Modeling and Optimization of Resource Allocation in Supply Chain

University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 5-2013 Modeling and Optimization of Resource Allocation in Supply Chain Management Problems Qi Yuan qyuan3@utk.edu Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Industrial Engineering Commons, and the Operational Research Commons Recommended Citation Yuan, Qi, "Modeling and Optimization of Resource Allocation in Supply Chain Management Problems " PhD diss., University of Tennessee, 2013 https://trace.tennessee.edu/utk_graddiss/1800 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange For more information, please contact trace@utk.edu To the Graduate Council: I am submitting herewith a dissertation written by Qi Yuan entitled "Modeling and Optimization of Resource Allocation in Supply Chain Management Problems." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Industrial Engineering Xiaoyan Zhu, Major Professor We have read this dissertation and recommend its acceptance: Xueping Li, Joseph Wilck, Frank M Guess Accepted for the Council: Carolyn R Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official student records.) Modeling and Optimization of Resource Allocation in Supply Chain Management Problems A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville Qi Yuan May 2013 c by Qi Yuan, 2013 All Rights Reserved ii To my dear parents and Huimin iii Acknowledgements First of all, I would like to express my deepest gratitude for the support and guidance of my major advisor Dr Xiaoyan Zhu I benefit a lot from her meticulous scholarship, straightforward personality and thoughtful kindness Also, I am indebted to Dr Xueping Li for his support all these years I appreciate Dr Joe Wick’s tremendous help for providing academic and industry related information Special thanks go to Dr Frank Guess, who always encourages me and gives me the confidence to better I would also like to mention Dr Alberto Garcia-Diaz, whose wonderful courses have brought me to a new horizon It has been a lifetime honor to learn from these great professors I am also thankful to Dr Fong-Yuen Ding, Dr Qingzhu, Dr Yuerong Chen, Mr Caiqiao Xu, Dr Gewei Zhang, Dr Laigang Song, Dr Dengfeng Yang, Dr Yan Chen, Dr Zhe Zhang, Dr Liang Dong, Mr Cong Guo, and Ms Zhaoxia Zhao Thanks to all of you for your time and encouragement, as well as your willingness to share You have made my adventure an easy journey Finally, it is hard to express in words the deepest love for my parents, who brought me up and always take care of me along the way I would also like to thank Ms Huimin Zhou, the most beautiful lady in the world, for making my life complete iv “You’ve got to find what you love.” – Steve Jobs v Abstract Resource allocation in supply chain management studies how to allocate the limited available resources economically/optimally to satisfy the demands It is an important research area in operations research This dissertation focuses on the modeling and optimization of three problems The first part of the dissertation investigates an important and unique problem in a supply chain distribution network, namely minimum cost network flow with variable lower bounds (MCNF-VLB) This type of network can be used to optimize the utilization of distribution channels (i.e., resources) in a large supply network, in order to minimize the total cost while satisfying flow conservation, lower and upper bounds, and demand/supply constraints The second part of the dissertation introduces a novel method adopted from multi-product inventory control to optimally allocate the cache space and the frequency (i.e., resources) for multi-stream data prefetching in computer science The objective is to minimize the cache miss level (backorder level), while satisfying the cache space (inventory space) and the total prefetching frequency (total order frequency) constraints Also, efforts have also been made to extend the model for a multi-level, multi-stream prefetching system The third part of the dissertation studies the joint capacity (i.e., resources) and demand allocation problem vi in a service delivery network The objective is to minimize the total cost while satisfying a required service reliability, which measures the probability of satisfying customer demand within a delivery time interval vii Contents Introduction and Research Topics 1.1 Introduction to resource allocation in supply chain management 1.2 Three topics