A typical optimization problem emanating from the actual optimization that is conducted on a routine basis at a leading retail store, Staples, Inc. is used to illustrate the concepts of Simulation-optimization. This is done by solving a transportation problem of 10 cities (a subset of the actual 40-city problem) using optimization principles. The same problem is then solved using probabilistic principles using the total traveled distance between two cities as a random variable (depending upon the route taken). In a typical Simulation-optimization problem, Monte Carlo simulation is used for simulation of random paths. In this paper, an alternate but efficient method is prposed and used based on the probability principles. This is based on the concept of capturing uncertainities ranging from 68.3% to 99.7% in the physical system based on (μ +/- σ, μ +/- 3)σ concept. The results from probabilistic optimization and the corresponding deterministic optimization are the compared. It is found that sumulation-optimization results in savings of cost as compared to traditional optimization

Liu, Laura

Putcha, Chandrasekhar

Putcha, Viswanath

English

European Scientific Journal (European Scientific Institute)

European Scientific Journal   August 2013  edition vol.9, No.23  ISSN: 1857 – 7881 (Print)  e - ISSN 1857- 7431 14  APPLICATION OF PRINCIPLES OF RELIABILITY ANALYSIS TO OPTIMIZATION PROBLEM IN TRANSPORTATION INDUSTRY    Viswanath Putcha Sr Analyst Transportation, Staples Inc., Framingham, MA, USA  Chandrasekhar Putcha, PhD, F.ASCE Professor, Department of Civil and Environmental Engineering,  California State University, Fullerton, USA Laura Liu, PhD Associate Professor, Department of Finance,  California State University, Fullerton, USA   Abstract: A  typical optimization problem emanating from the actual optimization that is conducted on a routine basis at a leading retail store, Staples, Inc. is used to illustrate the concepts of Simulation-optimization. This is done by solving a transportation problem of 10 cities (a subset of the actual 40-city problem) using optimization principles. The same problem is then solved using probabilistic principles using the total traveled distance between two cities as a random variable (depending upon the route taken). In a typical Simulation-optimization problem, Monte Carlo simulation is used for simulation of random paths. In this paper, an alternate but efficient method is prposed and used based on the probability principles. This is based on the concept of capturing uncertainities ranging from 68.3% to 99.7% in the physical system based on (µ +/- σ, µ +/- 3)σ concept. The results from probabilistic optimization and the corresponding deterministic optimization are the compared. It is found that sumulation-optimization results in savings of cost as compared to traditional optimization.  Keywords: Reliability, optimization, Simulation, Random, Transportation  1. Introduction This  paper deals with application of principles of simulation-optimization to problems related to transportation industry. In transportation industry, the main aim is to minimize the cost of transportating various comodities from several sources to several destinations with an with some European Scientific Journal   August 2013  edition vol.9, No.23  ISSN: 1857 – 7881 (Print)  e - ISSN 1857- 7431 15  practical constraints and associated cost for each of the path chosen. This is a typical LPP (Linear Programming problem) associated with transportation industry. The industry chosen for this study is Staples, Inc., a well known retail store with a good reputation. All this is related to deterministic optimization in which all the possible paths and the associated costs are  fixed quantities. In a probabilistic analysis, each path is consideed as a Random variable (RV) as the outcome is uncertain. The solution of a probabilstic analysis problem typically involves use of smulation principles such as Monte Carlo simulation. When these two pinciples (optimization and probability are combined, it leads to what is known as Simulation-Optimization. This paper considers a typical transportation industry problem (a subset of the actual problem used at Staples, Inc.) and solves it as a traditional deterministic optimzation problem as well as simulation-optimization problem (it’s probabilistic conterpart) and the results are compared. Essentially, this paper applies the concepts of reliability analysis to optimization problem with specific application to transportation industry (Putcha and Rao, 1991). 