************************************************!)
model "lexGP"
uses "mmxprs"
public declarations
! Decision variables for the number of products to make of each type
produceA,produceB: mpvar
! Deviational variables:
! There is a penalty for both under- and over-utilizing each department
surfeit_wiring, deficit_wiring: mpvar
surfeit_assembly, deficit_assembly: mpvar
! There is no penalty for surfeit in profit or in production of product B,
! only for deficits
deficit_profit, deficit_productB: mpvar
Goals: list of linctr or mpvar ! or also: list of any
end-declarations
produceA is_integer; produceB is_integer
! **** Production constraints:
! Meet or exceed profit goal of $200
! Profit for products A and B are $7 and $6 respectively
Profit:= 7 * produceA + 6 * produceB
Profit + deficit_profit >= 200
! Meet or exceed production goal for product B
produceB + deficit_productB >= 40
! Utilize wiring department:
! Products A and B require 2 and 3 hours of wiring, 120 hours are available
2 * produceA + 3 * produceB - surfeit_wiring + deficit_wiring = 120
! Utilize assembly department:
! Products A and B require 6 and 5 hours of assembly, 300 hours are available
6 * produceA + 5 * produceB - surfeit_assembly + deficit_assembly = 300
! Objective configuration
Goals:=[deficit_profit, deficit_productB, surfeit_wiring + deficit_wiring,
surfeit_assembly + deficit_assembly]
(! Cfg:= [objconfig("priority=4 abstol=0 reltol=0"),
objconfig("priority=3 abstol=0 reltol=0"),
objconfig("priority=2 abstol=0 reltol=0"),
objconfig("priority=1 abstol=0 reltol=0")] !)
! Same as:
forall(i in 1..4) Cfg+=[objconfig(5-i,1,0,0)]
! Uncomment this line to try out the effect of inversing priority order:
! forall(i in 1..4) Cfg(i).priority:=i
! setparam("XPRS_VERBOSE", true)
! Minimize deviations, in priority order
minimize(Goals,Cfg)
! **** Solution reporting
declarations
SolStat,SolvStat: array(integer) of string ! Status messages
end-declarations
SolvStat:: ([XPRS_SOLVESTATUS_UNSTARTED, XPRS_SOLVESTATUS_STOPPED,
XPRS_SOLVESTATUS_FAILED, XPRS_SOLVESTATUS_COMPLETED])
["The solve has not been started.", "Optimization has been interrupted.",
"Optimization has run into a nonrecoverable problem and failed.",
"Search completed."]
SolStat:: ([XPRS_SOLSTATUS_NOTFOUND, XPRS_SOLSTATUS_OPTIMAL,
XPRS_SOLSTATUS_FEASIBLE, XPRS_SOLSTATUS_INFEASIBLE, XPRS_SOLSTATUS_UNBOUNDED])
["No solution available.", "An optimal solution has been found.",
"A solution that is not proven optimal is found.",
"No solution exists.", "The problem is unbounded, if feasible."]
writeln("Problem solve status: ", SolvStat(getparam("XPRS_SOLVESTATUS")),
" Solution status: ", SolStat(getparam("XPRS_SOLSTATUS")),
" Objectives solved: ", getparam("XPRS_SOLVEDOBJS"))
if getparam("XPRS_SOLVESTATUS")=XPRS_SOLVESTATUS_COMPLETED and
getparam("XPRS_SOLSTATUS")=XPRS_SOLSTATUS_OPTIMAL and
getparam("XPRS_SOLVEDOBJS")=4 then
writeln('Production plan:')
writeln('Product A: ', produceA.sol, ' units')
writeln('Product B: ', produceB.sol, ' units')
writeln('Profit: $', Profit.sol)
if deficit_profit.sol > 0:
writeln('Profit goal missed by $', deficit_profit.sol)
if deficit_profit.sol > 0:
writeln('Profit goal missed by $', deficit_profit.sol)
if deficit_productB.sol > 0:
writeln('Contractual goal for product B missed by ', deficit_productB.sol, ' units')
if surfeit_wiring.sol > 0:
writeln('Unused wiring department hours: ', surfeit_wiring.sol)
if deficit_wiring.sol > 0:
writeln('Wiring department overtime: ', deficit_wiring.sol)
if surfeit_assembly.sol > 0:
writeln('Unused assembly department hours: ', surfeit_assembly.sol)
if deficit_assembly.sol > 0:
writeln('Assembly department overtime: ', deficit_assembly.sol)
else
writeln('Problem could not be solved')
end-if
end-model
|
(!*******************************************************
Mosel Example Problems
======================
file markowitzmo.mos
````````````````````
Markowitz portfolio optimization
A multi-objective quadratic programming example
-- Display of the optimal frontier as SVG graph --
In Markowitz portfolio optimization there are two objectives:
to maximize reward while minimizing risk (i.e. variance).
