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.

• 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 x_i represent the number of items bought at price COST_i.
• The quantity bought is given by x= ∑_ix_i , with a total price of ∑_iCOST_i·x_i
• MIP formulation :

  ∑i bi = 1

  x1 ≤ B1· b1

  Bi-1· bi ≤ xi ≤ 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.

Incremental pricebreaks: when buying a certain number of items we get discounts incrementally. The unit cost for items between 0 and B₁ is C₁, items between B₁ and B₂ cost C₂ 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+Ci· (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 x_i (i=1,..3) for the number of items bought at price COST_i.
• Total amount bought: x = ∑_i x_i
• Constraint formulation:

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

  x3 ≤ (B3-B2)\cdot b3

  b1 ≥ b2 ≥ b3