2. Details of JDA software used at Staples, Inc. JDA Software Group, Inc is a leading global provider of supply chain management, merchandising and pricing solutions. Located in Scottsdale, Arizona, JDA empowers more than 6,000 companies of all sizes to make optimal decisions that improve profitability and achieve real results in the discrete and process manufacturing, wholesale distribution, transportation, retail and services industries. JDA Software has acquired many market leaders including i2 Technologies, Manugistics, E3, Intactix and Arthur  (http://en.wikipedia.org/wiki/JDA Software)              At Staples, the Retail distribution team uses the Transportation optimization software provided by JDA. Transportation management and logistics management solutions from JDA Software effectively manage the entire closed-loop transportation process — from long-range strategies and operational planning to day-to-day execution (JDA User Manual, Transportation and LogisticsManagementRelease7.5.2.2, http://www.jda.com/company/display-collateral/pID/492/). 2.1 Key functions of a transportation management module can:  Implement transportation plans driven by consumer demand and replenishment policies   Control costs and streamline productivity  Leverage logistics to support profitable growth  Improve trading partner relationships with effective collaboration   The JDA will give us the least cost carrier assignment based on the information provided. It optimizes the complex transportation problem and give us the comprehensive solution. At Staples, Inc the required information European Scientific Journal   August 2013  edition vol.9, No.23  ISSN: 1857 – 7881 (Print)  e - ISSN 1857- 7431 16  from vendors, carriers, PC miller, inventory at warehouse and warehouse resources allocation for the transportation planning are integrated in the JDA system. Based on the constraints that we provied to the software, it creates optimal loads satisfying our constraints with stop sequence.  2.2 Capabilities of JDA that helps our transportation:  • Provides multi-modal, multi-echelon logistics network design • Optimizes based on actual costs, distances and service levels • Qualifies modal tradeoffs with service level requirements • Honors private fleet and third-party carrier volume requirements • Evaluates multiple business scenarios to determine optimal transportation policies • Provides an optimization engine that is consistent with the daily planning engine to ensure realization of anticipated results • Develops green network strategies • Enables multi-site, inbound, outbound and backhaul load planning • Provides deep support for private fleet and third-party carrier environments • Supports complex, multi-modal planning and optimization • Provides advanced carrier selection and load optimization • Leverages real business and operational constraints for load planning and optimization • Includes optimization for load building and containerization, inventory-sensitive transportation and dock-sensitive routes • Includes a robust, interactive planner workbench • Provides robust rate management and cost allocation • Delivers seamless planning-to-execution transition            Not only just building the loads, its also gives performance measures that enable us to evaluate the transportation cost in different senarios and also to find the progress. The report generated will have the detailed information of the loads and the optimization [3].(This  information is taken from JDA website)  2.3 Problem Formulation of Optimization        The goal of our optimization is to reduce the transportation cost. Allocation of resources is the best way to get the optimization results. There are several constraints that will limit optimization. For exapmle, Some store are feasible with 48 footer trucks, some require lift gate, some stores need to be delivered in perticular time. Satisfying all the constraints JDA will give the results.      