This example plots several points on the optimal frontier using
a blended multi-objective approach, and shows that a point
computed using a lexicographic approach also lies on this frontier.
(c) 2022 Fair Isaac Corporation
author: S. Heipcke, June 2022
*******************************************************!)
model "markowitzmo (SVG)"
uses "mmxnlp", "mmsvg"
declarations
STOCKS = 1..5 ! Set of 5 stocks
RET: array(STOCKS) of real ! Historical mean return on investment
COV: array(STOCKS,STOCKS) of real ! Historical covariances of the stocks
x: array(STOCKS) of mpvar ! Percentage of capital to invest in stocks
Goals: list of linctr or nlctr ! Objective functions
ObjCfg: list of objconfig ! Configuration of objectives
end-declarations
RET::[0.31, 0.87, 0.31, 0.66, 0.24]
COV::[0.32, 0.70, 0.19, 0.52, 0.16,
0.70, 4.35, -0.48, -0.06, -0.03,
0.19, -0.48, 0.98, 1.10, 0.10,
0.52, -0.6, 1.10, 2.48, 0.37,
0.16, -0.3, 0.10, 0.37, 0.31]
! Constraints:
! Must invest 100% of capital
sum(i in STOCKS) x(i) = 1
! Objectives:
! Total expected return
Return:= sum(i in STOCKS) RET(i)*x(i)
Variance:= sum(i,j in STOCKS) x(i)*x(j)*COV(i,j)
! List of objectives
Goals:=[Return, Variance]
! setparam("XPRS_VERBOSE", true)
! Vary the objective weights to explore the optimal frontier
declarations
POINTS: range
SOLMEAN: array(POINTS) of real
SOLVAR: array(POINTS) of real
end-declarations
SCALEX:=10
forall(p in 0..20, w=p/20.0+if(p=0,0.0001,if(p=20,-0.0001,0))) do
! Negative weight to minimize variance
! maximize(Goals,[objconfig(0,w),objconfig(0,w-1)])
! Same as:
maximize(Goals,[objconfig("weight",w),objconfig("weight",w-1)])
if getprobstat=XPRS_OPT then
SOLMEAN(p):=Return.sol
SOLVAR(p):=Variance.sol
writeln("Solution for w=", w, ": ", SOLMEAN(p), " / ", SOLVAR(p))
else
writeln("No solution for w=", w)
end-if
end-do
! Now we will maximize profit alone, and then minimize variance while not
! sacrificing more than 10% of the maximum profit
ObjCfg:=[objconfig("priority=1 weight=1 reltol=0.1"),
objconfig("priority=0 weight=-1")]
! Same as:
! ObjCfg:=[objconfig(1,1,0.001,0.1), objconfig(0,-1)]
! or:
! ObjCfg:=[objconfig(1,1,0.001,0.1), objconfig("weight",-1)]
!)
