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Grocery Store Design
Victor Brown Barbara Gelpi Shane Griffith Courtenay Mayes
Carrie Perkins
April 25, 2005
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Abstract
Donald Knuth, a well-known person in computer science, used graph theory to design his kitchen. While
utilizing graph theory for his personal challenge, he discovered that the garbage can is a highly desirable
neighbor node; apparently everything led back to and wanted to be next to the trashcan!
Fortunately in our application of graph theory in creating a newly designed floor plan for you, we did not
have this challenge. Everything in our recommended floor plan leads to both increased revenues, for you,
our client, and to smiles on your valued customers faces. However, we did follow in his footsteps in a similar
way. We applied modern graph theory to design a more efficient grocery store. To design this grocery store,
two different applications of graph theory were used to find a shortest path for must-have grocery items:
Floyds algorithm, and a Heuristic algorithm. These must-have items have nearly 100% penetration in
both your existing store and our recommended store. Both premium priced items and impulse buying items
were placed along the calculated shortest path in our well-organized store. After comparing and contrasting
our suggested store to your existing store, we are confident that our store is a superior design and is highly
worthy of your consideration!
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1 The Challenge
Our challenge is as follows: to meet the requirements of the February 1, 2005, RFP of Gleasons Grocers
to create a grocery store floor plan to increase profit while meeting the needs of customers. Our researchshows each week consumers spend an average of 1.5 hours grocery shopping during 2.5 trips to the grocery
store. Eighty-five percent (85%) of their purchases are to replenish staple items. Consumers have identified
quick and efficient shopping as one of the most desired traits in their evaluation of grocery stores. Degree
of loyalty is affected by shoppers evaluation of stores and we wish to design a floor plan to appeal to both
Large Basket Shoppers (LBS) and Frequent Shoppers (FS).
1.1 The Reasoning
Eighty-one percent (81%) of shoppers come into a grocery store with either a physical or mental shopping
list. Lets assume that every shopper wishes to find and purchase those items in as little time as possible.
In effect, taking the shortest path from the store entrance to the desired items and back to the entrance.
In our research, we identified milk, meat, paper products, frozen foods, and breads/cereals as the most
popular must-have items on a customers list. Additionally, most shoppers will buy items not included on
their list. For all intent and purposes, we will call these items impulse buys. Thinking logically, one may
perhaps conclude that the more items that a customer passes while shopping for must-haves, the more
impulse buys a customer will make. However, the Law of Diminishing Returns comes into play here. Forcing
exhausted customers past all items in a store is a negative experience which would prevent return visits.
Store design must strike a balance to invite return visits while maximizing the opportunities for impulse buys.
Considering this reasoning, our recommended floor plan has as many items as possible along the shortest
path. One may think that placing must-haves in close proximity to one another would be optimal. There
are several problems with this theory. One problem is: if the must-have items are in close proximity to
one another, there will be little to no impulse buying. Although this may be acceptable to the customer, it
would have minimal benefit to you, the store owner. Another problem is: some must-have items need to
be placed in designated areas. For example, it is preferable that milk and meats are placed in the rear or on
the side of stores in refrigeration units due to the electricity costs for running the units and accessibility for
stocking. Our research also shows a consumer trend to category rather than product purchasing. This
means a shoppers list will tend to think more of a meal rather than the individual components of the meal.
Grouping products by category, as our floor plan does, allows flexibility to meet the needs of both category
and product shoppers.
1.2 The Set-Up
Using graph theory, along with the considerations mentioned above, we designed what we believe to be a
new and improved grocery store floor plan. After careful consideration, we determined that the vertices of
this plan would include the end of the shelves and the middle of an aisle. If a customer is at the end of a
shelf and another shelf is next to the customer (across the aisle), this scenario represents the same vertex.
The edges include any aisle that takes a customer from one vertex to the next. After defining a vertex and
an edge, graph theory was utilized to calculate and determine a shortest path to our must-have items.