Objective: Minimization of outbound transportation cost for Staples.  Min Z = C1* X1 + C2* X2 + C3* X3 …………Cn*Xn  Where C1,C2,C3…Cn are the cost associated with the perticular store  European Scientific Journal   August 2013  edition vol.9, No.23  ISSN: 1857 – 7881 (Print)  e - ISSN 1857- 7431 17   And X1,X2,X3……Xn are the respective distances from the warehouse W to the store   2.3.1 Assumptions/Constraints: 1. A standard 53 footer truck can fit 30 pallets or 2600 cube 2. A standard 48 footer truck can fit 26 pallets  3. Depends on the geographical location of the store, they may require Lift gate at delivery 4. Total number of trucks available in a perticular day is limited and vary each day 5. A driver must take a layover of minimun 10 hours after eleven hours of drive 6. Delivery must be done in the store hours only and they vary from store to store 7. Some stores are have special requirement/equipment 8. No delivery on Sunday 9. Wearhouse is open 24x7 10. Orders are available at 2 am in the morning at wearhouse each day 11.  Some stores require delivery at Crossdock  12.  Special orders for the stores should also be routed with the existing orders 13. No order should left unrouted at any cost 14. Distance between points/stores is calcuated by PC Miller 2.3.2  Actual Example problem :        A typical transportation problem encountered on a particular day (depending on the demand) resulted in the following optimization table (Table 1) using JDA software.  Table 1 Actual results for 40 cities obtained using JDA optimization software Load stops PltDL Miles Manu Cost 00005537826M 4 10 357 $755  00005537830M 3 10 417 $806  00005537833M 4 7 406 $745  00005537837M 4 14 435 $846  00005537841M 4 9 407 $804  00005537845M 4 0 278 $521  00005537847M 4 10 428 $886  00005537851M 5 11 901 $1,697  00005537855M 7 8 780 $1,655  00005537861M 5 6 574 $1,104  00005537866M 4 7 368 $808  00005537870M 5 7 391 $856  00005537875M 4 8 418 $872  00005537879M 5 7 949 $1,801  00005537883M 9 10 1324 $2,444  European Scientific Journal   August 2013  edition vol.9, No.23  ISSN: 1857 – 7881 (Print)  e - ISSN 1857- 7431 18  00005537888M 4 10 445 $917  00005537892M 4 10 362 $656  00005537979M 4 12 251 $801  00005537987M 5 9 249 $822  00005537992M 7 8 408 $858  00005537997M 4 9 188 $672  00005538001M 5 12 396 $862  00005538005M 4 13 195 $684  00005538009M 4 8 247 $761  00005538013M 7 8 557 $1,130  00005538019M 3 15 180 $609  00005538022M 5 9 195 $447  00005538026M 3 11 256 $744  00005538029M 5 8 225 $506  00005538033M 5 9 149 $357  00005538037M 6 10 372 $814  00005538042M 4 11 214 $711  00005538046M 5 13 328 $706  00005538050M 8 8 409 $888  00005538056M 4 11 179 $669  00005538060M 3 11 196 $625  Total  348  $32539  The total cost for this transportation of various commodities from various sources to required destinations is $32539. 2.3.3 Example Problem (with Manual Adjustments) For some practical reasons we have to do manual adjustments for the optimization. As is well known, it is difficult to attain global optima. Making changes in the software based on the constraints of the actual problem resulted in getting a better optima. The corresponding results are shown in Table 2 below. Table 2 Results for 40 cities obtained after making (manual adjustments)  using JDA optimization software Load  Stops Pallets Miles Total Cost 00005537826M 4 10 357 $755  00005537837M 5 8 488 $1,048  00005537847M 4 10 428 $886  00005537855M 9 6 785 $1,476  00005537870M 4 6 391 $856  00005537892M 4 13 354 $758  00005537979M 4 12 251 $801  00005537987M 5 9 249 $822  00005538009M 4 8 247 $761  00005538013M 5 7 557 $1,130  00005538026M 3 11 256 $744  European Scientific Journal   August 2013  edition vol.9, No.23  ISSN: 1857 – 7881 (Print)  e - ISSN 1857- 7431 19  00005538840M 4 15 180 $669  00005538850M 7 10 383 $814  00005539332M 4 10 504 $1,022  00005539336M 5 11 960 $1,798  00005539341M 5 9 393 $891  00005539346M 5 12 803 $1,507  00005539350M 4 7 407 $745  00005539355M 4 6 538 $986  00005539359M 9 6 1028 $1,870  00005539364M 4 7 368 $808  00005539368M 4 10 445 $917  00005539378M 5 7 305 $661  00005539383M 8 6 400 $869  00005539389M 4 9 201 $689  00005539393M 4 9 346 $741  00005539398M 5 12 345 $716  00005539402M 4 13 184 $679  00005539408M 5 8 186 $429  00005539412M 4 8 194 $423  00005539415M 6 10 376 $822  00005539420M 5 9 149 $357  00005539424M 4 11 214 $711  00005539428M 4 11 190 $674  00005539432M 4 8 166 $662 Total  348  $30,497   As can be seen from Table 2, the optimum cost obtained now is lower than the previous optima. 