Division of Research Graduate School of Business Administration University of Michigan February 1973 ESTABLISHING A RETAIL CHAIN STORE BRANCH: A PERT APPLICATION Working Paper No. 71 by Nesa Labbe4Ju Assistant Professor of Statistics University of Michigan FOR DISCUSSION PURPOSES ONLY None of this material is to be quoted or reproduced without the express permission of the Division of Research.

12KG 'P PEs ~ BACKGROUND This paper is the product of research done by the author during the late fall of 1971.' The write up was postponed until the final results were obtained so that the predictions could be evaluated and the implementation observed. This is a true case; therefore, on request of the management.of the chain store, names, exact times, and network details are omitted. ABSTRACT This paper illustrates how a quantitative scheduling approach, PERT, was successfully used in the evaluation of the establishment of a retail store branch prior to its construction. Important questions such as job sequencing, critical jobs, and approximate starting time for construction were successfully answered by means of the PERT model.

"; CONTENTS Introduction Setting up the PERT network BUCKET Algorithm for Generating the Critical Path 1 1 4

TABLES 1. PERT Activities for Establishing a Chain Store 8-9 2. Summary of Results Using the Triangular Distribution in Generating Activity Times 10 3. Sample Output of the PERT Analysis 11-12 4. Observed Critical Activities for the Establishment of the Retail Store 13

FIGURES 1. PERT network for establishing a chain store. (Activity times are assumed to be Beta distributed, and their expected times are used to establish the critical path.) 2. PERT network for establishing a chain store, (Activity times are assumed to be triangular distributed and are generated.) 14 15

Introduction The establishment of a retail store is not a routine procedure. It is a very time-consuming and challenging responsibility for marketing experts. The degree of difficulty of this challenge depends on a host of factors,the most important of which is past experience. The specialists who are involved with the evaluation of the potential establishment of a new branch for a chain store encounter economic and time factors which are very complex and difficult to describe. This paper illustrates how a quantitative scheduling approach, PERT, was successfully used in the evaluation of the establishment of a retail store branch prior to its construction. Important questions such as job sequencing, critical jobs, approximate starting time for construction, and others were successfully answered by means of the model. This is a true case; therefore, on request of the management of the chain store, names, exact times, and network details are omitted. Setting Up the PERT Network The PERT method is a management-oriented technique for the scheduling of a series of nonrepetitive jobs so that the entire project can be completed in the shortest amount of time, The PERT method, as it relates to the establishment of a chain store branch, serves to consider a series of interrelated activities in order to determine their optimum planning and scheduling for the attainment of the desired goals.

-2 - The preliminary step before executing the PERT algorithm- is to develop the network model, which represents the interrelationships between activities and the general flow of these activities. (See Table 1 and Figures 1 and 2.) This development is accomplished by the following four-step procedure: Step 1: Development of the logical sequence in which the activities must be performed to complete the project. An-activity is the performance of a specific task and involves the effort of manpower. Step 2: Drawing the arrow-diagram network, using activities and events. An event is represented by a node, which is the occurrence of a checkpoint in time. The node represents the accomplishment of a task rather than the performance of a task or activity. The event represents the start of activity(s) or the completion of activity(s). Step 3: Assignment of activity times. Qualified people who'had dealt in the past with the establishment of retail stores supplied three typical activity times: optimistic time (OT), most probable time (MPT), and most pessimistic time (PT). Step 4: Numbering of nodes so that (1) each job has a unique set of nodes, (2) all activities entering a node have identical followers, (3)all activities leaving a node have identical predecessors, (4)a node represents the complete relationship between all entering and exiting jobs, and (5) the nodes are in ordered

