Do you have an Uncle Louie? Yep - we all do! You know what I mean - this guy:

When my wife and I were planning to get married, we had all sorts of big decisions to make. Where would our future home be? How many kids would we have? Joint bank account or individual? Through all of our deliberation, we almost forgot the most important decision of all. Who would have to sit with Uncle Louie at the wedding reception?!

Louie is the standard uncle in a large Italian family. The life of the party - inventor of our great Thanksgiving tradition -

Of course, Uncle Louie is not the only problem:

So... what to do? Luckily, working for SAS and being an optimization guy, I had the idea to build a mathematical model (integer program) to optimize the seat assignments to minimize the overall

A model for this problem was already presented in the paper

Here is the idea. Let's assume that you can collect (or simply know) a measure of unhappiness if guest \(g\) were to be seated with guest \(h\). Let's call this value \(a_{gh}\). The higher the value, the more unhappy guests \(g\) and \(h\) will be with your decision. Let \(G\) define the set of guests, \(T\) define the set of tables available, and \(S\) define the number of seats available at each table. The unhappiness of a table is defined as the maximum unhappiness of all pairs of guests at a table. The goal is to assign guests to tables so as to minimize the total unhappiness of all the table assignments.

To model this, let's define a binary decision variable \(x_{gt}\), which will be chosen to be 1 if guest \(g\) is assigned to table \(t\). Let \(u_t\) define the unhappiness of table \(t\). Then, we can formulate the optimization model as the following mixed integer linear programming (MILP) problem: \(\begin{align} & \text{minimize} & \sum_{t \in T} u_t \notag \\ & \text{subject to} & \sum_{t \in T} x_{gt} & = 1 & & g \in G \\ & & \sum_{g \in G} x_{gt} & \leq S && t \in T \\ && u_t & \geq a_{gh}(x_{gt} + x_{ht} - 1) && t \in T, g \in G, h \in G \text{ such that } g < h\end{align} \)

Constraint (1) ensures that every guest is assigned to exactly one table. Constraint (2) restricts each table to at most \(S\) guests. Constraint (3) defines \(u_t\), the unhappiness of table \(t\), by forcing it to be the maximum \(a_{gh}\), for all pairs of guests \((g,h)\) that are seated together (i.e., if \(x_{gt} = x_{ht} = 1\), then \(u_t \geq a_{gh}\)).

Using the OPTMODEL procedure in SAS/OR, we can now translate the algebra to code as follows. In order to protect the innocent, we show here the same unhappiness data from the paper (not the real data collected from my family). For the paper, they used the equivalent of random data, by setting \(a_{gh} = |g-h|\).

The final result is an assignment of guests to tables (those where \(x_{gt} = 1\) in the optimal solution) and is stored in the data set

Even for a medium-sized wedding, solving this problem with standard MILP techniques (branch-and-cut) can be quite difficult. The inherent symmetry in the model causes most tree-based search methods to spend a good deal of time doing redundant work in order to find an optimal solution. Notice, in this case, that the table identifier is arbitrary. Given a solution, if you move everybody originally at table 1 to table 2 and also move everybody originally at table 2 to table 1, the new solution has the same total unhappiness.

When we solve this problem with our standard MILP solver, it takes quite some time to get a good solution. The progress is shown in the log below (using SAS/OR 13.2). The sum of unhappiness (the objective) is listed in the log under 'BestInteger'. Any feasible solution must have total unhappiness greater than or equal to the 'BestBound'. After 1 hour of computing time, the standard algorithms are still far from finding an optimal solution and proving optimality (as designated under 'Gap').

Now, by simply using the following decomposition option, the wedding assignment problem can be solved much more quickly:

The automated DECOMP option

In the end, the wedding went off without a hitch. Uncle Louie was relatively tame for most of the night, and we received all positive reviews for the seat assignments. Of course, we forgot to account for this guy...

Louie is the standard uncle in a large Italian family. The life of the party - inventor of our great Thanksgiving tradition -

*The Galati Fart Game*(I am not kidding). So, when it came time to assign Uncle Louie to a particular table for the wedding reception, we had a small dilemma. Some family members wanted to be seated with Louie and very much enjoyed his company. Others would probably have rejected the invitation if they had been seated with Louie.Of course, Uncle Louie is not the only problem:

- Aunt Boo had a falling out with Aunt Paula back in the 70s over a leaked muffin recipe and has never forgiven her,
- Tommy Six Toes (second cousin) is a wanted felon in three states while my good friend, James, is a police detective,
- Auntie Sue enjoys throwing her food at people (the adults don't like this, the kids do),

So... what to do? Luckily, working for SAS and being an optimization guy, I had the idea to build a mathematical model (integer program) to optimize the seat assignments to minimize the overall

*unhappiness*of the guests.**The Optimal Wedding Seat Assignment Problem**A model for this problem was already presented in the paper

*Dippy – A Simplified Interface for Advanced Mixed-Integer Programming.*Here is the idea. Let's assume that you can collect (or simply know) a measure of unhappiness if guest \(g\) were to be seated with guest \(h\). Let's call this value \(a_{gh}\). The higher the value, the more unhappy guests \(g\) and \(h\) will be with your decision. Let \(G\) define the set of guests, \(T\) define the set of tables available, and \(S\) define the number of seats available at each table. The unhappiness of a table is defined as the maximum unhappiness of all pairs of guests at a table. The goal is to assign guests to tables so as to minimize the total unhappiness of all the table assignments.

