Optimize Your In-store Holiday Shopping Route

Remember the last time you were not tired after holiday shopping in a busy mall, bulk shopping at a discount-club store, or weekend shopping at a crowded grocery store?

(source links: http://info.museumstoreassociation.org, http://binaryapi.ap.org)

Hopefully, the simple and preliminary 'operations research (O.R)' analysis in this post can help make the experience a little less stressful and maybe even let us enjoy a bit of retail therapy that we can't get online.  The shopping optimization problem addressed here is simple one, albeit with a twist. This post was triggered by an observation made after recent shopping expeditions to a nearby Costco outlet.

We have a long shopping list of L line-items (i) of various quantities N(i) that we have to pick up in a store. How should we optimally reorder our shopping list to reflect the sequence in which we pick up these items?




Version 1
The simple version aims to minimize total shopping time. For example, we can formulate this as an instance of the so-called traveling salesman problem (TSP) that finds the shortest sequence that starts at the entrance, visits each of the L 'cities' once, then the checkout counter, and ends at the store exit (entrance). A google search shows this 2009 research paper by researchers at UPenn, which shares the technical details and interesting results for this version.

To simplify our problem, we assume that the topology of the store is prior knowledge, and we can roughly pre-assign items in the same aisle to the same 'city'. The resultant problem is to find the order of visits to the aisles of interest to us. If the aisles are neatly arranged as node intersections in a rectangular grid, then we simply have to find the shortest rectilinear "Manhattan" distance tour in this grid. Depending on the type of the store, the store-layout itself may be optimized by the retailer based on the observed foot-traffic and 'heat-map' data. For example, it may be designed to retain the customer in the system to increase chances of purchase (exploratory tour, store selling expensive luxury/fashion products), or quickly out of the system (routine tour, obligatory products like groceries), or based on some other goal. Store space and layout problems are known to be commercially useful optimization problems in the retail industry.

Picture linked from an iTunesApp (Concorde) page of Prof. Bill Cook that solves TSPs :)

Version 2
Let us additionally assume that product attributes have an impact on our order. Suppose one or more items in our list is located in the frozen section at a grocery store, then we may prefer to get there at the end of our tour. Luckily, many stores appear to anticipate this (?), and locate this section towards the end of the store. Thus if 'maximum freshness' or 'minimum damage' is an additional consideration, this may alter the TSP route and the final ordering of items in our shopping list.

Version 3
This was the problem that was of immediate interest to me today. Costco sells stuff in bulk, so the items tend to be heavy, and considerable energy is expended moving the cart through the store. At the risk of mangling undergrad physics, let us proceed with our analysis...
Let μ be the (constant) coefficient of rolling friction between the wheels of the shopping cart and the store-floor. Then the work required to move mass m through distance d = force x displacement =μmgd, where g is the (constant) acceleration due to gravity. Thus a squeaky-wheeled cart will make us work extra hard. Speed of shopping is a consideration if we want to keep track of power (force x velocity). Covering the same tour length in half the time requires twice the power, i.e. the rate at which our body expends stored energy.

Carry versus Cart (reinventing the wheel)
If we had zero rolling friction, technically the only work involved is in lifting items and putting them in our cart. We can assume that this work is independent of our order of visit and is a fixed quantity. On the other hand, if we are not working with a wheeled-cart, but carry our stuff in a market-basket or shopping bags, then we have to raise the potential energy of the items to height 'h' (mgh) all the way until checkout, and also overcome the sliding friction between our shoes and the floor, which would probably be higher than the rolling friction, as illustrated below.

(link source http://static8.depositphotos.com)

Why is a shopping experience increasingly stressful over time?
As we add items to our cart, the work done per unit time (power) required increases. In the continuous case, this problem closely resembles our 'optimal snow shoveling problem' analyzed during a February snow storm. The accumulated objective function cost there increases as the square of the distance. Here, we look at the discrete situation. When we visit city 'i' to pick up N(i) items each having mass m(i), we instantaneously add mass N(i) * m(i) that we have to lug around until checkout. After picking up k-1 items, my total work done will approximately be:
μg * sum (i = 0,... k-1) W(i) * d(i, i+1), where i = 0, represents the store entrance, m(0) represents the mass of an empty shopping cart, and
W(i) = sum(j = 0, .., i) N(j) * m(j) = total mass carried from city i to i+1.

