Assortment Optimization for Boundedly Rational Customers
M. Farhani a,∗ , C.G. Chorus b , and Y. Maknoonc
a,b,c Faculty of Technology, Policy and Management, Delft University of Technology, Netherlands
a M.Farhani@tudelft.nl, b C.G.Chorus@tudelft.nl, c M.Y.Maknoon@tudelft.nl∗ Corresponding author

Extended abstract submitted for presentation at the 11th Triennial Symposium on
Transportation Analysis conference (TRISTAN XI)
June 19-25, 2022, Mauritius Island

Keywords: Assortment optimization, bounded rationality, decoy effects, mobility on-demand
services.

1

INTRODUCTION

In the last decade, app-based mobility services have received considerable attention as an interface
between the service provider and the customer. The foundation of the app-based system isbased on the real-time interaction between the booking system and the user’s choice. Upon
customers’ arrival, a request is submitted to the platform. The platform then offers a travelmenu (i.e., assortment) to the user to choose from. Inferring customer preferences and responding
accordingly plays a vital role in app-based mobility services. One of the central decisions is whichalternatives to include in the list of offered assortment to each arriving passenger.
Assortment optimization relies on choice models to estimate individuals’ behavior. It is
initiated by van Ryazin and Mahajan in their seminal paper which uses the multinomial logit(MNL) model for modeling customer behavior (Ryzin & Mahajan (1999)). It then developed forvarious choice models most of which assume that customers are rational agents. However, studies
in marketing, economics, psychology (e.g., Simon (1957) and Hauser (1978)), and transportation(Di & Liu (2016)) have revealed that individuals’ decisions may deviate from perfect rationality
due to their cognitive limitations and biases. Such limitations and biases result in surprisingoutcomes which are known as boundedly rational/ irrational behaviors.
There are some cognitive assumptions that can account for humans’ bounded rationality,
such as reference-dependency of preferences, decoy effects, and lack of information or attention.
The theory of reference dependency, pioneered by Tversky & Kahneman (1991), denotes that areference point influences individuals’ preferences between given alternatives. This phenomenon
can explain cognitive biases in many choice situations, like transportation (Van de Kaa (2010)).
Decoy effects, first studied by Huber et al. (1982), are choice reversals caused by the compositionof choice sets. Such effects occur when adding a new alternative (i.e., decoy alternative) ina choice-set may increase the probability of choosing one of the former options (i.e., target
alternative). Decoy effects have been acknowledged in various choice contexts (Rooderkerk et al.
(2011)).
Despite the evolution of assortment optimization studies, bounded rationality is still largely
overlooked in this research area. Most of the assortment optimization studies use the notion of
consideration sets, proposed by Simon (1957), to capture humans’ bounded rationality causedby their lack of information or attention. Reference-dependency of preferences and choice-set
composition effects have been scarcely integrated into assortment planning frameworks.
TRISTAN XI Symposium

Original abstract submittal

M. Farhani, C.G. Chorus, and Y. Maknoon

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This study incorporates reference-dependency of preferences and decoy effects in the assortment optimization model using the Random Regret Minimization model. We propose a greedy
algorithm that is computationally tractable to find the optimal assortment when customers’ behavior follows RRM. We have tested our algorithm for micromobility services. The results showthat our proposed algorithm can find the optimal solution for all studied instances. Moreover, we
compare the planned assortments against the widely used multinomial logit model to examine
the effects of reference-dependency and choice set-dependency on the assortment decisions. Our
results indicate that these behavioral phenomena have significant impacts on the optimal choiceset, so they need to be taken into account by those who want to offer a menu of options to theircustomers.

