Dr. Chad Broman1 Dr. Tiffany Love2
1 Department of Applied Psychological Machine Learning, Cranberry-Lemon University, Pittsburgh, PA, USA
2 Department of De-Gamification, Cranberry-Lemon University, Pittsburgh, PA, USA
Once every modern relationship reaches the level of cohabitation, each couple must determine the optimum amount of household chore responsibilities. Historically in our relationship, this issue has been avoided by minimizing the total amount of responsibilities by avoiding having any kids and only taking care of plants and the occasional stray cat . Unfortunately for our relationship, Tiffany adopted a dog Franky which requires much more work and cleaning. This strain has resurfaced the unsolved problem of who should do the dishes, an issue which was previously solved with the approximation formulated in . Because of the increased stress on our own responsibilities, we as a couple will address this issue as a transportation problem using various linear programming techniques. Through this study, we determined that the Hungarian algorithm was not suitable to include Chad’s side hustle (game streaming) or Tiffany’s (Etsy knitting shop). In optimizing side hustle money, the Simplex method turned out to be too complicated to do by hand and the interior point method was cooler and a better way of determining that Chad should do the dishes.
Keywords: Transportation Problem, Household Chores, Assignment, Relationships, Interior Point Method, Optimization
When Tiffany adopted Franky three weeks ago she said she would take care of all of the extra responsibilities involved . Ever since, she has struggled to follow the agreed upon approximation determined in  which called for a 70-30 Tiffany-Chad dishes responsibility. Enforcing Chad’s responsibility in doing 30% of the dishes has historically proven to be difficult and has only been reliably accomplished using operant conditioning techniques . As Tiffany has lost more time to do the dishes as well as other previously agreed upon household chores it has become more apparently obvious that a new optimization approach is required to determine who does what chore, and how often; especially the dishes, which both of us particularly hate doing.
The transportation problem is a type of problem typically applied to various economic problems involving determining how much or what to produce at different factories with different cost functions associated with transportation and production. Once all constraints are determined as well as the associated cost functions, linear programming techniques can be used to find the optimum production to meet a fixed demand. This includes demands that could never be met due to a shortage in production capacity. Due to our recent developments of Etsy/game streaming side hustles, there has been a recent shortage in time to complete many of these chores, not to mention Tiffany’s dog. Our dog is just a puppy and Chad plays with him too. Until Franky learns to stop going in my man cave, we are likely to be in a labor shortage and I didn’t adopt him.
Traditional relationships, aka marriages, have often assigned 100% of the household chores to the wife/brood mother and 0% to the husband/provider. This assignment structure is no longer optimal. With most modern relationships now dual income and dual household responsibility, the 100-0 split of responsibility is no longer feasible outside of Utah. Typically, many couples tackle this problem by minimizing responsibilities such as opting for cats and plants instead of dogs or kids. However, eventually, someone’s going to fall in love with a dog from a shelter Rebecca posted on facebook a few Thursdays ago and now we have five times the amount of cleaning responsibilities. Because Franky needed a home, the tradeoff in additional work was largely seen to be worth it.
Due to the previously low workload and non-contentious nature of chore responsibility, work assignment issues rarely had to be calculated at the edge of our constraint functions and a rigorous approach was not required outside of in-law visits or during our dissertation writing periods in which we really let our place go . According to recent projections regarding Chad’s IRacing obsession , the shortage in household labor is expected to plummet nearly as fast as Tiffany’s due to her knitting based Etsy shop. With a high demand on personal freetime, the solution must satisfy Karush-Kuhn-Tucker (KKT) Conditions so that no matter the outcome there will be no whining because it will be an irrefutably optimum solution unlike last time .