To model this, let's define a binary decision variable \(x_{gt}\), which will be chosen to be 1 if guest \(g\) is assigned to table \(t\). Let \(u_t\) define the unhappiness of table \(t\). Then, we can formulate the optimization model as the following mixed integer linear programming (MILP) problem: \(\begin{align} & \text{minimize} & \sum_{t \in T} u_t \notag \\ & \text{subject to} & \sum_{t \in T} x_{gt} & = 1 & & g \in G \\ & & \sum_{g \in G} x_{gt} & \leq S && t \in T \\ && u_t & \geq a_{gh}(x_{gt} + x_{ht} - 1) && t \in T, g \in G, h \in G \text{ such that } g < h\end{align} \)

Constraint (1) ensures that every guest is assigned to exactly one table. Constraint (2) restricts each table to at most \(S\) guests. Constraint (3) defines \(u_t\), the unhappiness of table \(t\), by forcing it to be the maximum \(a_{gh}\), for all pairs of guests \((g,h)\) that are seated together (i.e., if \(x_{gt} = x_{ht} = 1\), then \(u_t \geq a_{gh}\)).

**PROC OPTMODEL and DECOMP**Using the OPTMODEL procedure in SAS/OR, we can now translate the algebra to code as follows. In order to protect the innocent, we show here the same unhappiness data from the paper (not the real data collected from my family). For the paper, they used the equivalent of random data, by setting \(a_{gh} = |g-h|\).

%macro WeddingPlanner(num_guests=., max_tables=., max_table_size=.); proc optmodel; num num_guests = &num_guests; num max_tables init &max_tables; num max_table_size = &max_table_size; if(max_tables=.) then max_tables = ceil(num_guests / max_table_size); set GUESTS = 1..num_guests; set TABLES = 1..max_tables; set GUEST_PAIRS = {g in 1..num_guests-1, h in g+1..num_guests}; var x{GUESTS,TABLES} binary; var unhappy{TABLES} >= 0; min MinUnhappy = sum{t in TABLES} unhappy[t]; con AssignCon{g in GUESTS}: sum{t in TABLES} x[g,t] = 1; con TableSizeCon{t in TABLES}: sum{g in GUESTS} x[g,t] <= max_table_size; con TableMeasureCon{t in TABLES, <g,h> in GUEST_PAIRS}: unhappy[t] >= abs(g-h) * (x[g,t] + x[h,t] - 1); solve with milp / maxtime=3600; create data Assignment from [guest table]= {g in GUESTS, t in TABLES : x[g,t] > 0.5} x[g,t]; quit; %mend WeddingPlanner; %WeddingPlanner(num_guests=70,max_table_size=6); |

`Assignment`.Even for a medium-sized wedding, solving this problem with standard MILP techniques (branch-and-cut) can be quite difficult. The inherent symmetry in the model causes most tree-based search methods to spend a good deal of time doing redundant work in order to find an optimal solution. Notice, in this case, that the table identifier is arbitrary. Given a solution, if you move everybody originally at table 1 to table 2 and also move everybody originally at table 2 to table 1, the new solution has the same total unhappiness.

When we solve this problem with our standard MILP solver, it takes quite some time to get a good solution. The progress is shown in the log below (using SAS/OR 13.2). The sum of unhappiness (the objective) is listed in the log under 'BestInteger'. Any feasible solution must have total unhappiness greater than or equal to the 'BestBound'. After 1 hour of computing time, the standard algorithms are still far from finding an optimal solution and proving optimality (as designated under 'Gap').

