Bottleneck Resource Allocation in Manufacturing

1. Bottleneck Resource Allocation in Manufacturing Mgmt. Sci. Vol. 42, No. 11, 1996 Anantaram Balakrishnan, Richard L. Francis & Stephen J. Grotzinger Smeal College of Business Administration, The Pennsylvania Univ. Dept. of Industrial and Systems Engineering, Univ. of Florida. 자동화실험실 장양자

2. Contents • Introduction • Bisection Method for Bottleneck Resource Allocation • Assembly-Release Planning Problem • Model Extensions • Conclusion 97년 2학기 정기 세미나

3. Introduction • Resource Allocation Problem • Choosing integer valued levels for various activities to maximize the minimum return or productivity • 응용 분야: line balancing, raw material allocation, maintenance planning, storage allocation 97년 2학기 정기 세미나

4. Monotonic Resource Allocation Problem • X; n-vector of integer activity levels xj • Si; total availability of resource i • gi(X); amount of resource i consumed by the activity levels in X, resource usage function (nondecreasing) • fj(xj); productivity of activity j (permit nonmonotonic) • F(X); system’s productivity =min{fj(xj); j=1 to n} • B=(Bj); lower bound of activity • C=(Cj); upper bound of activity 97년 2학기 정기 세미나

5. Research Review • Jacobsen(1971); discrete allocation problem with one linear resource constraint, marginal allocation approach 적용 • Proteus & Yormark(1972); improved bisection search algorithm • Brown(1979); efficient solution procedures for knapsack-sharing problems with piecewise linear and nonlinear productivity ftns. • Tang(1988); O(mn2) algorithm for a model (gi(X) are all linear, productivity fj(xj) strictly increasing linear ftn.) • Francis and Horak(1994); fast bisection search method 97년 2학기 정기 세미나

6. Bisection Method for Bottleneck Resource Allocation (1/3) • translation to nondecreasing ftn. (Brown, 1991) • X; feasible sol. • X*; optimal feasible sol. • z*=F(X*) • F(B)≤F(X) ≤z*≤F(C) • z∈[F(B), F(C)] 97년 2학기 정기 세미나

7. Bisection Method for Bottleneck Resource Allocation (2/3) • fj(.)의 inverse가 존재한다면 • inverse가 존재하지 않는 경우에는 bisection search를 이용하여 구한다. • ε이 주어지면, 일때, ε-OFS라고 한다. • Francis and Horak(1994), Y(z)=(yj(z))는 feasible sol. iff z≤z* 97년 2학기 정기 세미나

8. Bisection Method for Bottleneck Resource Allocation (3/3) • LB=F(B), UB=F(C) on z* • z=(LB+UB)/2 • F(X)≥z, feasible sol. X가 존재하는지 확인한다.(Y(z)=(yj(z))가 제약식을 만족하는지) • 만족하면 Y(z)는 feasible sol. z가 lower bound • 만족하지 않으면 z가 upper bound • UB-LB≤ε이 될 때까지 반복한다. • Y(LB)가 ε-OFS가 된다. • simple, computationally robust, easy to code 97년 2학기 정기 세미나

9. Assembly Release Planning Problem (1/5) • Grotzinger and O’Connor (1993) • how many parts to procure in each period • how to allocate these parts to different products (parts allocation decision) • how many component sets for each product to release for assembly in each period (assembly release decision) • based on estimates of probable future demands, service level requirements, inventory considerations • Assemble to forecast environment • n products, m part types, (p+1) procurement lead time, L assembly lead time 97년 2학기 정기 세미나

10. Assembly Release Planning Problem (2/5) • nonlinear optimization model • linear program for a “feasibility subproblem” incorporating the part allocation and assembly release decision • cumulative availability • actual service level for a product j at time t • target service level; τjt • service level difference = difference between the actual and target service level • Model; Maximize the minimum service level difference over all products and time periods • 만약 optimal이 nonnegative면 target service level을 만족 97년 2학기 정기 세미나

11. Assembly Release Planning Problem (3/5) • qjt; # of units of product j • aij; # of part i used per unit of product j • Sit; cumulative supply of part i at time t • Aj,L-1; cumulative availability of product j in period (L-1) • Aj,L+t=Aj,L-1+ (qj0+….+qjt) • Dj,L+t; cumulative demand for product j from time period 0 to period (L+t) • fjt (qj0+….+qjt); service level difference for product j at time (L+t) = Prob.{Aj,L-1+ (qj0+….+qjt)≥ Dj,L+t} - τj,L+t • Q=(qjt); product release qty. • (Q)=min{fjt (qj0+….+qjt): j=1 to n, t=0 to p} 97년 2학기 정기 세미나

12. Assembly Release Planning Problem (4/5) • xjt=qj0+….+qjt ←coupling constraints 97년 2학기 정기 세미나

13. Assembly Release Planning Problem (5/5) • coupling constraints • LB=F(0), UB=F(U)=min{fjt(ujt)} • wjt(z)=min{xjt:z≤fjt(xjt), xjt=0,1,…,ujt} • yj0(z)=wj0(z) • yjt(z)=max{yj,t-1(z), wjt(z)} =min{xjt:z≤fjt(xjt), xjt=yj,t-1(z), yj,t-1(z)+1,…,ujt} 97년 2학기 정기 세미나

14. Bisection Algorithm for ARP Problem (1/2) • Step 0: initialization • β←0;γ ← U;LB ← F(0);UB ← F(U); • Step 1: Search Process • REPEAT • z ←(LB+UB)/2; • compute w(z) in [β,γ]; • FOR j=1 to n • Set yj0(z) ←wj0(z); • FOR t=1 to p 97년 2학기 정기 세미나

15. Bisection Algorithm for ARP Problem (2/2) • Set yjt(z) ←max(yj,t-1(z), wjt(z)); • IF Y(z) satisfies the resource constraints • THEN Set LB ←F(Y(z)) and β ←Y(z); • ELSE Set UB ←z and γ ←Y(z); • UNTIL(UB-LB)≤ε • Step2: ε-optimal solution • FOR j=1 to n • Set qj0=yj0(LB); • FOR t=1 to p • Set qjt=yjt(LB)-yj,t-1(LB); 97년 2학기 정기 세미나

16. Model Extensions • Precedence Constraints • multi-stage production • cumulative release qty. of product j at time t ≥ cumulative release qty. of product j’ at time t’ • xjt≥xj’t’ for all <(j’,t’), (j,t)> • Part Substitutability • 서비스 수준을 높일 수 있으나 제품 비용을 증가시키고 품질, 신뢰성을 감소시킬 수 있다. • 원하는 서비스 수준에 대하여 총 대체 부품 수를 최소화하는 모델 97년 2학기 정기 세미나

17. Conclusion • 기본 모델을 coupling, precedence constraint, substitutability를 포함하는 모델로 확장하고, bisection method를 이용하였다. • discrete variables, nonlinear productivity functions, nonlinear resource constraints • determining the smallest bottleneck activity vector Y(z) that achieves the target objective value z is relatively easy • the resource-usage functions satisfy the monotonic resource-feasibility requirement • finding a feasible completion for any bottleneck activity vector X that is resource feasible is relatively easy • Y(z)의 feasibility만 확인하면 되기 때문에 효율이 좋다. 97년 2학기 정기 세미나