maximize(Goals,ObjCfg)
m0:=Return.sol
v0:=Variance.sol
writeln("Solution for config=", ObjCfg, ":\n", strfmt("",20), m0, " / ", v0)
! Plot the optimal frontier and mark the individual point that we calculated
svgaddgroup("GrS", "Variance vs mean return", 'grey')
svgsetstyle(SVG_STROKEWIDTH, 0.25)
svgaddgroup("GrM", "Max profit, then variance", 'darkred')
forall(p in POINTS) svgaddpoint("GrS", SOLMEAN(p)*SCALEX, SOLVAR(p))
svgaddline("GrS", sum(p in POINTS) [SOLMEAN(p)*SCALEX, SOLVAR(p)])
svgaddpoint("GrM", m0*SCALEX,v0)
svgaddtext("GrS",0.5,8, 'Return on investment vs variance')
svgsetstyle(svggetlastobj, SVG_COLOR, SVG_BLACK)
svgsetstyle(svggetlastobj, SVG_FONTSIZE, "5px")
svgsetgraphscale(10)
svgsetgraphviewbox(0,0,9,9)
svgsetgraphpointsize(0.5)
svgsetgraphlabels('Expected return', 'Variance')
! Optionally save graphic to file
svgsave("markowitz.svg")
! Display the graph and wait for window to be closed by the user
svgrefresh
svgwaitclose("Close browser window to terminate model execution.", 1)
end-model
|
(!*******************************************************
Mosel Example Problems
======================
file multiobjknapsack.mos
`````````````````````````
Multi-objective knapsack example
(c) 2022 Fair Isaac Corporation
author: S. Heipcke, June 2022
*******************************************************!)
model "multiobjknapsack"
uses "random", "mmxprs"
parameters
N = 15 ! Number of items
NOBJ = 2 ! Number of goals
MAXW = 10 ! Maximum weight that can be carried
end-parameters
declarations
R=1..N ! Set of items
WEIGHT: array(R) of real ! Weight per item
OBJS = 1..NOBJ ! Set of objectives
VALUE: array(OBJS,R) of real ! Value of items for each objective
take: array(R) of mpvar ! Whether to select an item
TotValue: array(OBJS) of linctr ! Objective functions (goals)
ObjCfg: array(OBJS) of objconfig ! Configuration of objectives
end-declarations
! Generate random weights and two random value metrics
setmtrandseed(123)
forall(i in R) WEIGHT(i):= mtrand_int(1, 4)
forall(o in OBJS, i in R) VALUE(o,i):= mtrand_int(1, 6)
! Decision variables for each item
forall(i in R) take(i) is_binary
! Total weight cannot exceed maximum weight
WLimit:= sum(i in R) WEIGHT(i)*take(i) <= MAXW
! Define the objectives
forall(o in OBJS) TotValue(o):= sum(i in R) VALUE(o,i)*take(i)
! Configuration of multiple objectives:
! * Distinct priority values: solved as pre-emptive multi-obj problem where
! a higher value indicates higher priority, that is, to be treated earlier
! * Equal priority values: solved as Archimedian multi-obj problem
ObjCfg(1).priority:=2
ObjCfg(2).priority:=1
! Solve the problem
setparam("XPRS_VERBOSE", true) ! Enable solver logging
setparam("XPRS_MULTIOBJLOG", 1) ! Configure multi-objective logging:
! 0=none, 1=summary, 2=detailed
maximise(TotValue,ObjCfg)
! Solution reporting
! if getprobstat=XPRS_OPT and getparam("XPRS_SOLVEDOBJS")=NOBJ then
if getparam("XPRS_SOLVESTATUS")=XPRS_SOLVESTATUS_COMPLETED and
getparam("XPRS_SOLSTATUS")=XPRS_SOLSTATUS_OPTIMAL then
writeln('Problem was solved to optimality.')
write('Items selected:')
forall(i in R | take(i).sol=1) write(i, ' ')
writeln("\nTotal weight:", WLimit.act)
writeln('First objective:', TotValue(1).sol)
writeln('Second objective:', TotValue(2).act)
elif getprobstat=XPRS_INF and getparam("XPRS_SOLVEDOBJS")=1 then
writeln('Failed to solve first objective.')
else
writeln('Solved first objective but failed to solve second objective.')
end-if
end-model
|