2

METHODOLOGY

In this research, we employ the Generalized Random Regret Minimization (G-RRM) model
proposed by Chorus (2014) to model customer choice behavior. We aim to select the optimal
assortment of alternatives from the given universal set Ω including N alternatives so that the
expected profit per customer is maximized. In our problem setting, alternative i is defined by
the bundle of M attributes (xi1 , ..., xiM ). The random regret of alternative i is composed outof a systematic regret Ri and an i.i.d. random error ϵi , RRi = Ri + ϵi . The systematic regretof alternative i is defined as the sum of the binary regrets that are associated with bilaterallycomparing the attributes of alternative i with each of the other alternatives in the choice set.
Thus, the systematic regret of alternative i in assortment s is written as below:
X

Ri (s) =MXln γ + exp [βm .(xjm − xim )] − ln(1 + γ)(1)j∈s, j̸=i m=1

Where βm and γ denote the weight of attribute m and the regret weight, respectively. Also,
the choice probability of alternative i is defined by Equation (2), where V0 denotes the constantattraction value of the no-purchase option.
πi (s) =exp (−Ri (s))
P
V0 + j∈s exp (−Rj (s))(2)

Let pi denote the associated profit of alternative i. Therefore, the assortment optimization
problem is formulated as follows.
Xmax
pi .πi (s)
(3)
s⊆Ω

i∈s

The RRM model (1) is a reference-dependent choice model in which each offered alternativeacts as a reference point for other options. It is also a convex function that is steeper in the loss
than in the gain domain, thus (1) represents semi-compensatory and loss-aversion behavior. Lossaversion indicates that people are more sensitive to losses than gains, and semi-compensatory
behavior suggests that the poor performance of an alternative in terms of some attributes can not
be completely compensated with the good performance of other equally important attributes.
Besides, the reference-dependent convex regret function enables the RRM model to predict decoy
effects (See Chorus (2014) and Guevara & Fukushi (2016) for more details).

2.1

Solution method

Each optimization on Problem (3) could require the evaluation of all potential subsets of theuniversal set. Generally, assortment optimization is a combinatorial problem. It has been proventhat the unconstrained assortment optimization under either MNL or NL choice model when
dissimilarity parameters change in (0 1] (Davis et al. (2014)) can be solved in polynomial time.
TRISTAN XI Symposium

Original abstract submittal

M. Farhani, C.G. Chorus, and Y. Maknoon

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Otherwise, the assortment optimization is an NP-hard problem. We propose a greedy algorithm
to solve Problem (3). Algorithm 1 represents the pseudo-code of the proposed algorithm.
Algorithm 1 Greedy Algorithm for the Assortment Optimization Problem
Initialization: Set S := {s ⊆ Ω; |s| = 2}, s∗ := ∅, O∗ := 0
for s ∈ S P
do
Os := i∈s pi .πi (s)while |s| ≤ N do P
k := argmax
pi .πi (s)j∈Ω\s i∈s∪{j}POk :=pi .πi (s)i∈s∪{k}if Ok ≥ Os thens ← s ∪ {k}Os ← Okelse
break;
end ifend while
if Os ≥ O∗ thens∗ ← sO∗ ← Osend ifend forreturn s∗

3

RESULTS AND DISCUSSION

The described methodology is coded with Python. As our running example, we consider an
online vehicle-sharing service that provides two types of transportation services: (i) bike (s = 1)
and (ii) scooter (s = 2). In this problem, alternative i is defined by three attributes: service type(si ), the walking distance to the pick-up location (di ), and associated price (pi ). We assume that
there is a universal set including ten potential options as described in Table (1). The service
provider offers each customer a set of alternatives so that the expected revenue per request is
maximized.
Table 1 – Universal set of potential alternatives.
No.
Service type (s)Walking distance (d)Price (p)

1
1
100
25

2
1
110
24

3
2
120
25

4
1
130
18

5
1
135
17

6
2
140
17

7
2
145
16

8
1
150
12

9
2
155
19

10
2
300
20

To examine the proposed algorithm’s performance and compare the RRM and MNL models,
we generated multiple scenarios based on the parameters of the regret and utility functions in
the RRM and MNL models. The regret weight (γ), weights of the service type (βs ), walking
distance (βd ), and price (βp ) are defined in Table (3)). We employ Equation (1) for the RRM
model. For the MNL model, we assume that the deterministic utility of alternative i is definedby ui = βs .si + βd .di + βp .pi . We presume that the attraction value of the no-purchase option
under both RRM and MNL models is 1.
TRISTAN XI Symposium