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2 Shortest Path Calculation Methodology
To calculate the shortest path from the entrance to our five items and back to the entrance, we employed a
two-step strategy. The first step applied Floyds Algorithm to determine the shortest path between all pairsof vertices. After calculating this shortest path, a heuristic algorithm involving the six vertices (entrance
plus five) was utilized calculating the shortest circuit.
2.1 Floyds Algorithm
We created an adjacency matrix representing the distance in feet between each of the connected vertices and
labeled this matrix as M(0). If vertex i was not connected to vertex j, then we entered for M(0)i,j .
To calculate M(1)i,j, we determined the shortest path from vertex i to vertex j allowing the vertex 1 to
be used as an intermediate step if necessary. Then, for M(2)i,j , we determined the shortest path from vertex
i to vertex j allowing vertex 1 or vertex 2 or both to be used as intermediate steps. In general, M(x)i,j
represented the shortest path from vertex i to vertex j allowing any vertex or combination of vertices from 1
to x to be used as intermediate steps. If our graph had n vertices, then the entries ofM(n) would represent
the shortest distance for each pair of vertices using any vertex (or vertices) as intermediate steps. Each
calculation was performed as follows:
M(x)i,j = min[(M(x 1)i,j), (M(x 1)i,x + M(x 1)x,j)].
We completed this calculation n3 times to find M(n) [1].
The algorithm describing this process is shown below [8] .
1 n rows[W]
2 D(0) W
3 for k 1 to n
4 do for i 1 to n
5 do for j 1 to n
6 do d(k)ij min(d
(k1)ij , d
(k1)ik + d
(k1)kj )
7 return D(n)
For W = (wij), where
Wij =
0 if i = j,
the weight of the directed edge (i, j) if i = j and (i, j) E,
if i = j and (i, j) / E.
2.2 Heuristic Algorithm
Now that we know the shortest distance between any pair of vertices, we created a new adjacency matrix
which only includes six vertices (the entrance and our five must-have items). To calculate the shortest circuit
through the store beginning at the entrance, we applied a heuristic algorithm. In this algorithm, we simply
selected the shortest path to an unvisited vertex in the matrix. It is not possible to prove that this is the
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shortest circuit, but we believe it is reasonable to expect that a shopper would select a circuit in the same
manner.
3 Current Floor Plan
For the existing store, we used a model of a Kroger store that is located in Eastern Kentucky. We chose this
store due to its commonality in layout and placement of stock.
The must-have items and the entrance are positioned as follows:
Entrance - Vertex 1
Meat - Vertex 15
Milk - Vertex 20
Paper Products - Vertex 23
Breads/Cereals - Vertex 45
Frozen Foods - Vertex 49
Having applied the above mentioned algorithms to this floor plan, we found the following adjacency
matrix. This matrix shows us the distance in feet from any of our must-have items or the entrance to any
other of these vertices.
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Adjacency Matrix for Heuristic Algorithm
Our Current Floor Plan
1 15 20 23 45 49
1 0 162 263 209 105 174
15 162 0 115 247 66 112
20 263 115 0 132 158 112
23 209 247 132 0 213 117
45 105 66 158 213 0 178
49 174 112 112 117 178 0
Using the above methods, the shortest path beginning and returning to the entrance (Vertex 1) passing
through our five important items is:
1 44 45 15 16 17 49 18 19 20 21 22 23 59 58 52 50 48
46 44 1This path would require a customer to walk 736 feet with 14 intermediate vertices.
4 Our Recommended Floor Plan
In the new store, we placed our must-haves in locations around the store that would be cost-efficient and
convenient for the owners and employees of the store.
We positioned the must-have items and entrance as follows:
Entrance - Vertex 1
Milk - Vertex 15
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Meat - Vertex 18
Paper Products - Vertex 32
Frozen Foods - Vertex 38
Breads/Cereals - Vertex 40
This positioning of the items in relation to the other items and the entrance produced this adjacency
matrix for our recommended floor plan.
