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.


less than B₁	COST₁ each
≥ B₁ and < B₂	COST₂ each
≥ B₂ and < B₃	COST₃ each

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 b_i = 1

x₁ ≤ B₁· b₁

B_i-1· b_i ≤ x_i ≤ B_i· b_i 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.

∑_i b_i = 1
x₁ ≤ B₁· b₁
B_i-1· b_i ≤ x_i ≤ B_i· b_i for i=2,3

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

CBP_i = CBP_i-1+C_i· (B_i-B_i-1) for i=1,2,3
Constraint formulation:

∑_iw_i=1

TotalCost = ∑_i CBP_i· w_i

x = ∑_i B_i· w_i

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)
```

CBP₀=0
CBP_i = CBP_i-1+C_i· (B_i-B_i-1) for i=1,2,3

∑_iw_i=1
TotalCost = ∑_i CBP_i· w_i
x = ∑_i B_i· w_i

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:

(B_i-B_i-1)\cdot b_i+1 ≤ x_i ≤ (B_i-B_i-1)\cdot b_i for i=1,2

x₃ ≤ (B₃-B₂)\cdot b₃

b₁ ≥ b₂ ≥ b₃

(B_i-B_i-1)\cdot b_i+1 ≤ x_i ≤ (B_i-B_i-1)\cdot b_i for i=1,2
x₃ ≤ (B₃-B₂)\cdot b₃
b₁ ≥ b₂ ≥ b₃

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