— 3 — sequence with the beginning node smaller than the ending node. Figures 1 and 2 represent the simplified'version of the final network that was developed for the chain store. But some analysis of relations between activities is necessary. Originally, the'activity from node 9 to 37 (analysis of general labor conditions.) was a predecessor to the activities from node 17 to node 26. (architectural analysis of space and design requirements for the entire.... outtet) and from node 22 to node 31 (preparation of advertising budget.'and strategy), After a lengthy discussion, however, it was agreed that these'three activities were unrelated for the principal reason that pr'exvailing pay scales were not part of the activity of the "analysis of general labor conditions". It was also agreed that the departmental estimation of gross margins (activity from node 9 to node 14) did not depend on the analysis of number, policies, and location of wholesalers (activity from node 12 to node 14) since the demand for most of the product-s under consideration was fairly high and since it was known that wholesalers did not charge high prices. The company already possessed- a' record of the wholesalers and their policies in that location. It also seemed logical.to assign selling areas to departments (activity from node 15 to node 17) only after the products to be sold were selected (activity from node 14 to node 15) because different products require both different areas and different locations in the store.. For exaple, jewelry, candy, paperback books, and the kinds of goods conducive to impulse buying are often located near doors so that people passing in and out can be exposed to them. Because few people will buy furniture or appliances on an

~~~ ~ i -4 -impulse basis, these types of goods can be buried:ri the heart of the. department store, several floors up. A nondetailed listing (including the.necessary dutnmy activities for Step 4)3: of all activities is given in Table 1. BUCKET Algorithm for Generating the Critical PathA simple algorithm, called a "BUCKET"' algorithm"WaSwrdevised and programmed in FORTRAN to define the critical path and "the-^iearly and late starting times for the activities. 4 detailed write up and listing of the program is summarizedas- follows: N.L. Wu, Business Programming in FORTRN IV (Dubuque, Iowa: C. Brown Company, 1973) Wm.

-5 -Initially, the classical Beta distribution,was considered for activity times, and the expected times were used for scheduling and for developing the critical path. (Results are shown in Figure 1.) However, a statistical analysis of past data collected by the firm made it clear that this approach might lead to-an underestimate of the due date. A Chi-Square test indicated that a triangular distribution better fit the activity times; therefore, 100 triangular variates were generated for each activity, given the OT, MPT, and PT times of each activity, and consequently 100 due dates and critical paths were generated. These results are summarized in Table 2o The early start (ES) and early finish (EF) times -are-b7-tained through the BUCKET method as follows. The early start time is that time at which an event can be started if all events prceeding it have been started and completed as early as-possiblec The Forward Scan Rule as outlined here is the simplest way of calculating the ES for all activities Give each node a BUCKET value: BUCKET (I) Initialize these BUCKET(I)s to zero Now for each activity K proceed as follows: BUCKET (JNODE) '- BUCKET (INODE)+TIME (INODE, JNODE) if and only if BUCKET(JNODE)' > BUCKET(JNODE) ES (K) =BUCKET (INODE) The latest finish (LF) time is that at which an event must be accomplished in order to avoid slippage in the project-completion date. For our retail store the project-completion date was