NOTE: The presolved problem has 852 variables, 29062 constraints, and 88620 constraint coefficients. NOTE: The MILP solver is called. Node Active Sols BestInteger BestBound Gap Time 0 1 0 . 0 . 2 NOTE: The MILP solver's symmetry detection found 36 orbits. The largest orbit contains 24 variables. 0 1 0 . 0 . 3 0 1 0 . 0 . 10 NOTE: The MILP solver added 25 cuts with 186 cut coefficients at the root. 100 33 0 . 9.0000000 . 27 200 65 0 . 10.0000000 . 40 300 97 0 . 10.4743428 . 52 400 131 0 . 11.0000000 . 64 483 162 1 283.0000000 11.0000000 2472.73% 75 500 171 1 283.0000000 11.0000000 2472.73% 76 600 204 1 283.0000000 11.1803643 2431.22% 87 -- snip -- 46600 43154 12 94.0000000 12.9549906 625.59% 3573 46700 43246 12 94.0000000 12.9573416 625.46% 3582 46800 43338 12 94.0000000 12.9614975 625.22% 3591 46896 43430 12 94.0000000 12.9618351 625.21% 3599 NOTE: CPU time limit reached. NOTE: Objective of the best integer solution found = 94.Fortunately, in the suite of SAS/OR solvers (embedded in PROC OPTMODEL) we also offer the DECOMP algorithm, which provides an alternative approach for solving specially structured MILP problems. In this particular case, DECOMP can automatically detect the symmetry, reformulate the algebraic model into what is known as the Dantzig-Wolfe reformulation, aggregate the symmetric pieces of the model, and use a specialized algorithm under the hood to solve the MILP. For more details on the methodology, consult the Decomposition Algorithm chapter in SAS/OR 13.2 User's Guide: Mathematical Programming (or contact me - the ideas behind this algorithm are the basis of my PhD thesis).

Now, by simply using the following decomposition option, the wedding assignment problem can be solved much more quickly:

solve with milp / decomp=(method=set); |

NOTE: The presolved problem has 852 variables, 29062 constraints, and 88620 constraint coefficients. NOTE: The MILP solver is called. NOTE: The DECOMP method value SET is applied. NOTE: All blocks are identical and the master model is set partitioning. NOTE: The Decomposition algorithm is using an aggregate formulation and Ryan-Foster branching. NOTE: The problem has a decomposable structure with 12 blocks. The largest block covers 8.31% of the constraints in the problem. NOTE: The decomposition subproblems cover 852 (100.00%) variables and 28992 (99.76%) constraints. NOTE: The deterministic parallel mode is enabled. NOTE: The Decomposition algorithm is using up to 4 threads. Iter Best Master Best LP IP CPU Real Bound Objective Integer Gap Gap Time Time NOTE: Starting phase 1. 1 0.0000 70.0000 . 7.00e+01 . 0 0 10 0.0000 56.5000 . 5.65e+01 . 0 0 20 0.0000 44.6000 . 4.46e+01 . 0 0 30 0.0000 35.8618 . 3.59e+01 . 0 0 40 0.0000 25.3660 . 2.54e+01 . 0 0 50 0.0000 16.6091 . 1.66e+01 . 0 0 60 0.0000 4.3861 . 4.39e+00 . 0 0 70 0.0000 0.0271 . 2.71e-02 . 0 0 71 0.0000 0.0000 . 0.00% . 0 0 NOTE: Starting phase 2. 76 0.0000 112.0000 112.0000 1.12e+02 1.12e+02 0 0 80 0.0000 112.0000 112.0000 1.12e+02 1.12e+02 0 1 90 0.0000 112.0000 112.0000 1.12e+02 1.12e+02 1 1 -- snip -- 460 46.0000 58.0000 58.0000 26.09% 26.09% 157 90 470 46.0000 58.0000 58.0000 26.09% 26.09% 165 94 480 50.0000 58.0000 58.0000 16.00% 16.00% 176 99 483 52.0000 58.0000 58.0000 11.54% 11.54% 178 100 485 58.0000 58.0000 58.0000 0.00% 0.00% 180 101 Node Active Sols Best Best Gap CPU Real Integer Bound Time Time 0 0 26 58.0000 58.0000 0.00% 180 101 NOTE: The Decomposition algorithm used 4 threads. NOTE: The Decomposition algorithm time is 101.69 seconds. NOTE: Optimal. NOTE: Objective = 58.

**Technical side-note**The automated DECOMP option

`method=set`is a hidden option that detects a set partitioning type of constraint system (as we see in this model). In order to explicitly define the block structures for your model (one subproblem for each table \(t\)), you could use the`.block`suffix for each constraint. For more details, consult the DECOMP documentation. The syntax to use explicit construction for this model is the following:for{t in TABLES} do; TableSizeCon[t].block = t; for{<g,h> in GUEST_PAIRS} TableMeasureCon[t,g,h].block = t; end; solve with milp / decomp=(method=user); |

**Photo Credit**(for the wedding photo): The Wiebners**Disclaimer**: The characters in this story are all real. This is not a dramatization.
## 4 Comments

What about the integer for dudes who taught you how to do a duck under and did not make the cut. I would like to see the data on how this list was developed.

The alternative method would be in-vivo JIT Brownian Motion Optimization, also known as the guests figure it out ;). Humans are really good in clustering and de-clustering based on relationships.

I wonder what the probability of the concerned persons ever even showing up are.

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