考慮商品數量折扣之聯合補貨問題 consider quantity discounts for joint replenishment...
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考慮商品數量折扣之聯合補貨問題 Consider quantity discounts for joint replenishment problem. 研究生 : 王聖文 指導教授 : 楊能舒 教授. Reporting process. Joint replenishment problem. The joint replenishment objective adjusts to the replenishment cycle between different products to avoid additional ordering costs. - PowerPoint PPT PresentationTRANSCRIPT
考慮商品數量折扣之聯合補貨問題 Consider quantity discounts for joint replenishment problem研究生 : 王聖文
指導教授 : 楊能舒 教授
Reporting process
Introduction
Background and Motivation
Research of objective
Research Process
Literature
Quantity discounts
Particle swarm
optimization
Joint replenishment problem
Research Method
Mathematical model
Solving process
Experiment and
Analysis
Instance calculus
Analysis of results
Conclusion
Programming approach
Particle swarm
optimization
Research Plan
Discount percentage
Numbers of item
The joint replenishment objective adjusts to the replenishment cycle between
different products to avoid additional ordering costs.
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Joint replenishment problem
Quantity discounts
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Order quantity
Price
Consider quantity discount
s
Establish heuristic method to solve the joint replenishment problem considering quantity
discounts
Consider quantity discounts for joint replenishment problem
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Research Motivation and objective
Related literature
Joint replenishment problem
Particle swarm optimization
Establish heuristic method
Experiment parameters set
Analysis of results
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Consider quantity discounts for joint replenishment problem
Research Process
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
No. Author Year Method ResultThe
relationship between this
study
1J.Kenny, R .C. Eberhart
1995
Propose particle swarm optimization and explain the feature and concept of this algorithm.
Particle swarm algorithms consider a few parameters and fast convergence of the solution space, suitable for solving large problems.
Use the feature of particle swarm optimization for solving joint replenishment problem.
2 Goyal, S.K. 1973
Propose the method to find out the upper and lower bounds of the optimal ordering cycle.
By changes in cycle multiplier derived upper and lower bounds of the optimal ordering cycle.
Use the method proposed by Goyal to set the upper and lower bound of basic cycle.
3 Goyal, S.K. 1974
Propose the method to find out upper and lower bounds of the optimal ordering cycle and give an example.
Changes in cycle multiplier will affect the total cost.
Similar to Goyal(1973), there is example for reference.
No. Author Year Method ResultThe
relationship between this
study
4 Silver, E.A. 1976
Derivate cycle multiplier formula and set upper and lower bounds of cycle multiplier also proposed sorting indicators.
Silver proposed a simple method can get a good solution performance.
Use the method proposed by Silver to set the cycle multiplier lower and upper bound as well as to determine the basic ordering cycle.
5
Shi, Y.,Eberhart, R.
1998Propose weight added to the particle swarm optimization .
By adjusting the weights to change the solution space of global search or local search.
Consider the weight can increase the accuracy of search solutions.
6I.K.MOON,S.K.GOYAL,B.C.CHA
2008
Using genetic algorithms try
to solve consider quantity discounts for joint
replenishment problem.
.
Provide a method to solve consider quantity discounts for joint replenishment.
Mentioned joint replenishment model for considering quantity discounts.
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
No. Author Year Method ResultThe
relationship between this
study
7Kaspi, M. & Rosenblatt, M.J.
1983The extension of Silver (1976).
Find out the relationship of cycle multiplier and frequency.
Another method for solving joint replenishment problem.
8Kaspi, M. & Rosenblatt, M.J.
1991The extension of Kaspi, M. & Rosenblatt, M.J. (1983).
Add frequency upper and lower bounds, and the same amount range. Proposed the RAND algorithm.
Another method for solving joint replenishment problem.
9Goyal, S.K. & Belton,
A.S.1979
Another indicator to modify the method proposed by Silver (1976)
Claimed to use the indicators to get a better solution.
Increase the selection of indicators。
Problem description
• Consider quantity discounts joint replenishment problem for single supplier to multi-retailers.
• Objective is minimize the total cost.
