Price breaks

All items discount: when buying a certain number of items we get discounts on all items that we buy if the quantity we buy lies in certain price bands.

Formulation with binary variables or Special Ordered Sets of type 1 (SOS1):

• Define binary variables b_i (i=1,2,3), where b_i is 1 if we pay a unit cost of COST_i.
• Real decision variables xp_i represent the number of items bought at price COST_i.
• The quantity bought is given by x= ∑_ixp_i , with a total price of ∑_iCOST_i·xp_i
• MIP formulation :

  ∑i bi = 1

  xp1 ≤ B1· b1

  Bi-1· bi ≤ xpi ≤ Bi· bi for i=2,3

  where the variables b_i are either defined as binaries, or they form a Special Ordered Set of type 1 (SOS1), where the order is given by the values of the breakpoints B_i.

Formulation as piecewise linear expression:

• Specification as list of piecewise linear segments with associated intervals:

  ⋃i=1,2,3 [Bi-1,Bi[ : COSTi·x

  Implementation with Mosel:

  TotalCost:= pwlin(union(i in 1..3) [pws(B(i-1), COST(i)*x)])

• Specification as list of points:

  ⋃i=1,2,3 [Bi-1,COSTi·Bi-1,Bi,COSTi·Bi]

  Implementation with Mosel:

  TotalCost:= pwlin(x, union(i in 1..3) [B(i-1), COST(i)*B(i-1), B(i), COST(i)*B(i)])

Incremental pricebreaks: when buying a certain number of items we get discounts incrementally. The unit cost for items between 0 and B₁ is COST₁, items between B₁ and B₂ cost COST₂ each, etc.

Formulation with Special Ordered Sets of type 2 (SOS2):

• Associate real valued decision variables w_i (i=0,1,2,3) with the quantity break points B₀ = 0, B₁, B₂ and B₃.
• Cost break points CBP_i (=total cost of buying quantity B_i):

  CBP0=0

  CBPi = CBPi-1+COSTi· (Bi-Bi-1) for i=1,2,3

• Constraint formulation:

  ∑iwi=1

  TotalCost = ∑i CBPi· wi

  x = ∑i Bi· wi

  where the w_i form a SOS2 with reference row coefficients given by the coefficients in the definition of the total amount x.
  For a solution to be valid, at most two of the w_i can be non-zero, and if there are two non-zero they must be contiguous, thus defining one of the line segments.
• Implementation with Mosel (is_sos2 cannot be used here due to the 0-valued coefficient of w₀):

  Defx := x = sum(i in 1..3) B(i)*w(i)
  makesos2(My_Set, union(i in 0..3) {w(i)}, Defx)

Formulation using binaries:

• Define binary variables b_i (i=1,2,3), where b_i is 1 if we have bought any items at a unit cost of COST_i.
• Real decision variables xp_i (i=1,..3) for the number of items bought at price COST_i.
• Total amount bought: x = ∑_i xp_i
• Constraint formulation:

  (Bi-Bi-1)\cdot bi+1 ≤ xpi ≤ (Bi-Bi-1)\cdot bi for i=1,2

  xp3 ≤ (B3-B2)\cdot b3

  b1 ≥ b2 ≥ b3

Formulation as piecewise linear expression:

• Specification as list of slopes with associated intervals:

  ⋃i=1,2,3 [Bi-1,Bi[ : COSTi

  Implementation with Mosel (only points of slope changes are specified, start value is 0):

  TotalCost:= pwlin(x, union(i in 1..2) [B(i)], union(i in 1..3) [COST(i)])

• Specification as list of piecewise linear segments with associated intervals:

  ⋃i=1,2,3 [Bi-1,Bi[ : CBPi-1+COSTi·(x-Bi-1)

  Implementation with Mosel:

  TotalCost:= pwlin(union(i in 1..3) [pws(B(i), CBP(i,1)+COST(i)*(x-B(i-1)))])

• Specification as list of points:

  ⋃i=0,1,2,3 [Bi,CBPi]

  Implementation with Mosel:

  TotalCost:= pwlin(x, union(i in 0..3) [B(i), CBP(i)])