Original abstract submittal

M. Farhani, C.G. Chorus, and Y. Maknoon

4∗∗
Table (2) shows the optimal assortment under the RRM (SRRM) and MNL (SMN L ) models as∗well as the proposed assortment by our algorithm (SAlg ). The expected revenue of the proposed∗∗∗
assortments by the RRM (RRRM), MNL (RM
N L ), and algorithm (RAlg ) are also shown by
this table. Under all scenarios, our proposed algorithm finds the optimal assortment, which is
obtained by the complete enumeration.

Table 2 – Optimal RRM and MNL assortments for different scenarios.
No.
1
1.1
1.2
1.3
2
2.1
2.2
2.3
3
3.1
3.2
3.3

γ
0
0
0
0
0.5
0.5
0.5
0.5
1
1
1
1

βs
0.5
10
0.5
0.5
0.5
10
0.5
0.5
0.5
10
0.5
0.5

βd
-0.9
-0.9
-10
-0.9
-0.9
-0.9
-10
-0.9
-0.9
-0.9
-10
-0.9

βp
-2.9
-2.9
-2.9
-10
-2.9
-2.9
-2.9
-10
-2.9
-2.9
-2.9
-10

∗
∗
SRRM
& SAlg{1,2,3,10}{1,2,3,4,5}{1,10}{1,2,3,10}{1,2,3,10}{1,2,3,4,5}{1,2,3,10}{1,2,3}{1,2,3,10}{1,2,3,4,5}{1,2,3,9,10}{1,2,3}∗∗
RRRM& RAlg
18.67
24.98
24.93
18.86
18.61
23.51
21.36
18.46
18.62
21.40
20.43
18.43

∗
SM
NL
{1,2,3,4,5,6,9,10}{1,2,3,4,9,10}{1,2,3,9,10}{1,2,3,4,5,6,9,10}{1,2,3,4,5,6,9,10}{1,2,3,4,9,10}{1,2,3,9,10}{1,2,3,4,5,6,9,10}{1,2,3,4,5,6,9,10}{1,2,3,4,9,10}{1,2,3,9,10}{1,2,3,4,5,6,9,10}

∗
RM
NL
16.13
17.04
18.56
16.08
16.13
17.04
18.56
16.08
16.13
17.04
18.56
16.08

We use this example to show that the RRM model can replicate the decoy effect, resulting
from the reference- and choice set- dependency of preferences. To this end, we consider the
optimal RRM assortment for Scenario 3.2 and calculate choice probabilities of the offered alternatives in the absence and presence of alternative 10. Let s1 = {1, 2, 3, 9} and s2 = {1, 2, 3, 9, 10},
Table (3) denotes the choice probabilities of the offered alternatives for s1 and s2 . Following the2)
proposed approach by Guevara & Fukushi (2016), we define θi = ππii (s(s1 ) for alternative i to assess
the impact of alternative 10 on market share of alternative i. If θi > 1 then alternative 10 is adecoy for alternative i.
Table 3 – Decoy effect in the optimal RRM assortment for Scenario 3.2

πi (s1 )πi (s2 )(s2 )θi = ππii (s1)

410.2718
0.272
1.003

2
0.228
0.239
1.048

3
0.189
0.209
1.102

9
0.099
0.141
1.422

10
0.0002-

CONCLUSIONSThis study develops an assortment optimization framework that incorporates humans’ bounded
rationality. The proposed bounded rational framework leads to different assortment and purchasedecisions made by the supply and demand sides, respectively. We explain these differences inlight of the different behavioral assumptions of our model. We propose a greedy algorithm thatcan solve the problem. In the next phases of this research, we will apply the algorithm to bigger
choice sets and elaborate on mathematical proofs.
TRISTAN XI Symposium

Original abstract submittal

M. Farhani, C.G. Chorus, and Y. Maknoon

5

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TRISTAN XI Symposium

Original abstract submittal