Article Summary: Heuristic for Assort-Packing and Distribution

This report summarizes an article that explores the application of an algorithm-based decision-making process in the fashion industry, specifically focusing on the assort-packing and distribution problem (APDP). The article identifies the challenge of efficiently distributing a wide variety of fashion products while minimizing lead times and costs. It formulates the APDP as a linear programming problem and presents a heuristic algorithm as a solution approach. The heuristic approach reduces computational complexity by generating pack configurations and selecting the best ones based on the sum of squared differences between pack configuration and article size distributions. The results of this approach, compared to exact and benchmark algorithms, demonstrated its efficiency, particularly for large-scale instances. The article concludes that the proposed algorithm is more efficient for real-world APDP cases, outperforming other approaches.

Running head: ARTICLE SUMMARY 1
Article Summary
Name of Student
Institution affiliation

ARTICLE SUMMARY ` 2
Article Summary
Introduction
Technological advancement in the present world has led to the adoption of algorithm-
based decision-making in fashion industry operations. In the competitive world of the fashion
efficient and logical operations in distribution and effective reverse and forward logistics designs
for fashion commodities. Distribution of a wide variety of fashion products is the challenge
facing the fashion industry with a reduction of distribution lead time of great concern, which
leads to the adoption of the assort-packing where packaging is done considering stores where
they will be distributed. Many operations involved in packaging lengthen distribution lead time,
such as unpacking, repacking, and product assortment. The summary defines the problem,
mathematical model, the approach to solve the problem and the results from the article.
Problem Definition
The distribution of the products, which are varied and involves many operations is the
key problem facing the fashion industry. The adoption of an algorithm to help in the decision
making process is applied in order to minimize the costs while maximizing profits within the
constraints of the company. The assort-packing packing problem is complex as it requires the
consideration of pack configurations, store demands, and factory constraints. The adoption of the
assort-packaging and distribution problem (APDP) to minimize the over- and under-shipments to
stores considering shipment schedule and business requirements by the fashion industry has
enabled to keep up with competitors in the whole industry. The heuristic algorithm is used to
implement the assort-packaging and distribution problem and solve it in a timely manner (Sung

ARTICLE SUMMARY ` 3
& Jang, 2018). The assort-packaging and distribution problem is examined, its basic
optimization formulated together with its complexity analysis, and then its solution using the
heuristic approach determined as discussed in summary.
Mathematical Model
Solving an APDP which is a linear programming problem which is solved systematically
following the three steps; Decision variables which are mathematical symbols representing a
firm’s activity levels, Objective function which is a relationship that is linear in mathematics that
defines the firms’ objectives in accordance to the decision variables and Model constraints. The
model constraints shows relationships (linear) of the firm’s decision variables that represents the
restrictions on the company from the area of operation.
i. The decision variables are S, B and K which represents the stores, different assort-
packing configurations packs and a set of possible box capacities respectively.
ii. Objective function is to minimize over-shipment and under-shipment to stores. It’s
given by;
minimize ∑
s ∈S
∑
s ∈ S
(uis +ois)
iii. Constraints are;
a. Over and under-shipment calculated on comparison of distribution amount and
demand
∑
i ∈I
yib xbs−ois +uis=dis ∀ i∈ Ι , ∀ s ∈ S
b. Possible capacity of the assort-pack

ARTICLE SUMMARY ` 4
c. Ensure each assort-pack has only one possible capacity
d. Limits total number of boxes
e. It sets highest number of boxes to be received by a store
f. It restricts one or more boxes to a store
g. The number of distributed quantity is restricted not to surpass the quantity
produced.
Where;
Ιis items indexed with i
disquantity demand for i
M represents maximum boxes
Q represents maximum boxes received by a store
Pi represents production for i
xbs number of boxes for assort −pack ID b distributed ¿ store x

ARTICLE SUMMARY ` 5
yib number of products for item i∈assort − pack ID b
tbk 1 if assort −pack ID b has its capacity value withk ∧0 otherwise
ois
represents
−shipment of item i ¿ store s
uis represents under−shipment of itemi ¿ store s
Solving the APDP involves again making the non-linear equations (b) and (g) linear by
making adjustments on the equations model. The APDP is NP-Hard, which simply means that it
is a hard problem. In order to solve the NP-hard problem, an NP-complete problem is expressed
from the NP-hard which is a constrained short path problem and them finding a least cost path
from the manufacturer to the store while satisfying the constraints. However, there does not exist
an algorithm that is polynomial-time for a real life APDP thus gives computational intractability
for large APDP hence the need for development of a new approach that gives a good feasible
solution for a large APDP in the shortest time possible.
New approach to solving APDP
Computational issues observed in the basic model, especially for an industry-scale
problem where large possible configurations are needed to be resolved using a new two-phase
heuristic approach. The approach reduces the complexity in computation of the APDP by
decreasing the number of searches on the configurations. The heuristic approach first generates
possible pack configurations based on the way the problem is inherent – the article demand
distribution is known and identical. The approach is based on the conjecture – there is similarity
between the article demand and the size distribution of optimal pack configuration in the general
APDP. The algorithm uses the following steps:

ARTICLE SUMMARY ` 6
1. Create starter configurations that meet constraints due to capacity not exceeding optimal
store demand.
2. Calculation of sum of squared difference (SSD) between pack configuration and article
size distributions of each pack configuration.
3. Categorize the configurations of the packs by the sum items in the pack and select h
configurations with h the smallest sum of squared differences in each group.
4. Create the candidate group with selected pack configurations.
The steps are summarized and implemented in the capture below.
The example shows three groups of possible box capacities, step 4 contains final candidate
solutions of pack configurations where h is 2, two smallest SSDs sum of squared differences in
every group selected and six different pack configurations are found.

ARTICLE SUMMARY ` 7
Results
A solution to the reduced optimization model that identifies the best pack from the
groups of candidate pack configurations and then makes the decision regarding the optimal
allocation to stores is found. The decision variables are the amount of pack configurations
distributed to the store, whether a pack configuration is selected, less or more than, over-
shipment and under-shipment indicated by; xrs, zr, dis ois and uis respectively. The model is shown
below.
minimise ∑
s ∈S
∑
i∈ I
(uis+ois)
Subject to
The solutions to small, medium, and large-scale instances for the new heuristic
approach were compared to those of a basic exact approach and a benchmark algorithm. The

ARTICLE SUMMARY ` 8
three main factors influencing the performance of the algorithms were varied; the distribution of
store-article demand, number of sizes, and the number of stores, (Sung & Jang, 2018). The
outcomes showed that the new heuristic approach found solutions for all instances with the
lowest average CPU time. The benchmarked algorithm did not provide all solutions. At the same
time, the exact approach gave out all solutions, and the concluded results showed that both
algorithms were efficient for small-scale instances as they all provided an average CPU time of
less than one minute. Considering the medium-scale instances, still, the new heuristic approach
gave better than the exact and benchmark algorithm as the exact approach could not give results
within time limits, and the benchmark algorithm gave results with an average CPU time of over
29 minutes.
Conclusion
In conclusion, large-scale instance results suggested that the proposed algorithm is
more efficient than the benchmark algorithm when compared, although they all were better than
the exact approach. With the variation of the variables, it shows that the new approach
outperforms the other previous approaches like the exact and the benchmark algorithm used
earlier. The new approach is efficient for solving APDP in real word cases.

ARTICLE SUMMARY ` 9
References
Sung, S. W., & Jang, Y. J. (2018). Heuristic for the assort-packing and distribution problem in the
fashion apparel industry. International Journal of Production Research, 56(9), 3116-3133.