3. Constraint development and Weight Selection
The first step to developing an optimum chore solution is to define the constraints and weights of each production system (i.e. Chad and Tiffany). First the time and subjective effort cost will be created to determine the individual cost for each of us to do each chore. Next, weekly and monthly schedules will be used to determine time constraints. Finally, the monetary value of each party’s speculative post-work side hustles will be included to add positive values to the spare time saved for each party. Once structured, the transportation network will create not only a node structure shown in figure 2 but a cost matrix defining a system of equations defining the constraints. By the time the system is defined, there will be an objective set of rules to find the optimum solution so that Chad won’t try and get out of the dishes like he so often tries to .
3.1 Chore Cost
It’s a difficult task to create objective comparative metrics for the amount of effort it takes to complete an unpaid task around the house, but it’s not impossible.  developed and tested two useful metrics to determine the existential dread and drop in after work rest caused by doing chores. By measuring the Sunday Scary Metric (SSM), the existential dread caused by day to day burnout can be quantified by determining how hard it is to fall asleep on a Sunday night before a work day. For pure physical exhaustion, the Normalized Estimated Inverse Sabbath (NEIS) can be used to quantify the exhaustion caused by each chore. Averaged together, each metric can be used to measure the cost of doing the chore for each party.
Likewise independently gathered historic Tiffany data  and historic Chad data  has been used to create a responsive multivariate predictor on how long it will take each person to complete each task based on time-series mood models . Contrary to the simplistic model, evidence has shown that though Chad will complete many chores faster than Tiffany for far less of a cost in time, NEIS, and SSM; the work in cleaning, and dish washing tasks is occasionally deemed to be not good enough and requires Tiffany to re-complete the task. Many times, the dishes are completely fine and Tiffany just doesn’t trust our dishwasher to finish the job. To compensate for this issue, a ‘re-do’ penalty cost is applied accordingly weighted from the findings in .
3.2 Time Constraints
We both have full 9-5 jobs and on average keep our work to 40 hours a week outside of our occasional crazy weeks and traveling for conferences and other work trips. Once we both finished schooling, after hours free time expanded only to contract due to Chad’s periodic Vegas bender getaways with friends, and fishing trips. While constrained by frequent discord hangouts with his guy friends I don’t like because they keep me up often until 3:30am, Chad’s day to day time is generally open. Likewise, Tiffany’s time becomes quickly limited due to the frequency of Love Island episodes quickly airing and the hour long phone chats to discuss the latest dating game show outcome. The episodes extend seemingly indefinitely as Tiffany will often look at her phone, miss something, and rewind throughout each episode. After seasonally adjusted for better fishing weather and reality show season airing, Chad on average has 3-6 free hours a day depending on environmental conditions while Tiffany has 4 free hours a day. A cross analysis showed that Chad’s time would have dropped another hour if he had kept that boat  while another study suggested such a boat would have reduced the time of travel to the good fishing spots and was a great deal once adjusted for current inflation .
3.3 Side Hustles
While each of us has plenty of time to individually take care of each household chore to even include taking Franky out for walks, each of our respective side hustles have taken up the remainder of our free time and more. Tiffany has taken up the hobby of knitting and has begun selling fingerless gloves and personalized knitted laptop bags on Etsy. An independent and not at all biased business analysis accomplished prior to this study  has determined the profit per labor hour which is shown in table 1 when all of the materials are accounted for. According to Tiffany, they take a lot longer and more materials only because of the complicated design which will soon catch on and raise the price.
|Item||Number Sold||Labor (Hrs/Item)||Profit/Hr|
Chad has recently begun playing and streaming races on IRacing, a semi-professional formula 1 racing game and hyper realistic race simulator. It’s so cool, they LIDAR the tracks so it’s exactly to real life specifications, real drivers use it, that’s why the tracks are so expensive, it’s not a scam. He does okay at it, but spends more money on the digital race tracks and hardware than he’d make if he were to monetize his streaming platform. He hasn’t told me how much he spends on that stupid game but I’ve checked and he usually only has a few viewers on his stream so he can’t be making that much money off of it. According to Chad, if he wins some eight hour race this Sunday, he should get a lot more viewers and might start practicing with professional formula 1 racers.
