1 Mathematical Programming Integer Programming. 2 Common Types of IP’s and IP Constraints Two...

1

Mathematical ProgrammingMathematical Programming

Integer Programming

2

Common Types of IP’s and Common Types of IP’s and IP ConstraintsIP Constraints

• Two common types of IP’s– #1: Capital budgeting– #2: Set covering

• Four common IP constraints– #3: Binary Variables & Logical Conditions– #4: Fixed costs– #5: Minimum order or purchase size– #6: Quantity discounts

3

#1: Capital Budgeting#1: Capital Budgeting• Make a set of “go/no-go” decisions concerning a

pool of potential projects• Projects consume common resources

– Typically place capital demands in different time periods

– May require other resources such as staff time or expertise, possibly in different periods

– Often have logical linking constraints (can only do project #4 if you do project #6, etc.)

• When there is just one resource constraint it is known as a “knapsack problem”

4

#2: Set Covering#2: Set Covering• Choose least cost set of “facilities” that serves

(covers) every customer adequately.• Example: Company wants to locate fewest

number of service centers such that all customers in region are within X miles of a service center.

• Example: A city has eight districts; each can house an ambulance. How many ambulances are needed and where should they be placed to make sure all districts are within 3 minutes of an ambulance?

5

Set Covering (cont.)Set Covering (cont.)

• Decision variables– Xi = 1 if ambulance placed in district i– Xi = 0 if no ambulance placed in district i

• Objective– minimize number of ambulances– i.e., minimize number of Xi’s that equal 1

• 8 Constraints, 1 for each neighborhood– for every neighborhood, at least one of the

neighborhoods within a 3 minute drive must have its Xi = 1

6

Data for Ambulance Problem:Data for Ambulance Problem:Travel Time Between DistrictsTravel Time Between Districts

#1 #2 #3 #4 #5 #6 #7 #8

#1 0 4 4 6 8 8 8 10

#2 3 0 3 4 7 6 9 9

#3 3 3 0 2 5 3 5 9

#4 6 3 2 0 3 2 5 5

#5 6 8 4 3 0 5 2 4

#6 9 6 4 2 4 0 3 2#7 8 12 6 5 2 3 0 2#8 10 9 7 4 4 2 2 0

Fro

m N

eigh

borh

ood

To Neighborhood

7

Ambulance Problem:Ambulance Problem:Mathematical FormulationMathematical Formulation

Min X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8

s.t. X1 + X2 + X3 >= 1

X2 + X3 + X4 >= 1

X2 + X3 + X4 >= 1

X3 + X4 + X5 + X6 >= 1

X4 + X5 + X7 >= 1

X3 + X4 + X6 + X7 + X8 >= 1

X5 + X6 + X7 + X8 >= 1

X6 + X7 + X8 >= 1 and all Xi binary

8

#3: Binary Variables and #3: Binary Variables and Logical ConditionsLogical Conditions

• Suppose could place an ambulance in only one of the first three districts:– X1 + X2 + X3 <= 1

• Suppose needed an ambulance in at least one of the first three districts– X1 + X2 + X3 >= 1

• Exactly one– X1 + X2 + X3 = 1

9

Binary Variables and Binary Variables and Logical Conditions (cont.)Logical Conditions (cont.)

• Suppose could only do project #2 if project #1 is funded. I.e., cannot have X2 = 1 if X1 = 0. Then add constraint:– X1 - X2 >= 0

• Suppose could only do project #2 if either project #1 or project #3 is funded.– X1 - X2 + X3 >= 0

• Etc.

10

Can Apply This Idea Can Apply This Idea to Constraintsto Constraints

• If must meet at least 1 of 2 constraints:– X1 + 2X2 + 3X3 <= 15 or– 2X1 + X2 + X3 <= 10

• Create binary variables Yi & constraints– M Y1 >= X1 + 2X2 + 3X3 - 15– M Y2 >= 2X1 + X2 + X3 - 10– where M is any very, very large number

• If constraint i violated, then Yi must be 1• Ensure both constraints are not violated

– Y1 + Y2 <= 1

11

Extend to GeneralExtend to GeneralSets of ConstraintsSets of Constraints

• Suppose must meet K of L constraints:– gi(X1, X2, X3, ..., Xn) <= 0, i = 1, 2, 3, ..., L

– Create binary variables Yi such that Yi = 1 if constraint i is violated.

