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Introductory examples

Problem name and type, featuresDifficulty
approx Approximation: Piecewise linear approximation **
SOS-2, Special Ordered Sets, piecewise linear approximation of a nonlinear function
burglar MIP modeling: Knapsack problem: 'Burglar' *
simple MIP model with binary variables, data input from text data file, array initialization, numerical indices, string indices, record data structure
chess LP modeling: Production planning: 'Chess' problem *
simple LP model, solution output, primal solution values, slack values, activity values, dual solution values
pricebrai All item discount pricing: Piecewise linear function ***
SOS-1, Special Ordered Sets, piecewise linear function, approximation of non-continuous function, step function
pricebrinc Incremental pricebreaks: Piecewise linear function ***
SOS-2, Special Ordered Sets, piecewise linear function, step function

Further explanation of this example: 'Applications of optimization with Xpress-MP', Introductory examples (Chapters 1 to 5) of the book 'Applications of optimization with Xpress-MP'[download all files]

Source Files

Data Files


   Mosel Example Problems

   file pricebrinc.mos
   Incremental pricebreaks formulated with SOS2   

   (c) 2008 Fair Isaac Corporation
       author: S. Heipcke, Sep. 2006

model "Incremental pricebreaks (SOS2)"
 uses "mmxprs"
  NB = 3                                ! Number of price bands
  BREAKS = 0..NB
  COST: array(1..NB) of real            ! Cost per unit within price bands
  w: array(BREAKS) of mpvar             ! Weight variables
  x: mpvar                              ! Total quantity bought
  B,CBP: array(BREAKS) of real          ! Break points, cost break points
 DEM:= 150                              ! Demand
 B::  [0, 50, 120, 200]
 COST:: [0.8, 0.5, 0.3]
 CBP(0):= 0
 forall(i in 1..NB) CBP(i):= CBP(i-1) + COST(i) * (B(i)-B(i-1))

! Objective: total price
 TotalCost:= sum(i in BREAKS) CBP(i)*w(i)  

! Meet the demand
 x = DEM 

! Definition of x
 Defx:= x = sum(i in BREAKS) B(i)*w(i)

! Weights sum up to 1
 sum(i in BREAKS) w(i) = 1

! Definition of SOS2
! (we cannot use 'is_sos2' since there is a 0-valued coefficient)
 makesos2(union(i in BREAKS) {w(i)}, Defx)

! Solve the problem

! Solution printing
 writeln("Objective: ", getobjval, 
         " (price per unit: ", getobjval/DEM, ")")
 forall(i in BREAKS)
  writeln("w(", i, "): ", getsol(w(i)), " (price per unit: ", 
   if(i>0, CBP(i)/B(i), CBP(i)), ")")


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