3.1  Simulation Optimization Problem For the purpose of this research to illustrate the concept of Simulation optimization, a problem of transporting material to 11 cities ( a subset of the actual 40 city problem used at Staples, Inc.) is used here. First, traditional deterministic optimization is performed using JDA software followed by Probabilistic optimization. 3.1.1  Deterministic Optimization For the ease research, we took a sample of 11 orders based in OH and WV. The order set shown and the necessary information for those orders is shown in table 3 below.        European Scientific Journal   August 2013  edition vol.9, No.23  ISSN: 1857 – 7881 (Print)  e - ISSN 1857- 7431 20  Table 3 Request of orders for the chosen cities for deterministic optimization Source ID Source City State Pallets Cube Destination ID Destination State RDC00994 HAGERSTOWN MD 8 696 STR01202 ATHENS OH RDC00994 HAGERSTOWN MD 10 885 STR00851 FERMONT OH RDC00994 HAGERSTOWN MD 10 873 STR00840 ELYRIA OH RDC00994 HAGERSTOWN MD 9 782 STR00753 NEW PHILADALPHIA  OH RDC00994 HAGERSTOWN MD 10 829 STR00591 WOOSTER OH RDC00994 HAGERSTOWN MD 8 680 STR00451 COLUMBUS OH RDC00994 HAGERSTOWN MD 9 763 STR00403 COLUMBUS OH RDC00994 HAGERSTOWN MD 10 877 STR00609 BECKLEY WV RDC00994 HAGERSTOWN MD 10 879 STR00607 CHARLESBURG WV RDC00994 HAGERSTOWN MD 10 847 STR00432 CHARLESTON WV RDC00994 HAGERSTOWN MD 10 869 STR00352 CHARLESTON WV  We took this order set and optimized using the JDA. These orders were grouped together and for the simplicity there is no return pallets included. All the loads are one way trips. This will help us to give variation in the cost. The results were tabulated and it shows the number of stops in each truck load, number of pallets included, miles and the total cost.  The results are tabulated as below in Table 4 below Table 4 Results for Deterministic optimization using JDA Load Stops Pallets Miles Total cost 00005258907M  4 29 492 $1060 00005258911M  3 20 426 $776 00005258914M  4 28 473 $792 00005258918M  4 27 402 $840 Total    1301 $3468                             3.1.2  Probabilistic optimization (Simulation Optimization) The procedure used is as follows: 1. Calculate the statistical parameters of the random varaible (Td) which can be considered a Random Variable (RV). These statistical parameters are - Mean value -µ and standard deviation -σ). 2. Calculate µ -  σ, µ + σ, µ -  2 σ, µ  + 2 σ, µ -  3 σ, µ  + 3 σ values of Td. There are total of 6 values but there are 3 sets  [ µ -  σ, µ + σ], [µ -  2 σ, µ  + 2 σ,], [µ -  3 σ, µ  + 3 σ]. The reason for choosing these three is because each of these three sets because these capture uncertainties of  68.3%, 95.4% and 99.7% respectively which cover almost all practical ranges. Hence results European Scientific Journal   August 2013  edition vol.9, No.23  ISSN: 1857 – 7881 (Print)  e - ISSN 1857- 7431 21  obtained using these ranges, are practical and correspond to a realistic situation. 3. Pick the cities corresponding to each of these 6 values. 4. Pick those cities and run the software which will give us the optimal path. Using the above detailed procedure, the following steps are performed as solution of probabilistic optimization (simulation –optimization). It is to be noted that, the suggested method is used instead of traditional Simulation method such as – Monte Carlo Simulation (Ang and Tang, 2007), and the method used elsewhere (Putcha and Rao1991; Putcha and Tadi, 2010)  is because this is a large number of random samples is required for Monte Carlo Simulation for it to be realistic. Further, the method proposed in this paper captures most of the uncertainties  up to 99.7%. For the same data set, we are now calculating the distances between the cities. The stores were arranged in the order of their zip codes.  