-6 -determined by the early finish time of the last activity. The Reverse Scan Rule as outlined here is the simplest way of calculating the LF for all activities: Give each node a BUCKET value: BUCKET(I) Initialize the BUCKET(I)s to the project-completion date Now for each activity K, starting with the last activity, proceed as follows: BUCKET(INODE) '-BUCKET (JNODE)-TIME(INODE, JNODE) if and only if BUCKET (INODE)' BUCKET(INODE) LF (K) =BUCKET (JNODE)' After all early start and late finish times are calculated-for each activity (K), the early finish (EF) and late start (LS) for these activities can be obtained as follows: EF (K) =ES (K) +TIME (K) LS (K) =LF.(K)-TIME (K) The total slack time (TOTSL) for each activity (K) is the difference between the early start and the late start of the activity; TOTSL indicates how long an activity may be postponed without affecting the due date. TOTSL (K) =LS (K)-ES (K) The critical activities are those activities which cannot be delayed in order to finish the job as soon as possible. Therefore, if the total slack of an activity is zero, that activity is a critical activity.

A sample outputput of the PERT analysis for the retail" stor& is given in Table 3. The observed critical activities for' the "case-are listed in Table 4. The output of this atalysis was used by management in determining where slack time would be available and where potential problems might develop. The activities on the critiecal path were carefully scheduled, and, in order to minimize delays, manpower was assigned accordingly.

TABLE 1 PERT Activities for Establishing a Chain Store (All Dummies are Included) Nodes Act t Optimistic Most Probable P I J Time (OT) Tie ( ActivitTy I J Time (OT) Time (MPT) issimisti< ime (PT) 1 2 Dummy 0 0 1 4 Dummy 0 0 1 5 Dummy 0 0 1 6 Dummy O 0 1 7 Dummy 0 0 1 8 Dummy 0 0 1 9 Analysis of retail trade potential 48 64 1 29 Research on availability of credit investigation 8 8 2 29 Research on number, kinds, and practices of financial institutions of the area 8 16 4 9 Analysis of potential customer drawing area 48 64 5 9 Area population analysis 16 24 6 9 Analysis of area-purchasing power 24 32 7 9 Analysis of competitive situation 16 24 8 9 Analysis of area's growth trends 16 24 9 11 Detailed demand analysis by department or product groups 16 48 9 14 Estimation of gross margins by department 32 48 9 37 Analysis of general labor conditions 32 48 11 12 Detailed demand forecast by department or product groups 16 48 12 14 Analysis of number, policies, and locations of wholesalers 0 0 14 15 Selection of product groups to be carried 0 0 15 16 Dummy 0 0 15 17 Assignment of selling areas to departments 64 96 16 17 Assignment of nonselling space for administrative functions 16 32 17 19 Dummy O 0 17 20 Dummy 0 0 17 22 Dummy 0 0 0 0 0 0 i 96 1 8 32 96 32 48 32 32 -96 64 96 96 0 0 0 192 64 0 0 - O

TABLE 1 (Continued) PERT Activities for Establishing a Chain Store (All Dummies are Included) Nodes Optimistic Most Probable iJesA~tivity Time (OT) Time (MPT) 17 23 Dummy 0 0 17 24 Dummy 0 0 -17 26 Architectural analysis of space and design requirements for the entire outlet 416 480 17 30 Preparation of merchandise budget planning 200 256 17 31 Obtain estimates of wages and salaries expenses 1 2 17 36 Setting up the organization structure 16 24 19 31 Estimation of delivery costs and policies 8 16 20 31 Setting pricing policies 0 0 22 31 Preparation of advertising budget and strategy 16 24 23 31 Design layouts for departments 128 192 24 31 Setting of operating-hours policy 1 2 26 27 Dummy 0 0 26 28 Dummy 0 0 26 29 Site-location analysis 48 64 27 29 Estimation of construction costs 8 16 28 29 Estimation of equipment costs 8 16 29 31 Preparation of a capital-budgeting program 24 32 30 31 Establishment of inventory-control system 3 5 31 33 Estimate operating costs. 24 32 31 37 Obtaining bids for the physical plant 8 16 33 35 Dummy 0 33 37 Preparation of proforma income statements for next five years 24 32 35- 37 Preparation of first-year-cash:f-ow 24 32 36 37 Manpower planning 64 128.Pessimlstic Time (PT) 0 0 576 400 3 48 24 0 32 256 3 0 0 96 24 24 96 16 96 24 0 96 96 160 I I I I I r mi I re I I I'' a cY - --- ---------- -: - --- -- Ir ~ I _m