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Joint replenishment problem (Not consider quantity discounts )
Joint replenishment problem (Consider quantity discounts )
Programming approach
Analysis of results
Heuristic method
Find the optimal replenishment strategies
Single item replenishment problem (Consider quantity discounts )
Research steps
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Compare Programming approach and heuristic methods
Mathematical Symbol Description
• Di: Demand for items• hi: Items i per unit holding cost ratio• S: Major ordering cost• si: Minor ordering costs• Ci: Unit price of item i• ki: Integer number that determines the
replenishment schedule of item i• T: Basic cycle• TC: Total cost
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Mathematical model with quantity discounts
Single items replenishment problem
Joint replenishment problem
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Programming approach to solve joint replenishment problem.(Not consider
quantity discounts)
• S=4000Item 1 2 3si 1000 1000 1000hi 0.2 0.2 0.2Di 12000 1200 120Ci 500 500 500
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Programming approach to solve joint replenishment problem.(Not consider
quantity discounts)• Objective function :
• T 、 ki 、 yij are decision variables• ki at least one of 1 (basic cycle)• The other items cycles is the integer multiple of the basic cycle.
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Programming approach to solve joint replenishment problem.
(Not consider quantity discounts)
basic cycle T
0.0872
(k1,k2,k3) (1,2,5)
TC 6090767
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Programming approach to solve Single item replenishment problem (Consider
quantity discounts )
• D=120000• h=0.2• S=100
Ordering quantity
Unit price
Q<5000 35000≦Q<1000
02.96
Qi≧10000 2.92
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Programming approach to solve Single item replenishment problem (Consider
quantity discounts )• Originally objective function :
• changed to
• yj is binary ， Indicates whether to use a discounted price j ， j=1,2,3
• Qj 、 yj are decision variables• ，
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Programming approach to solve Single item replenishment problem (Consider
quantity discounts )
• The result of programming approach solving is the minimum TC for $ 354,520 occurred when the order quantity Q is 10000.
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
0 5000 10000 15000 20000345000
350000
355000
360000
365000
370000
Total cost
總成本 TCEOQ
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Programming approach to solve Single item replenishment problem (Consider
quantity discounts )
Programming approach to solve joint replenishment problem.(Consider
quantity discounts)• S=4000
Item 1 2 3si 1000 1000 1000hi 0.2 0.2 0.2Di 12000 1200 120
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
C1 C2 C3Qi<500 500 500 500
500≦Qi<1000 470 480 475Qi≧1000 440 460 450
Quantity discount table
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Programming approach to solve joint replenishment problem.(Consider
quantity discounts)
Programming approach to solve joint replenishment problem.(Consider quantity
discounts)• Originally objective function :
• changed to• yij is binary. Denote the items i whether use discounted prices j. i=1,2,3 ， j=1,2,3
• T 、 ki 、 yij are decision variables.•
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Programming approach to solve joint replenishment problem.(Consider quantity
discounts)Basic cycle
T0.0926
(k1,k2,k3) (1,9,4)
(C1,C2,C3) (440,460,500)
TC 6047011
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Analysis of results
• Programming approach example shows that it is feasible to solve small quantity discounts problem .
But large quantity discounts problem programming approach can not be solved.
Find another new algorithm for solving
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Optimal solution
(k1,…,ki,T,V1,…,Vi)(k1,…,ki,T,V1,…,Vi)
(k1,…,ki,T,V1,…,Vi)
(k1,…,ki,T,V1,…,Vi)(k1,…,ki,T,V1,…,Vi)
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Particle swarm optimization
Mathematical Symbol Description:• Xid ： Position of the particle i on d-th
iteration.• Vid ： Speed of the particle i on d-th iteration.• Pid ： The best position of the particle i in d
iterations.• Pgd ： The best position of all particle i in d
iterations.• Cj ： Learning coefficient.• ω ： Weight.• ωmax ： Weight maximum.• ωmin ： Weight minimum.• Rj ： Independent random variable. The range is [0,
1].• Vmax ： The maximum allowable speed when the
particle update.
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Decision variables are ki and basic cycle T
ki and basic cycle T set to particle position
Use particle swarm optimization to minimize the total cost.
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Heuristic method
Particle velocity update formula ：
Particle position update formula ：
Solving
Particle speed limit ：
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Heuristic method
• According to the method proposed by Goyal (1973 & 1974) set the upper and lower bounds of ki and basic cycle T.
• According to the method proposed by Silver(1976) to find out k1T is the basic cycle.