3.4 Constraint Equations
Once every consideration was calculated according to the analysis from [8-15] the objective cost of each of us doing each chore is given a Hassle Unit (HU) based on the SSM and NEIS from . Because Chad really enjoys IRacing and Tiffany really enjoys knitting, those assignments are given negative HUs. The estimated HU’s show things like; how Chad is less efficient at Dishes and cleaning according to , Chad is the better cook because I don’t under season like Tiffany does , there is no way Tiffany could figure out how to get the mower started, and Franky is Tiffany’s dog though I will help from time to time if she really is in a bind. The corresponding values are shown in table 2 and each task has a set number of HUs required until completion.
|Taking Care of Franky||2.03||15.82|
4. Optimization Methodology
This paper will utilize three different algorithms to handle the Assignment problem presented and defined in the previous section. The Hungarian method will be used to match one person to each of the necessary tasks (excluding IRacing and Knitting which are boundless and not immediately necessary). Next, we’ll attempt to solve the optimization problem by hand using the Simplex method and then attempting to computationally solve using the Interior point method.
4.1 Hungarian method
The Hungarian method is a straightforward way to hand assign one job to one person. It works by structuring a square matrix of tasks to be assigned to one person each. Then the method subtracts the lowest value of each element in a row until the matrix has at least one zero in each row. The zeros of each row correspond to an assigned task. Unfortunately, this method does not take into account the amount of money produced by our side hustles but that doesn’t matter much because Tiffany doesn’t make that much money on her knitting anyway. Additionally, any money made by Chad at IRacing is purely speculative that he wins the big race later this week and that creates twitch followers and is unlikely to compensate for his gaming rig.
4.2 Simplex Algorithm
In the Simplex Algorithm the task costs and production constraints are constructed into a single matrix as shown below in equation 1 where A and b form the system of equations Ax = b of the constraints plus the cost matrix C.
If there are only two constraint equations, it can be solved graphically by selecting the best vertex produced by the system of equations. As discussed in the previous section, we have far more than two constraint equations. In our case, the matrix can be solved by introducing new artificial variables and solving through pivoting around columns until the matrix is reformed enough to solve by hand.
4.3 Interior Point Method
Alternatively to solving the equations by hand, the simplex method can be approximated non-linearly with computers using the Interior-Point method. The Interior Point method works by starting at a point which can solve the system of equations in the region of possibility that may or may not be optimized. Then it iterates on some rip-off of Newton’s method until it reaches an inner vertex point which the simplex method would theoretically solve for. Instead of including all of that math or our code, below in figure 3 is a CCL picture of a line traveling from an inner-point in a multidimensional feasible constraint space towards the optimum inner vertex. The picture alone is reason enough to believe this method is rock solid and should be believed.
5. Results and Discussion
According to the results of the various algorithms tested, Chad is going to do the dishes. Guided by every reasonable metric it’s going to be Chad and not me, I put it in table 3 to clarify. When we included the effects of my Etsy shop through the Inner-Point method I was the clear winner here because it’s a way better hobby than IRacing and I’m too busy taking care of Franky. Even in the Hungarian method where he less efficiently did the dishes, according to the calculated HUs, he still had to do the dishes because according to the math, he’d rather scrub caked on mozzarella and sugar off of plates than take Franky out for a walk and is even worse at vacuuming . Because he’s so awful at cleaning, the decision was clear-cut, objective, and passed KKT conditions. Unfortunately, we were unable to solve the system of equations by hand using the Simplex algorithm because we kept screwing up our column pivots and getting different answers.
|Method||Who should do the dishes|
Tiffany says we were unable to solve for the system of equations by hand but she just didn’t like the outcome. Each time she had to do the dishes and said we messed up one of the column pivots. Likewise I’m not completely convinced that we correctly created the right dummy variables transforming the Hungarian Method’s Matrix from a rectangular to a square like we needed to. I’m not even sure that was a correct application of the algorithm. Some experts believe this is only the case because the algorithm is assigning the amount of time to take care of Franky equally even though it’s an extra responsibility that Tiffany accepted by adopting him.
