Suppliers frequently offer quantity discounts to their customers to entice large orders. Consider the following incremental quantity discount schedule.

Units ordered ................ Price per unit
1−1,000 .............................. $50.00
1,001−5,000 ....................... $50.00 for the first 1000 units and $45.00 for units ordered between 1,001 and 5,000
≥ 5,000 ................................ $50.00 for the first 1000 units, $45.00 for the next 4,000 units,
                                                and $40.00 for any units ordered beyond 5,000

With the help of binary variables, an incremental quantity discount schedule can be incorporated into a mixed integer program that determines order size. Let Q be a decision variable that represents the total amount purchased. Let qi be a decision variable that represents the total amount purchased within interval i of the price schedule. Thus, Q will equal q₁ + q₂ + q₃. Also, let Yi be a binary variable that equals 1 if Q lies in any higher interval than interval i of the price schedule (i.e., if Q would yield a lower marginal price than the price in interval i), and 0 otherwise. For example, if 5,200 units are ordered, Q = 5,200, q₁ = 1,000, q₂ = 4,000, q₃ = 200, Y₁ = 1, and Y₂ = 1. On the other hand, if only 3,200 units are purchased, Q = 3,200, q₁ = 1,000, q₂ = 2,200, q₃ = 0, Y₁ = 1, and Y₂ = 0.

Using these definitions, the purchasing cost in the objective function will then equal 50q₁ + 45q₂ + 40q₃. One of the new constraints added to the program will be Q = q₁ + q₂ + q₃. In addition to nonnegativity and binary constraints, what additional four constraints need to be added to ensure that the correct price is charged for the amount purchased? Define M as a very large number, and use it in one of the new constraints.