INTEGER AND MIXED- INTEGER OPTIMIZATION

Transcription

1 34 INTEGER AND MIXED- INTEGER OPTIMIZATION Conceptually, it s the same as linear optimization: Continue formulating everything using linear functions: Minimize/Maximize a linear function Subject to linear equality and inequality constraints New: some or all variables may be required to be integers. This is a real game changer as far as the expressivity of the models is concerned. Examples from the handout. No free lunch: these problems are way harder to solve than linear optimization problems.

2 Forcing constraints, consequences. If binary choice x has been selected, then y must also be selected. Same as: x = 1 implies y = 1. (If x = 0, it doesn t affect y.) With a linear inequality: x y. Algebraic proof? Geometric proof? Example: facility location problem: We are given n possible facility locations and a list of m clients to be served from these locations. There is a fixed cost c j of opening a facility at location j, while the cost of serving client i from location j is d ij. Select which facilities to open and assign clients to facilities, minimizing the total cost. Tricky part: the implicit constraint that we can only serve a client from a location if we open a facility at that location. 35

3 Linearization Some problems naturally give rise to nonlinear expressions involving binary variables. Nonlinearity is still not allowed in integer linear programs, so we have to linearize. E.g., how do we linearize the constraint z = xy (x, y, z {0,1})? Very common! Interpretation: z must also be chosen if and only if x and y are both chosen 36

4 Linearization E.g., how do we linearize the constraint z = xy (x, y, z {0,1})? Very common! Interpretation: z must also be chosen if and only if x and y are both chosen x + y z 1 y + z 0 x + z 0 How do we come up with this stuff? Just suss it out (not always practical), or Draw the feasible points, describe the bounding polytope (dim<=3), or Use the variable forcing technique from the previous slide. 37

5 Linearization, a note of caution Linearize the constraint z = xy (x, y, z {0,1})? x + y z 1 y + z 0 x + z 0 The above is not the only solution, in fact, we could have done this with one fewer inequalities: 2z x + y z + 1. The latter solution is worse (in a well-defined sense that we will see very soon), even though it uses fewer inequalities! Shows (a little) the dangers of just trying to eyeball constraints. 38

6 : COVERING, PARTITION, PACKING 39 Definition by picture: given cover partition packing a T i x 1 i a T i x = 1 i a T i x 1 i a ij = 1 if item i is in set j, 0 otherwise x j = 1 if set j is selected, 0 otherwise

7 COVER, PARTITION, PACKING The names don t matter, but you have to understand the concepts. Exercise: In each of these scenarios, what kind of constraint do we have (cover, partition, etc.)? (More than one might be applicable.) 1. USPS has to route its trucks to pick up mail from mailboxes. 2. Each nodes in an ad-hoc wireless network needs an operating frequency. Those physically close cannot have the same frequency because of interference. 3. We need to select a committee from the department s faculty members. Some faculty cannot stand each other, and must not be in the same committee. 4. The same with many committees. Every faculty can be in at most one committee, but enemies cannot be in the same one. 40

8 Permutations Some problems require that we decide on the order of things (jobs, tasks, etc.) For n things to order, we can have n 2 binary variables x ij = 1 if thing i is the jth one in our ordering, and 0 otherwise. Implicit constraints: i x ij = 1 j and j x ij = 1 i. Every item gets a position, every position is filled by exactly one item. Surprisingly(?), this works way better than using only n integer variables with values from {1,, n} to encode the order. 41

9 42 MODELING Variables with N possible values. Suppose we have a decision variable x that should take its value from a given set {v 1,, v N }. x = j v j y j j y j = 1 y j 0,1 j If the set of allowable values is {0,1,, N}, then a binary representation can be more economical x = j 2 j y j y j 0,1 j

10 AN EXAMPLE WITH MANY COMPLICATING CONSTRAINTS A job sequencing problem with setup times A machine can perform m operations, each requiring a unique tool. The machine has a magazine that can hold B < m of these tools. Loading/unloading tool j takes s j (setup) time. Only one tool can be loaded or unloaded at a time. At the start of the day, the magazine is empty. At the start of the day there are n jobs waiting to be done using the machine. Each job takes a number of operations; completing job i means carrying out a set of operations J i. Before the job can be commenced, all tools required for the job must be loaded. We want to determine the optimal job sequence to minimize the time required to complete all of them. 43

11 Setup cost, aka. fixed charge. In applications when we incur an initial constant cost for setting up an activity, in addition to the (linear) cost of the activity. Examples: Powering up a system that has a unit cost per time once running. Service cost: fixed cost of call, plus an hourly rate. Cost is c 0 + c 1 x if x > 0, otherwise 0. Suppose we know x M in the feasible region. Then: Cost is c 0 y + c 1 x, with y {0,1} and x My. 44

12 45 MODELING Either-or constraints. (Aka. disjunctive constraints) We are given two constraints, a T x b and c T x d. We need at least one of them to be satisfied, but not necessarily both. Note that this is no longer convex, so we cannot hope for a simple LP formulation. Suppose we know that a T x and c T x are bounded from above on the feasible region. See why this works (for some sufficiently large number M): a T x b + ym c T x d + 1 y M y {0,1} Exercise: we can do the same, when at least K out of N constraints must hold. How? (Use a separate y i {0,1} for each constraint )

13 Semi-continuous variables x is either 0 or has a (continuous) value in [a, b] for some given 0 < a < b. E.g., output of a generator cannot be arbitrarily low, it is within a positive minimum and maximum, otherwise it is turned off. ay x by, y {0,1} 46