In other words, the 'cost' accrued while traversing any pair of cities also depends on the items we picked up earlier, i.e., it is no longer memory-less and varies depending on the cities visited earlier.

Result: our shopping gets increasingly more tiring over time

If one of the items in our shopping list involve a heavy item, then our optimal solution may no longer be the shortest time tour.  We now have to solve a minimum-effort TSP. Operations Researchers have also looked at methods for solving a path-dependent TSP in the past. A simple heuristic approach would be to break the trip into two tours. The earliest items stay with us the longest, so we first find the optimal sequence through the light items, followed by the shortest tour through the heavy items. We also have to organize the shopping cart carefully enough to ensure that our light items do not get squished by heavy products, and our ice-cream doesn't melt. I'm sure there are better algorithms than the one provided here.

Recommendation
If we want to burn calories while shopping then a good way to do that, based on our previous discussion is (a) carry our items, (b) pick up the heaviest items first, and (c) take the longest route to maximize energy expenditure (d) walk briskly. This health-and-fitness article provides ten tips for exercising that includes these ideas. For the rest of us who are looking to minimize effort, we simply do the opposite:

We sort our shopping list items in decreasing order of expected weight (frozen stuff goes to the bottom too), while also ensuring that the aisles of adjacent items in the list are close to each other, and we are not revisiting aisles or zig-zagging much.  Some swapping may be required to find improved sequences. Many of us have probably evolved into efficient shoppers over time, and naturally take these steps and more, but if there an opportunity for improvement that the 'science of better' can uncover for us to make our shopping less stressful, let's take it.

Happy Holidays!

Update 1: December 17, 2013
IBM Smarter Retail article says:
"..... Most recently, Lowe’s has offered the customer a product location functionality that is integrated with their wish list. According to Ronnie Damron, .....Whether customers are browsing the store for ideas or searching for a specific item in a hurry, we think the Product Locator feature will simplify the shopping process, creating a better experience that encourages customers to come back again and again.”

Solve TRP, Drive Tesla

So the awesome and hyped Tesla Model S is beating Mercedes, BMW, and Audi in sales despite the $70K price tag. One problem is the current lack of charging stations required to recharge over long trips. CNNMoney reports:
"CNNMoney took a Tesla Model S on a test drive from Washington D.C. to Boston. The all-electric car drove 450 miles with two stops to recharge at Tesla's Supercharger stations in Delaware and Connecticut"



A second problem is that unlike conventional gasoline cars that can be refueled very quickly, charging can take a relatively long time, and adds significantly to your total travel time. In the near future, as the number of Teslas increase, the number of charging stations is likely to go up too. However, it will remain a sparse resource for some time. A future Tesla trip from New York to California will require many recharging stops, reminiscent of the old stagecoach problem during the Gold Rush era, which is used to illustrate the principles of Dynamic Programming.

Update: May30, 2013
Popular Mechanics reports:
"...the California-based EV automaker plans to expand its fast-charging "supercharger" network across the continental U.S. During today's announcement, Musk said New York to Los Angeles trips would be viable as soon as this winter. 

"When you get in a car you have the ability to go almost anywhere," Musk said. "That's what the supercharger network will do." 

The rollout will begin as soon as this summer, with the number of stations tripling to 27 nationwide by the end of June. In addition to increasing supercharger density along the already-established California and Mid-Atlantic coasts, Tesla will debut chargers in a handful of new metropolitan areas within the next few weeks, including Austin/Houston, Portland/Seattle, Boulder/Denver, and Chicago/Milwaukee. By the end of the year, Musk expects to to have "a couple hundred" superchargers spread across the U.S. and southern Canada, with as little as 80 to 90 miles between stations along the Altantic and Pacific coasts..."

Tesla Routing Problem
Given a road network G having a limited number of charging stations located at specific nodes, the Tesla Routing Problem (TRP) will have to:
Find the shortest feasible time path from origin O to Destination D in network G.

Feasibility requires the car to visit enough charging stations in a timely manner. Optimality requires that the sequence of visits be carefully selected such that the total time spent is kept to a minimum.