Table 5 Request of orders for the chosen cities for probabilistic  optimization S.no City ID STATE ZIP Distance in miles from W 1 ATHENS L1 OH 45701 291 2 FREMONT L2 OH 43420 365 3 ELYRIA L3 OH 44035 321 4 NEW PHILADELPHIA L4 OH 44663 262 5 WOOSTER L5 OH 44691 305 6 COLUMBUS L6 OH 43212 333 7 COLUMBUS L7 OH 43235 350 8 BECKLEY L8 WV 25801 294 9 CHARKSBURG L9 WV 26301 185 10 CHARLESTON L10 WV 25313 303 11 CHARLESTON L11 WV 25309 304   For the purpose of research, let’s assume store L11 is the mandatory delivery store. Now we need to find the optimal path that is feasible to make a delivery to that store. Let us call the warehouse location as “W” and the final destination as “D”. There are different paths that a truck can go to this destination.      The Distance from the W to D is a random variable and let’s say it “Td”. Consider different random paths between these two points. These order numbers were obtained by random number generator.        European Scientific Journal   August 2013  edition vol.9, No.23  ISSN: 1857 – 7881 (Print)  e - ISSN 1857- 7431 22  Table 6 Distances for cities for Probabilistic optimization using JDA Source “W” L L L L L L L L L L  “D” L11 Total Distance Td W 4 3 10 7 1 5 9 8 2 6 D 2091 W 8 3 7 10 6 5 2 4 1 9 D 1849 W 1 5 4 7 10 8 9 3 2 6 D 1671 W 9 3 1 7 6 8 5 10 4 2 D 2222 W 6 2 9 4 5 7 10 1 8 3 D 2183 W 5 10 7 8 6 3 2 9 4 1 D 2158 W 7 3 9 2 6 8 4 5 10 1 D 2104 W 10 7 4 2 9 1 5 6 8 3 D 2392 Calculating the Mean µ = ∑Td/8                                            = 2083.75 Standard Deviation SD =                                              = 225.33 Table 7  Range of Distances corresponding to the random variable of total distance between initial and final city (based on probability theory)  µ - s µ + s µ - 2s µ + 2s µ - 3s µ + 3s  1858.42 2339.08 1633.09 2534.41 1407.76 2759.74  Consider the µ - σ, the nearest Td value to this number is 1849. In this path, consider stores that make the solution feasible. In other words, using these points obtain a solution that is feasible with the real time constraints. By applying this method to all the µ and Sigma combinations we got the following results: Table 8 Results for Probabilistic Optimization  Combination Feasibility Total Cost in $ µ - σ Y 1440 µ + σ Y 1562 µ - 2σ Y 1292 µ + 2 σ Y 1562 µ - 3σ Y 1440 µ + 3σ Y 1562  4.0 Discussion of Results and Conclusions As can be seen from Table 4 and Table 8, the total optimal cost for deterministic optimization is $3468 while the range of optimal cost for probabilistic optimization is from $1292 to $1562 for the whole spectrum of uncertainties. This shows the efficacy of probabilistic optimization or what is known as simulation-optimization. While the procedure for simulation-optimization is little bit involved as it uses the concept of probability and statistics, it does result in saving of cost. Hence it is recommended that simulation-optimization be used for solving transportation problems.  European Scientific Journal   August 2013  edition vol.9, No.23  ISSN: 1857 – 7881 (Print)  e - ISSN 1857- 7431 23  Acknowledgement The authors wish to thank various people for their contribution to this paper,  Mr. Brian Tighe , Mr. Bill Ross and Ms. Mary Allain, and specially Mr. Daniel Mahon for  technical support in the software. The first author of this paper wishes to thank Dr Chandrasekhar Putcha for guding him to do research work related to this paper, and Mr. Brian Tighe, manager at Staples, Inc.  and  his parents and loved ones for their support and encouragement throughout his life.  References: Ang, A.H-S and Tang, W.H. (2007). Probability Concepts in Engineering Planning and Design, Vol. II – Decision, Risk and Reliability, John Wiley Publications. Putcha, C.S. and Rao, J.K. (1991). Probabilistic Network Analysis for Project Completion Time, Civil Engineering Systems, Vol. 8. Putcha, C.S. and Tadi, R.K. , (2010). A paper presented at the First International Conference on “Sustainability in Transportation Structures, held in Phoenix, AZ, November 17-19.                         

APPLICATION OF PRINCIPLES OF RELIABILITY ANALYSIS TO OPTIMIZATION PROBLEM IN TRANSPORTATION INDUSTRY

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APPLICATION OF PRINCIPLES OF RELIABILITY ANALYSIS TO OPTIMIZATION PROBLEM IN TRANSPORTATION INDUSTRY

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