,(ki=1)
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Heuristic method
• Step1 ： Initialization Randomly generated particles and randomly
assigned to the initial position and speed. k1=1 ， k2 、 k3 upper bound are
3.2969 、 10.5409,and 0.0408 ≦T ≦0.1091 。(k1,k2,k3) Basic cycle T (v1(k2),v2(k3),v3
(T))(1,2,3) 0.1090 (0.5412,2.4325,3.2516)
(1,4,2) 0.0706 (-1.1833,-2.5648,3.5421)
(1,3,9) 0.0957 (3.7685,-3.5453,1.0010)
(1,1,10) 0.1010 (2.5612,0.0025,-2.5671)
(1,2,5) 0.0752 (-1.2657,2.6570,-3.9627)
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Heuristic method
• Step2 ： Evaluation Evaluate each particle function value.
Randomly generated values of k, T and v into the objective formula.
(k1,k2,k3) Basic cycle T (v1(k2),v2(k3),v3(T))
Fitness value (TC)
(1,2,3) 0.1090 (0.5412,2.4325,3.2516) 6187419
(1,4,2) 0.0706 (-1.1833,-2.5648,3.5421)
6439054
(1,3,9) 0.0957 (3.7685,-3.5453,1.0010) 6190963
(1,1,10) 0.1010 (2.5612,0.0025,-2.5671) 6065844
(1,2,5) 0.0752 (-1.2657,2.6570,-3.9627)
6429491
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Heuristic method
• Step3 ： Update Pid According to the value obtained by Step2 update so far the best position of each particle.
• Step4 ： Update Pgd According to the value obtained by Step3 update so far the best position of group of particle.
The fitness value of each particle after iteration.
Update so far the fitness value of each particle.
the best position of each particle.(k1,k2,k3,T)
6187419 6187419 (1,2,3,0.1090)
6439054 6439054 (1,4,2,0.0706)
6190963 6190963 (1,3,9,0.0957)
6065844 6065844 (1,1,10,0.1010)
6429491 6429491 (1,2,5,0.0752)
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Heuristic method
• Step5 ： Randomly generated R1, R2 and update Xid, Vid.
Obtained Pid and Pgd from Step3,Step4 then update particles speed and position.
(k1,k2,k3)
Basic cycle T
(v1(k2),v2(k3),v3(T)) (R1,R2)
(1,1,11) 0.1091 (-0.6485,4,2.9174) (0.1057,0.5678)
(1,1,11) 0.1091 (-4,4,3.2312) (0.9512,0.7123)
(1,4,6) 0.1091 (3.7685,-3.5453,1.0010) (0.6717,0.0045)
(1,3,10) 0.0408 (2.3051,0.0023,-2.3104) (0.2130,0.7951)
(1,1,11) 0.0408 (-2.0411,4,-3.5432) (0.3016,0.4510)
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Heuristic method
(k1,k2,k3) Basic cycle T Fitness value (TC)
(1,1,11) 0.1091 6067180
(1,1,11) 0.1091 6067180
(1,4,6) 0.1091 6052318
(1,3,10) 0.0408 6827442
(1,1,11) 0.0408 6838908
Step6 ： Repeatedly step2 to step5, and stop when it reaches the
termination conditions.
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Heuristic method
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
• After repeated iteration, the total cost TC will gradually close to the optimal solution. Programmin
g approachHeuristic method
Basic cycle T
0.0926 0.0752
(k1,k2,k3) (1,9,4) (1,2,5)(C1,C2,C3) (440,460,5
00)(440,480,5
00)
TC 6047011 6051882
Analysis of results
• Compared with Programming approach, the calculate time of heuristic method is shorter than Programming approach significantly.
• And the convergence speed of heuristic method also faster than Programming approach, several iteration that can be get a good solution.
• Heuristic method is suitable to apply in large joint replenishment problem.
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Conclusion
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Heuristic method solve the example
proposedBy Goyal(2008)
Solving performance ?
Research plan
Introduction Research of
objective
Research Process
Literature Research Method
Research Method
- Instance calculus
Research Method
- Analysis of results
Conclusion Research plan
Programming
approach
Heuristic method
Small proble
m
The optimal solution can be obtained
.
The number of
calculation is less and
good solution can be
obtained.Large proble
m
Can not be
solved.?
Conditions set
Numbers of item
5 20
Discount percentage
10% 30%
…… …
Research plan
最後統整
參數分析
實際範例
撰寫程式
0 1 2 3 4 5 6 7
開始月份執行月數
Timetable
END