in this research direction 1.2.1 The MCNF-VLB problem 1.2.2 The data prefecthing problem 1.2.3 The joint capacity and demand allocation problem Structure of the dissertation 1.3 Resource Allocation in Supply Chain Distribution Network: the MCNFVLB Problem 2.1 Literature review on the MCNF problem and the MCNF-VLB problem 2.1.1 Existing MCNF problems 2.1.2 Extension to the MCNF-VLB problem 10 Mixed integer linear programming (MILP) models 11 2.2.1 The MCNF-VLB model 11 2.2.2 A numerical example 15 Computational testing with CPLEX 18 2.2 2.3 viii Erickson, R E., Monma, C L., and Veinott, A F (1987) Send-and-split method for minimum-concave-cost network 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(ri + Qi ; Qi ) − B ∂ri Qi ri = [Gi (ri + Qi ; Qi ) − Gi (ri ; Qi )] − [1 − Gi (ri + Qi ; Qi )] Qi ri +Qi ygi (y ; Qi )dy − Qi ri ¯i (x; Qi ) is a decreasing function of x Thus, ∂B(Q, r)/∂ri ≤ 0, which means ∂B(Q, r) is B ¯i (x; Qi ) = a decreasing function of ri Also, with Equation (3.5), it is easy to show that B ∞ ygi (y + x; Qi ) dy = √ ¯i (x; Qi ) ≤ √ B 2π¯ σi2 Li ∞ 2π¯ σi2 Li y exp ∞ y exp −(y+x−¯ µi Li ) 2¯ σi2 Li −(y+x−¯ µi Li )2 2¯ σi2 Li dy When ri approaches infinity, dy = √ exp 2π¯ σi2 Li −(x−¯ µi Li ) 2¯ σi2 Li ∞ y exp −y 2¯ σi2 Li dy, ¯i (x; Qi ) is less than the order of e−x With this knowledge in Equation (3.5), which means B Bi (Q, ri ) approaches zero as ri approaches infinity These imply the statements in the proposition 108 Proof of proposition By the proof of Proposition 1, ∂B(Q∗ , r∗ )/∂ri ≤ and ∂B(Q∗ , r∗ )/∂ri approaches zero when ri∗ approaches infinity for i = 1, 2, , N Then, by condition (2), λ∗1 > and approaches zero accordingly To ensure complementary condition (3), space constraint (3.2) must be binding whenever λ∗1 > Proof of proposition By Equations (3.6) and (1), the cache hit ratio of stream i for i = 1, 2, , N is Ri = + ∂B(Q, r)/∂ri Model (3.8) and model (3.11) have the same Lagrangian function as Equation (3.9), and thus the optimal solution of model (3.11) must satisfy condition (2), which is one of the KKT conditions for model (3.11) Therefore, ∂B(Q∗ , r∗ )/∂ri = −λ∗1 and Ri = − λ∗1 for all i = 1, 2, , N Proof of proposition The feasible set of model (3.11) is H = {(Q, r) : S(Q, r) = N i=1 (ri + Qi /2) = s0 ; F (Q) = N i=1 µ ¯i /Qi ≤ f0 ; and Q, r ≥ 0} It is easy to show that S(Q, r) − s0 and F (Q) − f0 are convex functions over the region of Q, r ≥ Because set {x : g(x) ≤ 0} is convex if function g(x) is a convex function and the intersection of convex sets is convex (Bazaraa et al 2006), set H is convex Proof of proposition From Equation (1), we have for all i, j ∈ {1, 2, , N } and i = j, ∂ B(Q, r) = [Gi (ri + Qi ; Qi ) − Gi (ri ; Qi )] > ∂ri Qi and ∂ B(Q, r) = ∂ri ∂rj Therefore, the Hessian matrix associated with function B(Q, r) with variables Q fixed is positive definite (Bazaraa et al 2006) By Proposition 4, the feasible set is convex Therefore, when variables Q are fixed, the corresponding model (3.11) is convex programming 109 Proof of proposition Substituting the expression of ρj = constraint (4.2), we can get its alternative form n i=1 m j=1 n i=1 ¯ j /µj into αij λi /µj = λ αij λi 1−e−(T −τij )(µj − n i=1 n i=1 αij λi ) ≥ r0 λi It is easy to show that the left-hand-side of the expression is an increasing function of µj Also, constraint (4.3) cannot be binding for all µj , otherwise constraint (4.2) will be violated If constraint (4.2) is non-binding under the optimal solution, we can decrease µj by a small amount ∆µj in a non-binding constraint (4.3), such that the feasibility of constraints (4.2) and (4.3) are maintained Thus, the total cost can be further reduced, violating the optimality of the solution As a result, constraint (4.2) must be binding under the optimal solution Proof of proposition When the demand allocation variables α are fixed, the feasible set of model P is H = {µ : g1j (µ) = n i=1 αij λi ≤ µj ; g2 (µ) = − n i=1 m j=1 αij λi 1−e−(T −τij )(µj − n i=1 n i=1 αij λi ) λi −r0 It is easy to show that g1j (µ) − µj and g2 (µ) + r0 are convex functions over the region of µ ≥ Because set {x : g(x) ≤ 0} is convex if function g(x) is a convex function and the intersection of convex sets is convex (Bazaraa et al., 2006), set H is convex The objective function of model P is linear, thus also convex Therefore, when demand allocation variables α are fixed, the corresponding model P with capacity allocation variables µ is convex programming (Bazaraa et al., 2006) A.2 The KKT and second order sufficient conditions ¯ For convenience, we denote S(Q, r) = S(Q, r) − s0 and F¯ (Q, r) = F (Q) − f0 Then the ¯ standard formats of space constraint (3.2) and frequency constraint (3.3) are S(Q, r) ≤ ¯ and F¯ (Q, r) ≤ 0, respectively The gradients of S(Q, r) and F¯ (Q, r) with respect to 110 ≤ (Q1 , Q2 , , QN , r1 , r2 , , rN ) are: ¯ ∇S(Q, r) = ∇F¯ (Q, r) = 1 , , , , 1, 1, , 2 µ ¯2 µ ¯N µ ¯1 − , − , , − , 0, 0, , , Q1 Q2 QN ¯ where vector ∇S(Q, r) has the first N elements of 1/2 and the last N elements of 1, and vector ∇F¯ (Q, r) has the last N elements of To obtain the second order sufficient conditions, first define the critical cone T (Q, r, λ1 , λ2 ) at solution (Q, r) for different cases as follows If the space and frequency constraints are both binding, then T (Q, r, λ1 , λ2 ) =        y∈ 2N : N i=1 yi /2 + 2N i=N +1 yi = and N i=1 −µ¯i yi /Q2 i = y∈ 2N : N i=1 yi /2 + 2N i=N +1 yi = and N i=1 −µ¯i yi /Q2 i ≤0 if λ1 > 0, λ2 = y∈ 2N : N i=1 yi /2 + 2N i=N +1 yi ≤ and N i=1 −µ¯i yi /Q2 i = if λ1 = 0, λ2 > y∈ 2N : N i=1 yi /2 + 2N i=N +1 yi ≤ and N i=1 −µ¯i yi /Q2 i ≤ if λ1 = 0, λ2 = if λ1 > 0, λ2 > If the space constraint is binding and the frequency constraint is non-binding, then T (Q, r, λ1 , λ2 ) =     y∈ 2N : N i=1 yi /2 + 2N i=N +1 yi = if λ1 >    y∈ 2N : N i=1 yi /2 + 2N i=N +1 yi ≤ if λ1 = If the space constraint is non-binding and the frequency constraint is binding, then T (Q, r, λ1 , λ2 ) =     y∈ 2N : N i=1 −µ¯i yi /Q2i = if λ2 >    y∈ 2N : N i=1 −µ¯i yi /Q2i ≤ if λ2 = 111 If the space and frequency constraints are both non-binding, then T (Q, r, λ1 , λ2 ) = 2N ¯ Because ∇S(Q, r) and ∇F¯ (Q, r) are nonzero and linearly independent of each other, all the feasible solutions satisfy constraint qualification (Bertsekas 1999) Since all of the feasible solutions satisfy constraint qualification, an optimal solution (Q∗ , r∗ , λ∗1 , λ∗2 ) must satisfy the following KKT conditions with respect to Lagrangian function (3.9) (Bazaraa et al 2006): ∂L(Q∗ , r∗ , λ∗1 , λ∗2 ) λ∗ µ ¯i ∂B(Q∗ , r∗ ) λ∗1 − 2∗ = 0, = + ∂Qi ∂Qi (Qi ) ∂B(Q∗ , r∗ ) ∂L(Q∗ , r∗ , λ∗1 , λ∗2 ) = + λ∗1 = 0, ∂ri ∂ri N λ∗1 ri∗ + i=1 N λ∗2 i=1 N i=1 i = 1, 2, , N − s0 = µ ¯i − f0 = Q∗i N Q∗ + i ri∗ Q∗i i = 1, 2, , N ≤ s0 ; i=1 µ ¯i ≤ f0 Q∗i j=1 + − Qi Q2i ∞ rj +Qj Qj (y − x) rj x ∂gi (y ; Qi ) dydx ∂Qi ∞ (y − ri − Qi )gi (y ; Qi ) dy ri +Qi ri +Qi ∞ (y − x)gi (y ; Qi ) dydx, ri x 112 (3) (5) (6) where ∂B(Q, r)/∂ri is given in Equations (1) and (1), and N (2) (4) λ∗1 , λ∗2 ≥ 0, ∂B(Q, r) = ∂Qi (1) assuming that partial derivative of gi (y ; Qi ) exists Furthermore, a solution (Q∗ , r∗ , λ∗1 , λ∗2 ) satisfying the KKT conditions (1) – (6) is a strict local minimizer if any nonzero vector y in the critical cone T (Q∗ , r∗ , λ∗1 , λ∗2 ) satisfies: y∇2 L(Q∗ , r∗ , λ∗1 , λ∗2 )yT > 0, (7) where the Hessian matrix is ∗ ∗ ∇ L(Q , r , λ∗1 , λ∗2 ) ∂ L(Q∗ , r∗ , λ∗1 , λ∗2 ) = , ∂xi ∂xj xi , xj ∈ {Q1 , Q2 , , QN , r1 , r2 , , rN } The detailed expression of the Hessian matrix ∇2xx L(Q∗ , r∗ , λ∗1 , λ∗2 ) can be obtained, but it is too complicated to analyze Thus, we ignore the details to save space 113 Vita Qi Yuan is a Ph.D candidate at the Department of Industrial and Systems Engineering, The University of Tennessee, Knoxville He is expected to complete his study in 2013 His research interests include supply chain modeling, network flow, and transportation optimization in the operations research area He is a student member of the Institute of Industrial Engineers (IIE) and Institute for Operations Research and Management Sciences (INFORMS) He worked as an operations research intern at Norfolk Southern Corporation, Atlanta, GA in 2012, and is now working as a supply chain research analyst at Niagara Bottling, a major beverage company in the Greater Los Angeles Area 114 ... efficiency of this new MCNF-VLB formulation 29 Chapter Resource Allocation in Data Prefetching: from a Supply Chain Modeling Perspective The data prefetching problem uses the concept in supply chain modeling. .. multiple supply and demand nodes can be easily transformed to the case of a single -supply and single-demand node It can be done by adding a single dummy supply node and a single dummy demand node,... respectively, and three demand nodes 4, and with demand of 5, 10, and 15, respectively We modify it to an equivalent single -supply and single-demand MCNF-VLB by adding a dummy supply node and a dummy demand