Total travel time = driving time + recharge time

This involves two decisions:
D1. Which recharging stations to visit en-route?
D2. How long to recharge at every pit stop?

Assumptions:
A1. The longer you recharge, the longer your available range (to a limit).
A2. All recharging stations are similar, and infinite capacity (no queuing!)
A3. Range is deterministic

Update on A2: The Tesla can charge on any outlet, but the time to recharge can vary depending on the source.

Clearly, optimally solving TRP involves the finding a solution to a Traveling Salesman Problem even in the special case of instantaneous recharge (Y=0).

(Updated September 2013: As mentioned in the comments, the general case is not a TSP in the network of charging stations. Here, the assumption is that stations are sparsely located and we visit each one along the way. The title is changed to reflect this distinction).

This is no different from the (milder) challenge faced by conventional car drivers in unfamiliar areas where gas stations may be hard to come by (see this old post). Charge cannot be carried in Jerry-cans. The  shortage of charging stations does make every long Tesla trip a non-trivial problem to manage. Tesla owners have to deal with a NP-Hard Optimization Problem (Heh) on every such trip, but in practice, it should not be difficult to find good quality, feasible TRP solutions.

The recharge time makes the TRP a bit more interesting. If two stations are close to each other, we don't have to recharge fully, and just need the minimum to reach the next stop - or perhaps we recharge fully, which allows us to reach a station that is further away but turns out to be more time-effective, overall. Clearly, these two decisions appear to be interlinked and may need to be jointly optimized. D1 is a binary decision variable X(i) - we either visit or don't visit a station at node X(i), whereas D2 is a semi-continuous variable Y(i) conditional on D1:

Y(i) ≤ U.X(i)
where U = maximum recharge time.

Another interesting and maybe even useful problem for Tesla-makers to solve is where to add new charging stations on the network G over time. What is the minimum number and location of stations required to guarantee feasible solutions to any TRP in the 48 states of continental US? How will stations deal with congestion in the future? and so on ...

I haven't driven a Tesla yet but I wouldn't mind relocating my office to inside one and telecommute.

Optimally Shoveling Snow

This post attempts to compare the efficiency of various approaches used to clear a rectangular area (L > B) of snow, using a good old manual shovel of width w, always moving in straight lines. We assume that the gradient of the area is zero, and that h inches of snow has fallen uniformly over the field. We also ignore the capacity of the shovel for now. The total snow that must be banked at the borders of the field is constant = LBh. Also total working distance traversed ~ LB/w.

Snow shoveling can be a grueling exercise for some, and public health departments always publish safe shoveling tips during the winter season. This is especially important for people with potential heart problems or back problems. Hopefully analyses like these will help (I'm sure people have looked at this before, but just in case ...).


Our objectives are to minimize:
1. Total work done = Total snow moved X distance moved.
2. Number of Stop-Starts with a fixed unit cost k incurred
3. Total work time in cold weather.

Model:
Given an area bx, and assuming we are moving along x, the snow nearest the starting point must be moved the maximum distance, and this distance linearly reduces to near-zero at the end of x. Thus (introducing an integral sign in this blog, yikes):
a) total work done = ∫wxdx = wx²/2
b) Unit stop-start cost = k

Approach 1: Shoveling east-to-west (along L, banking at the west end) or bi-directional traversing:
total number of passes =  B/w
total work done = (B/w) * wL²/2 + kB/w = BL²/2 + kB/w



Note that the effort level is independent of the shovel width. A wider shovel requires fewer passes but more effort per pass.

Approach 2: Shoveling south-to-north (along B, banking at north end) or traversing in both directions:
total cost = LB²/2 + kL/w

Thus, the second approach reduces the total work done by a factor L/B. However, this reduction is achieved at the expense of more stops (which may not be a bad thing from a health perspective; k may be negative or zero for some). If L = 2B, the second approach requires 50% less effort, but requiring twice as many passes. 

