Elements of the blending model

The model is described in terms of a number of constants specifying the various types of metals, raw materials, scrap, and ingots. In the instance data shown in Instance data for the blending problem (blending.dat), there are three metals, two raw materials, two kinds of scrap, and one kind of ingot. The model also defines ranges for each of the components. It then defines the cost of the various components in costMetal, costRaw, costScrap, costIngo. In the instance data, for example, the second raw material has a cost of 5. The data items Low and Up specify the production constraints and give lower and upper bounds on the quantity of each sort of metal in the alloy. For example, in the instance data, between 30% and 40% of the alloy must be the second metal. The next data items, percRaw, percScrap, and percIngo, specify the percentage of each metal in the sources. In Instance data for the blending problem (blending.dat), the second type of scrap contains 1% of the first metal, none of the second metal, and 70% of the third metal. Finally, the data alloy specifies the amount of alloy to be produced.

Instance data for the blending problem (blending.dat)

NbMetals = 3;
NbRaw = 2;
NbScrap = 2;
NbIngo = 1;

CostMetal = [22, 10, 13];
CostRaw = [6, 5];
CostScrap = [ 7, 8];
CostIngo = [ 9 ];
Low = [0.05, 0.30, 0.60];
Up = [0.10, 0.40, 0.80];
PercRaw = [ [ 0.20, 0.01 ], [ 0.05, 0 ], [ 0.05, 0.30 ] ];
PercScrap = [ [ 0 , 0.01 ], [ 0.60, 0 ], [ 0.40, 0.70 ] ];
PercIngo = [ [ 0.10 ], [ 0.45 ], [ 0.45 ] ];
Alloy  = 71;

The decision variables specify how much of each source is used in the alloy: the array p specifies the quantities of pure metals, array r specifies the quantities of raw materials, array s specifies the quantities of scrap, array i specifies the number of ingots. All variables are of type float except number of ingots, which are integers. The problem is thus a mixed integer-linear program. The instruction

dvar float m[j in Metals] in low[j] * alloy .. up[j] * alloy; 

is particularly interesting, since it shows how to specify the range of decision variables in a generic fashion. More precisely, the range of variables m[j] is given by the expression:

low[j] * alloy .. up[j] * alloy

Note also that the model uses the variables in array m as intermediary variables to represent the quantity of each metal produced.

There are two types of constraints in this problem.

• The forall constraint

  subject to {
    forall( j in Metals )
      ct1:
        m[j] == 
        p[j] + 
        sum( k in Raws )   PercRaw[j][k] * r[k] +
        sum( k in Scraps ) PercScrap[j][k] * s[k] +
        sum( k in Ingos )  PercIngo[j][k] * i[k];
      ct2:  
        sum( j in Metals ) m[j] == Alloy;
  }
  

  makes sure that the right amounts of metal are produced. The amount m[j] of metal j must be equal to the amount of pure metal p[j] added to the quantity of metal j contained in the raw materials, the scrap, and the ingots. The correct amount of metals are computed using the percentage of metals contained in the sources.

• The sum constraint

  
  sum(j in Metals) m[j] == alloy;
  

  makes sure that the various metals produced give the correct amount of alloy. The objective function in this model is rather simple. It consists of computing the price of each source from its unit price (e.g., costMetal) and the amount produced (e.g., p[j]).

Final Solution with objective 653.6100:
  p = [0.0467 0.0000 0.0000];
  r = [0.0000 0.0000];
  s = [17.4167 30.3333];
  i = [32];
  m = [3.5500 24.8500 42.6000];