Regardless Chad has to do the dishes and we correctly used the Inner-Point method which had the canned python library we both agreed was too complicated to pick apart and recode ourselves. There was too much going on to question the programming and we tested it with some dummy examples to make sure we used it properly. CHADS DOING THE DISHES!
So Chad’s doing the dishes, there’s a new method to objectively allocate who does what task around the house, and he can’t whine about it this time  because it meets KKT conditions. So what? I guess I’ll do the dishes. As soon as the algorithm spat out the answer Tiffany completely lost interest in this paper but still wants the credit. All I know is, as soon as we don’t have to clean up after Franky as much as we’ve been doing lately, I’m totally re-running these numbers. Man, this is total BS, I have to practice on the track or I won’t be competitive on the new German circuit on IRacing. Anyway, the Inner-Point method made my computer fan turn on for a while but it solved the problem of who should do the dishes in our more constrained environment.
7. Conflicts of Interest
Okay, conspiracy theory time. Tiffany’s grandparents bought all of her Etsy shop sales. That was totally the only reason I have to do the dishes because I don’t make money playing IRacing…yet!. They don’t even have laptops, why would they need those Zelda themed laptop covers? They don’t even play Zelda! Okay, I just needed to document this for the next paper when we re-decide who does the dishes.
- Love, Tiffany 2018 Minimum Maintenance and Reliability Study of Adopting Skittles the Cat :: Journal of Household Animal Husbandry
- Broman, Chad; Love, Tiffany 2014 Approximate Chore Assignment Algorithms for who Should do the Dishes :: Journal of Relationship Goals
- Love, Tiffany 2022 Minimum Maintenance and Reliability Study of Adopting Franky the Dog :: Journal of Household Animal Husbandry
- Love, Tiffany 2021 Operant Conditioning Methods to get Chad to do the Dishes :: Self Published
- Broman, Chad; Love, Tiffany 2017 Targeted Cleaning Standards that Prevent Ants; Methodology to Stay Hygenic while Dissertation Writing :: Journal of Grad Student Maidology
- Love, Tiffany 2022 IRacing Time Consumption: A Predictive Model of when My Boyfriend is Coming to Bed :: Journal of Relationship Goals
- Broman, Chad; Love, Tiffany 2021 Theoretical Conditions for Fair Chore Assignment and Chronic Complaint Prevention :: Journal of Relationship Goals
- Yelnets, Stanley III 2021 Markov Models for Ruining your Weekend: A Comparative Study :: Journal of Astrological Big Data Ecology
- Broman, Chad 2015 Why is Valentine’s Day so Important? A Time Analysis of Tiffany’s Relationship Expectations :: Journal of Psychological Machine Learning
- Love, Tiffany 2018 A Non-Parametric Model of my Boyfriend’s Chore Completion Times; How to know when to remind Chad to take out the Trash before the truck get here :: Journal of Relationship Goals
- Broman, Chad 2021 A Time-Series Analysis of my Girlfriend’s Mood Swings :: Journal of Astrological Big Data Ecology
- Love, Tiffany 2019 Cleaning Error Rates and how Often I have to Re-clean the Dishes after my Boyfriend “Finishes” :: Journal of Relationship Goals
- Love, Tiffany 2021 An Exploratory Analysis of my Boyfriend’s Fishing Obsession; A Jungian Approach :: Journal of Psychological Machine Learning
- Broman, Chad 2021 Financial Projections of Speedboat Investments in a Bull Market; The Opportunity Cost of not having an Extra 20 Horse Power :: Fishing Times [REJECTED]
- Broman, Chad 2022 Full Production Cost of my Girlfriends Elaborate Knitting Etsy Business :: Journal of Side Hustle Culture
- Broman, Chad 2020 The Cholula Correction Coefficient of My Girlfriend’s Bland Scrambled Eggs :: Journal of Budget Gourmet Dining
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If you missed out on the other Chad-Tiffany Papers don’t worry too much, they don’t relate to each other too much, but are still worth the read