-10 -TABLE 2 Summary of Results Using the Triangular Distribution in Generating Activity Times Distribution and Statistics for 100 Runs Classes Midclass Value Frequency Probability 875-905 890 1.01 905-935 920 3.03 935-965 950 11.11 965-995 980 15.15 995-1025 1010 21.21 1025-1055 1040 20.20 1055-1085 1070 16.16 1085-1115 1100 11.11 1115-1145 1130 1.01 1145-1175 1160 1.01 Smallest simulated duration: 881 time units. Largest simulated duration: 1147 time units. Mean simulated duration:. 1023.2. Standard deviation of the simulated duration: 52.1. Suggested (95% confidence) due date: 1095 time units. Actual due date was-1084 time units.

TABLE 3 Sample Output of the PERT Analysis ACTIVITY GENERATED* ES EF LS LF TOTAL CRIT I J TIME SLACK PATH 1 2 0 0 0 829 829 829 1 4 0 0 0 0 0 0 1 5 0 0 59 59 59 1 6 0 0 0 47 47 47. 1 7 0 0 0 61 61 61 1 8 0 0 0 67 67 67 1 9 53 0 53 31 84 31 1 29 8 0 8 835 843 835 2 29 14 0 14 829 843 829 4 9 84 0 84 0 84 0 5 9 25 0 25 59 84 59 6 9 37 0 37 47 84 47 7 9 23 0 23 61 84 61 8 9 17 0 17 67 84 67 9 11 38 84 122 84 122 * 9 14 37 84 121 148 185 64 9 37 59 84 143 971 1030 887 11 12 63 122 185 122 185 0 12 14 0 185 185 185 185 0 14 15 0 185 185 185 185 0 15 16 0 185 185 249 249 64 15 17 103 185 288 185 288 0 16 17 39 185 224 249 288 64 17 19 0 288 288 891 891 603 17 20 0 288 288 907 907 619 17 22 0 288 288 880 880 592 I -J I

TABLE' 3 (Continued) Sample Output of the PERT Analysis. _ lS; - --....... -.... ~ - - l- ~ -. — _ -. --. ----..... -- _..~__:.... ACTIVITY GENERATED* ES EF LS LF TOTAL CRIT I J TIME SLACK PATH 17 23 0 288 288 703 703 415 17 24 0 288- 288 906 906 618 17 26 482 288 770 288 770 0 * 17 30 378 288 666 523 901 235 17 31 1 288 289 906 907 618 17 36 29 288 317 913 942 625 19 31 16 288 304 891 907 603 20 31 0 288 288 907 907 619 22 31 27 288 315 880 907 592 23 31 204 288 492 703 907 415 24 31 1 288" 289 906 907 618 26 27 0 770 770 829 829 59 26 28 0 770- 770- 829 829 59 26 29 73 70- 843 770 843 0 27 29 14 770' 784 829 843 59 28 29 14 770' 784 829 843 59 29 31 64 843 907 843 907 0 * 30 31 6 666 672 901 907 235 31 33 45 907 952 907 952 0 31 37 22 907 929 1008 1030 101 33 35 0 952 952 968 968 16 33 37 78 952 1030- 952 1030 0 * 35 37 62 952 1014 968 1030 16 36 37 88 317 405 942 1030 625 I N) Activity times are generated using the triangular distribution

-13-..:: -*::004.;.:: ~TABLE 4 - Observed Critical Activities for the Establishment of the Retail Store (Dummy Activities are not Included) Activity Nodes I J Activity Description 4 9 Analysis of potential customer drawing area 9- 11 Detailed demand..analysis' by- department or product groups 11. 12 Detailed demand forecast:.'by department or product groups 12 14 Analysis of number, 'policies, and locations of wholesalers 14 15 Selection of product groups to be carried 15 17 Assignment of selling areas to departments 17 26 Architectural analysis of space and design requirements for the entire outlet 26 29 Site-location analysis 29 31 Preparation of a capital budgeting program 31 33 Estimate of operating costs 33 37 Preparation of proforma income statements for next five years Alternate 35 Critical Pat 37 h' Preparation of first-year cash flow

/ - / / ue iJLc u.LSLTUUtLedana their expected ti'mses areI-usedto establish the criticadi path.;' t =0+4M+P. -.; - at:. ':.: —: ' - ~4~ / / Q' I / / / II q 'S Figure 1. PERT network for establishing a chain store.

Activity t imes are assumed to...' be triangular distributed and: are generated. -' /.1 / / / * - - jw~7J fl* * )# * - -*- i: *.p l, j.i.,; --:,z, t..-;..,S - - --- er;. i.:.:~ +,. -..;;9 _ - ^:..-f -i --, -; P 1 *z4. - D;0w' ~~l-,L s 7,,,:~~~~~~~~~~~~~: _ _O~~ _critiz -Wft __ Mitt- 41.. 0/ I / ES S.. car ocr srVl ^ a, slrnetl.x or 8 m. a..... ELgffl t \ ' el ' - - \. -,en~~~~~~~~~~~~~~.t...:... v...,.-:-: I! n Figure 2. PERT network for establishing a chain store.

-. I ~~~~~~~~~~~~~~~~~~~~~~~~....... — 1-1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -- -l-,''. - -. I I -1. - I 1. -. - -1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0