Description

Minimizing Total Average Cycle Stock.

Large Model of Type : MIP

Category : GAMS Model library

Main file : stockcc.gms

$title Minimizing Total Average Cycle Stock (STOCKCC,SEQ=225)

$onText
Minimizing Total Average Cycle Stock.


Silver, E A, and Moon, I, A Fast Heuristic for Minimizing Total
Average Cycle Stock Subject to Practical Constraints. Journal
of the Operational Research Society 50 (1999), 789-796.

Keywords: mixed integer linear programming, cycle stock, inventory
$offText

Set
   nn 'items'             / n1*n48 /
   mm 'reorder intervals' / i1*i9  /;

Scalar N 'max total number of replenishments' / 100 /;

Parameter
   Y(mm)  'possible number of orders'
          /  i1   3, i2   6, i3   9, i4  12, i5  18
             i6  36, i7  52, i8  78, i9 156         /
   Dv(nn) 'demand rate times unit cost of item nn'
          /  n1     20.04,  n2     21.72,  n3      37.92,  n4      54.12
             n5     61.80,  n6     81.24,  n7      94.20,  n8     119.40
             n9    171.60, n10    208.80, n11     415.27, n12     470.23
            n13   1212   , n14   1393.2 , n15    1496.4 , n16    1600
            n17   1702.4 , n18   1714.5 , n19    1870.5 , n20    1977.8
            n21   2647.12, n22   3143.82, n23    4173   , n24    4347.78
            n25   4917   , n26   5048.3 , n27    5397.2 , n28    6692.4
            n29   6837.6 , n30   8003.1 , n31    8449.5 , n32    9152
            n33  13236.3 , n34  13970   , n35   15327.6 , n36   16368
            n37  19765   , n38  20470.3 , n39   23182.2 , n40   25026
            n41  31675.6 , n42  56734.2 , n43   69040.4 , n44   75192
            n45  97066.5 , n46  99803.2 , n47  105984   , n48  106465    /;

Variable
   x(nn)    'number of orders per unit time'
   z(nn,mm) 'discrete orders choices'
   obj      'objective variable';

Binary Variable z;

Equation
   defobj
   capacity
   defx(nn)
   defsos(nn);

defobj..     obj =e= sum(nn, 1.5*Dv(nn)/x(nn));

capacity..   sum(nn, x(nn)) =l= 3*N;

defx(nn)..   sum(mm, z(nn,mm)*Y(mm)) =e= x(nn);

defsos(nn).. sum(mm, z(nn,mm)) =e= 1;

x.lo(nn) = Y('i1');
x.up(nn) = Y('i9');

Model stock / all /;
* solve stock minimizing obj using minlp;

$onText
This is the alternative formulation that becomes an easy to solve MIP
model.

First we note that defsos means that exactly one nn index value is
matched with each mm index.

Second, defx says that x(nn) must be equal to Y(mm) corresponding
to this match.

Third, the objective term 1.5*Dv(nn)/x(nn) is therefore equal to
1.5*Dev(nn)/Y(mm) for the selected (nn,mm) match. Since the match
is defined by z, the objective can be redefined using z with the
coefficient 1.5*Dv(nn)/Y(mm), and x is no longer needed.
$offText

Parameter CostZ(nn,mm) 'cost for item nn with order schedule mm';
CostZ(nn,mm) = 1.5*Dv(nn)/Y(mm);

Equation defobjmip;

defobjmip.. obj =e= sum((nn,mm), Costz(nn,mm)*z(nn,mm));

Model stockmip / defobjmip, capacity, defx, defsos /;

solve stockmip miniminzing obj using mip;