– M Yi >= gi(X1, X2, X3, ..., Xn) for each i

– To ensure that no more than L-K constraints are violated, add constraint

– Y1 + Y2 + Y3 + ... + YL <= (L - K)

12

#4: Fixed Costs:#4: Fixed Costs:Ordering ShirtsOrdering Shirts

• Suppose ordering shirts from LL Bean costs $20/shirt plus a $5 shipping charge (regardless of the number of shirts ordered) and ordering shirts from Land’s End costs $19/shirt plus a $10 shipping charge.

• How express total cost as a function of– X1 = # of shirts ordered from LL Bean

– X2 = # of shirts ordered from Land’s End

13

Algebraic Formulation of Total Algebraic Formulation of Total Cost of Ordering ShirtsCost of Ordering Shirts

• Create two new binary variables– Y1 = 1 if order shirts from Bean; else 0– Y2 = 1 if order shirts from L.E.; else 0

• Add four constraints– M Yi >= Xi for i = 1 and 2– Yi <= M Xi for i = 1 and 2– M is some very large number (“Big M”)– Note: Yi = 1 if Xi > 0 and Yi = 0 if Xi = 0.

• Total cost of ordering shirts is Min 5 Y1 + 20 X1 + 10 Y2 + 19 X2

14

Fixed Costs:Fixed Costs:General FormulationGeneral Formulation

• Suppose incur a fixed cost FCi > 0 if a decision variable is greater than 0 (in addition to variable cost (ci) per unit)– Create a binary variable Yi

– Add constraints: M Yi >= Xi for each i

– Ensures Yi = 1 if Xi > 0

– Objective function becomes

Min FC1 Y1 + c1 X1 + FC2 Y2 + c2 X2 + ...

– Minimization ensures Yi = 0 if Xi = 0.

15

#5: Minimum Order #5: Minimum Order or Purchase Size or Purchase Size

• Suppose a book store has to order at least 20 copies of a title if it orders any.

• If X = # of copies ordered, how can this constraint be written? – Create binary variable Y such that Y = 1 if X > 0

and Y = 0 if X = 0.– Add constraints: – M Y >= X and (so if X > 0, Y = 1)– X >= 20 Y (if Y = 1, X >= 20)

16

General Formulation of Min. General Formulation of Min.

Order or Purchase SizeOrder or Purchase Size

• Suppose that if Xi > 0, it must be at least Li (I.e., cannot have 0 < Xi < Li)

– Create a binary variable Yi such that Yi = 1 if Xi > 0 and Yi = 0 if Xi = 0.

– Add constraints:

– M Yi >= Xi and (so if Xi > 0, Yi = 1)

– Xi >= Li Yi for each i (if Yi = 1, Xi >= Li)

17

#6: Quantity Discounts #6: Quantity Discounts

• Suppose first Li units cost Ci1; additional units can be purchased at Ci2 < Ci1.

– Create a binary variable Yi such that Yi = 1 if Xi > Li. Divide Xi into Xi1 and Xi2.

– Add constraints:

Xi2 <= M Yi and (if Xi2 > 0, Yi = 1)

Xi1 >= Li Yi for each i (if Yi = 1, Xi1 >= Li)

Obj fn: Min C11 X11 + C12 X12 + C21 X21 +

C22 X22 + C31 X31 + C32 X32 + ...

18

Quantity DiscountQuantity Discount Example Example (One Item; Multiple Breaks)(One Item; Multiple Breaks)

• Stock brokerage fees are– $27.50 fixed cost per order + – 0.0092 * $ amount up to $10K– 0.0066 * $ amount between $10K - $25K +– 0.0045 * $ amount over $25K.

• How express the brokerage fee as a function of the value of the order? Let– X1 = order amount up to $10K– X2 = order amount from $10K - $25K– X3 = order amount over $25K– Y1, Y2, Y3 are binary

19

Quantity Discount Quantity Discount Example SolutionExample Solution

• ConstraintsX3 <= M Y3 (if X3 > 0, Y3 = 1)

X2 >= 15,000 Y3 (if Y3 = 1, X2 >= 15,000)

X2 <= 15,000 Y2 (if X2 > 0, Y2 = 1)

X1 >= 10,000 Y2 (if Y2 > 0, X1 >= 10,000)

X1 <= 10,000 Y1 (if X1 > 0, Y1 = 1)

• Purchase – amount: X1 + X2 + X3

– cost: 27.50 Y1 + 0.0092 X1 + 0.0066 X2

+ 0.0045 X3

20

BreakBreak

21

Mathematical ProgrammingMathematical Programming

Nonlinear Programming

22

MaximinMaximin

23

MaximinMaximin

24

MaximinMaximin

25

MaximinMaximin

26

Minimale AbweichungMinimale Abweichung

27

Minimale AbweichungMinimale Abweichung

28

Minimale AbweichungMinimale Abweichung

29

BreakBreak