Adjacency Matrix for Heuristic Algorithm
Our Recommended Floor Plan
1 15 18 32 38 40
1 0 245 174 143 174 174
15 245 0 84 169 112 155
18 174 84 0 85 60 188
32 143 169 85 0 113 122
38 174 112 60 113 0 135
40 174 155 188 122 135 0
Using the above methods, the shortest path beginning and returning to the entrance (Vertex 1) passing
through our five important items is:
1 25 28 31 32 19 18 37 38 41 17 16 15 50 49 46 43 40
36 34 31 28 25 1
This path would require a customer to walk 728 feet with 17 intermediate vertices.
5 Summary
So what does all of this information mean to you?
Our design saves you money NOW!
Existing store design 277 feet by 210 feet = 58,170 square feet
Proposed store design 240 feet by 240 feet = 57,600 square feet
At the current construction cost of $250 per square foot, this difference of 570 square feet translates to aninstant savings of $142,500 per store!
Our design increases your revenue potential!
A list shopper (81% of your clients), , following the shortest path to the must-have items, passes 14
opportunities for impulse buying in your existing store. In our proposed design, the same customer passes
17 opportunities for impulse buying. This represents a more than 20% increase in revenue potential!
Our design keeps your customers happy!
A no-list shopper (19% of your clients) may browse the category-style floor-plan, providing the enriching
experience our research shows consumers are increasingly demanding. Additionally, there is sufficient aisle
space for strategically placed wings and shoppers or off-shelf displays (convenient for both category shopping
and increased revenue.)
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6 The Solution
Our design is practical.
Save money on construction while increasing revenue potential. Our mathematical design is innovative
and forward thinking.
The computerized model presented by Green Apple Consulting, Incorporated is one step away from
PDA usage on every shopping cart. Gleasons Grocers will lead the country in innovation, design and
customer satisfaction.
PDA usage is one-step away from the potential of RFID technology to track Inventory and provide
quicker check-out lines for Enterprise Resource Planning (ERP).
As our partnership progresses to its next stages, our advanced mathematical thinking will help us
re-define category decision trees and cross category purchasing dynamics. Our design is consumer-oriented.
Category shopping is on the rise (from 10 to 20% in the last year) and our flexible floor plan accom-
modates this consumer trend.
Consumers are increasingly looking to shopping for food to be an enriching experience and our flexible
floor plan easily leads to the next step of food-centric islands in the store.
The straight-forward, dynamic floor plan easily accommodates the new shoppers of tomorrow - men
and children of single parents.
Green Apple Consulting, Incorporated- Creating global systems while accommodating local differences.
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References
[1] Floyds algorithm. http://faculty.leeu.edu/ sgriffith/Downloads/Project/floyd.ppt.
[2] Grocery shopping becomes more spontaneous. http://igd.com/CIR.asp?menuid=33&cirid=1535.
[3] Grocery shopping with a wireless pda. http://gtreseachnews.gatech.edu/reshor/rh-ss03/pda.html.
[4] Module 8: Graphs, part II. http://www.seas.gwu.edu/ simhaweb/cs151/lectures/module8/module8.html.
[5] Path problems in directed graphs. http://www.csse.monash.edu.au/ lloyd/tildeAlgDS/Graph/Directed/.
[6] Shopping in the future: an enriching experience.
http://www.accenture.com/xd/xd.asp?it=afweb&xd=locations%5Cnews%5.
[7] Shortest path problems. http://cs.engr.uky.edu/ lewis/cs-heuristic/text/dynamic/shortest.html.
[8] Thomas H. Cormen et al. Introduction to Algorithms, chapter 25. Prentice-Hall, second edition, 2001.
[9] Mr. Gene King, Super Kroger Manager, Winchester, KY. Personal Interviews, February 2005.
[10] Mr. Michael Eddy, Assistant Manager, The Kroger Co., Clarksburg, West Virginia. Personal Interviews,
March 2005.
[11] Richard Neapolitan et al. Foundations of Algorithms, chapter 3. Jones and Bartlett Publishers, 1997.
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Appendix
The appendix contains full-size CAD drawings of both floor plans and matrices used when making calcula-
tions with Floyds Algorithm.
A full-size CAD drawing of the current floor plan.
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A full-size CAD drawing of the recommended floor plan.
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An adjacency matrix of the current floor plan made by calculations of Floyds Algorithm.