Approach 3: I observed that my Canadian neighbor divides up the rectangle into two areas, always starts from the middle, and creates banks on both sides. If we bank the snow at the south and north ends:
total number of passes =  2L/w
total cost = LB²/4 + 2kL/w

The third approach is twice as effort-efficient as the second approach, at the expense of making twice as many banks, plus an additional penalty of traveling to the mid point after each pass. On the other hand, you are also less likely to run into capacity problems, since the material handled per pass is the lowest. We can also verify that among all starting points chosen for approach-3, the mid-point (B/2) yields the minimal effort. In general, any of these approaches become preferable in terms of total cost, depending on the chosen objective function. For example, we could modify approach-3 and bank snow close to the ends the field (< B/2) at their respective (west or east) ends at the expense of adding more stop-starts.


Time Considerations

If the shape of the field changes, the approaches may change, but it appears that a good principle to follow is to limit the material moved between successive bankings. In particular, we want to identify the shortest Euclidean path between every point in the field to the nearest feasible bank, and purely from an effort perspective, shovel snow along a path to a feasible bank that is optimal for all points on that path. This seems to reduce the working duration taken as well: 
Suppose we maintain constant momentum (i.e. mass X velocity = constant) during a pass, our velocity will inversely proportional to the amount of snow accumulated (e.g. 1/wx). Total time per pass ∝  ∫wxdx = wx²/2, which is proportional to the effort.

For such a velocity model, approach-3 also appears to be optimal from the time perspective. If we assume constant velocity, then the three approaches take the same working time, ignoring the shovel-free travel time required for approach-3. Among these alternative time-optimal approaches, the third approach does the best on the effort-objective.


Lawn Mowing
Non-propelled, walking lawn mowers with bags? Regardless of the traversing method adopted, our effort (per our chosen model) increases as the square of distance until we empty our cut-grass. If we adopt approach-2 or 3, we have more banking opportunities.  However, approach-3 may result in a mowing pattern that won't look so nice, so approach 2 may be more preferable. Operating a self-propelled mower with built-in mulching may reduce our effort considerably.

Update 1: Post "Nemo" Shoveling, February 9, 2013
A truly optimal solution is to have a kind and wonderful neighbor. After three hours of efficient shoveling (h = 24 inches) and 40% completion, Kevin zipped in with his snow plough and finished the remainder in 5 minutes before going off to help others. I now have to find the best way to thank him.

Update 2: (WSJ, via @ThaddeusKTSim), February 10

 (source link: online.wsj.com)

Optimal Location of Speed Limit Signs

One of the nice things about the relatively more recent automobile GPS products is that they are able to inform us of the current speed limit. Sometimes, when we take our eyes of the road for a second to grab a soda, we may have driven past a speed limit sign and be unaware of the new speed limit. Of course, there are times when the GPS unit itself does not display the speed limit for certain areas, and at other times, it is off by 10 mph, perhaps due to recent road updates. Like any decision support system, the GPS unit cannot be a fail-safe backup for user negligence.

The problem of optimally locating speed limit signs on a network must have been studied and solved a long time ago especially since the analysis of traffic and transportation networks has long been popular research area among ORMS folks. Perhaps there exists a practical combinatorial optimization problem in terms of determining minimal/safest/least ambiguous ways of locating speed signs in the presence of scarce resources and budget limits. A partial list of assumptions and constraints include:
- Speed limits change in discrete quantities of 5mph or 10 mph
- Every driver will treat the last speed limit sign (or update) they saw on the network as the prevailing speed limit
- Every driver at every point in the road network must be in unanimous agreement on what the speed limit is. This is the ideal situation.
- In practice, perhaps the above requirement can be relaxed to stipulate that any non-trivial ambiguity must be resolved with a certain time or distance threshold whose value is location-specific
- Different types of speed limit signs are possible - they may be time-dependent (day/night) as well as location-dependent (school, bridge, tunnel) - Speed limit values can be fixed or variable ("smart roads"). This requirement can potentially inject a dynamic optimization aspect into the problem.

Perhaps a greedy method may be sufficient to generate a good answer to the (fixed value) speed-limit signpost location problem. There are of course, numerous other related location optimization problems on road networks:
locating advertisement hoardings, direction/information signs, emergency vehicles, detours, etc. All these models appear to be well studied in the literature. The proliferation of smart-phones can also have an impact on future 'smart road network' design in general. All in all, location, location, location science continues to be a very interesting sub-area of Operations Research.