Ex
isting
Floorp
lan
Adjancency
Ma
trix
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
4
4
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
1
0
39
32
30
30
39
3
9
2
39
0
19
15
3
19
0
19
4
32
15
0
26
39
39
5
19
26
0
23
6
39
23
0
23
7
23
0
23
8
23
0
41
41
9
41
0
41
10
41
0
41
11
41
41
0
30
12
30
0
21
26
13
21
0
21
13
13
14
26
21
0
23
66
15
23
0
23
66
66
16
23
0
23
66
66
17
23
0
23
66
66
18
23
0
23
66
66
19
23
0
23
66
26
20
23
0
23
26
26
21
23
0
38
26
22
38
0
71
23
71
0
62
32
24
62
0
36
47
36
25
36
0
13
39
36
47
26
13
0
38
45
36
39
27
38
0
38
56
38
38
56
28
38
0
19
4
1
36
45
29
30
19
0
43
3
6
41
30
30
43
0
17
66
2
3
31
39
39
17
0
15
15
32
15
0
13
23
33
15
0
13
23
34
13
13
0
13
13
35
23
13
0
15
30
36
23
13
0
15
30
37
15
15
0
15
15
38
30
15
0
15
30
39
30
15
0
15
30
40
15
15
0
15
15
41
13
30
15
0
42
13
30
15
0
43
66
66
66
0
6
6
44
39
41
36
23
66
0
66
23
45
66
66
6
6
0
66
46
56
36
41
2
3
66
0
66
23
47
66
66
66
0
66
48
38
45
23
66
0
66
23
49
66
66
66
0
66
50
45
38
23
66
0
66
23
51
66
66
66
0
66
52
39
36
56
23
66
0
41
23
53
41
0
68
41
41
54
26
26
68
0
32
55
41
32
0
32
41
56
26
26
32
0
68
57
41
68
0
41
41
58
47
36
39
23
41
41
0
23
59
32
36
47
41
23
0
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An adjacency matrix of the recommended floor plan made by calculations of Floyds Algorithm.
ProposedFloorplan
AdjacencyMatrix
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
2
6
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
1
0
64
64
45
45
49
2
64
0
26
24
23
3
26
0
26
38
30
4
26
0
26
45
38
5
26
0
32
53
30
6
32
0
32
30
30
7
32
0
26
30
56
8
26
0
26
38
45
9
26
0
26
30
30
10
26
0
23
23
19
11
23
0
26
38
15
12
26
0
26
30
26
26
13
26
0
23
38
15
14
23
0
26
23
19
15
26
0
26
30
30
16
26
0
26
38
45
17
26
0
32
30
56
18
32
0
32
30
30
19
32
0
26
53
30
20
26
0
26
45
38
21
26
0
26
3
8
30
22
64
26
0
24
2
3
23
45
24
0
53
30
24
45
24
53
0
30
25
49
30
30
0
3
2
32
19
26
38
23
32
0
64
34
27
23
38
32
6
4
0
34
28
19
3
4
34
0
45
45
19
29
45
30
45
0
86
45
30
30
45
45
86
0
45
31
19
45
45
0
56
56
19
32
53
38
56
0
105
54
33
38
53
56
105
0
54
34
19
54
54
0
38
38
26
35
38
0
75
53
30
38
36
38
75
0
53
30
38
37
30
30
53
0
30
38
30
30
0
30
30
39
30
30
53
0
30
40
30
30
0
30
30
41
30
30
30
0
53
42
30
53
0
75
38
38
43
30
75
0
53
38
38
44
30
30
30
53
0
45
26
38
38
38
38
0
26
46
38
38
26
0
49
49
19
47
38
56
49
0
105
49
48
56
38
49
105
0
49
49
19
49
49
0
38
38
19
50
30
45
38
0
75
38
51
45
30
38
75
0
38
52
19
38
38
0
26
26
19
53
23
30
26
0
56
30
54
30
23
26
56
0
30
55
38
30
38
19
30
30
0
23
23
56
26
15
19
23
0
36
57
19
15
26
23
36
0
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