NOMENCLATURE
3.2 Genetic Algorithm
GA is a search algorithm developed by Holland (1975) which is based on the mechanics of natural selection and genetics to search through decision space for optimal solutions. The metaphor underlying GAs is natural selection. In evolution, the problem that each species faces is to search for beneficial adaptations to the complicated and changing environment. In other words, each species has to change its chromosome combination to survive in the living world. In GA, a string represents a set of decisions (chromosome combination), that is a potential solution to a problem.
Each string is evaluated on its performance with respect to the fitness function (objective function). The ones with better performance (fitness value) are more likely to survive. Then the genetic information is exchanged between strings by crossover and perturbed by mutation. The result is a new generation with (usually) better survival abilities. This process is repeated until the strings in the new generation are identical, or certain termination conditions are met (Ahmed et al. 2014). A generic flow of GA is given in Fig.3.6. This algorithm is continued since the stopping criterion is reached. GAs is used in forming models to solve optimization problems.
Readers can find more details of GAs in Gen and Cheng (2000), Kaya (2009).GAs are different from other search procedures in the following ways (Chen, 2004): (1) GAs consider many points in the search space simultaneously, rather than a single point;
(2) GAs work directly with strings of characters representing the parameter set, not the parameters themselves; (3) GAs use probabilistic rules to guide their search, not deterministic rules. Because GAs considers many points in the search space simultaneously there is a reduced chance of converging to local optima. In a conventional search, based on a decision rule, a single point is considered and that is unreliable in multimodal space. GAs consists of four main sections. They are as
36
follows: Encoding, Selection, Reproduction, and Termination (Gen & Cheng, 2000;
Mitchell, 1996).
3.2.1 Working procedure of GA
The following steps summarize the working procedure of GA (Ahmed et al. 2014) i. Generate initial random population of chromosomes
ii. Evaluate the fitness f(x) of each chromosome x in the population
iii. Create a new population by repeating following steps until new population is complete
iv. Select two parent chromosomes from a population according to their fitness v. Crossover the parents to form a new offspring (children) with a probability.
vi. Mutate new offspring at each of its position in chromosome with a probability vii. Place new offspring in a new population
viii. Use newly generated population for next run of algorithm
ix. If the stopping criterion is satisfied or the generations are identical stop the generation and return the best solution in current population.
x. Go to step ii.
GA has certain advantages which are:
It can quickly scan a vast solution set. Bad proposals do not affect the end solution negatively as they are simply discarded.
The inductive nature of the GA means that it doesn't have to know any rules of the problem - it works by its own internal rules. This is very useful for complex or loosely defined problems and unconstrained optimization problem.
Genetic algorithms search parallel from a population of points so it can effectively explore many different branches of the tree at once and when a certain branch turns out to be non-optimal, abandon that search proceeding with other more likely candidates.
They do not tend to be easily trapped by local optima, due again to the parallelism of their approach.
The unconstrained nature of the objective function indicates a vast solution set and makes GA an ideal approach to solve this kind of problem within a short time with favorable solutions compared to other techniques.
37
The genetic algorithm approach is depicted in Figure 3.6.
Fig. 3.6 Flow chart of Genetic Algorithm’s working procedure (Ahmed et al. 2014) Based on these points, GA is considered as an appropriate technique for solving unconstrained optimization problems and has been successfully applied in many different areas to determine the optimal values of the decision variables. (Ahmed et al.
2014).
3.2.2 Parameters of GA
Crossover probability says how often crossover will be performed. If there is no crossover, offspring is exact copy of parents. If there is a crossover, offspring is made from parts of parents' chromosome. If crossover probability is 100%, then all offspring is made by crossover. If it is 0%, whole new generation is made from exact
Generate random population of chromosomes
Evaluate the fitness of each chromosome based on fitness function
Parents
Generate new chromosome Offspring
Meet stopping criteria?
Return this chromosome as the optimum solution Selection
Crossover Mutation
No
Yes
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copies of chromosomes from old population (but this does not mean that the new generation is the same!).
Mutation probability says how often will be parts of chromosome mutated. If there is no mutation, offspring is taken after crossover (or copy) without any change. If mutation is performed, part of chromosome is changed. If mutation probability is 100%, whole chromosome is changed, If it is 0%, nothing is changed.
Population size says how many chromosomes are in population (in one generation).
If there are too few chromosomes, GA has a few possibilities to perform crossover and only a small part of search space is explored. On the other hand, if there are too many chromosomes, GA slows down.
39 CHAPTER IV
MODEL DEVELOPMENT 4.1 Problem Identification
A vast amount of researches have already been conducted based on the nature of suppliers’ performance when received an order from the manufacturers/ retailers.
Random disruptions and random capacity of suppliers are continuously being addressed in the literature now and then. But the following gaps are observed in the literature on this subject:
1. Most of the papers addressing the issue of suppliers’ random availability deal with only one supplier. Because it becomes difficult to generate regenerative cycle when working with more than one supplier.
2. Though very few papers considered two or multiple suppliers they assumed the demand from the retailers/end customers as unit (=1) for simplicity. But in reality demand generated randomly at a constant average rate which more resembles to Poisson process. But working with Poisson process will lead to complex transient state probability for different supplier states (ON/OFF) when an order is placed.
3. Almost all of the disruption models did not consider the suppliers random capacity. It had always been assumed that supplier will always deliver the amount as ordered if he is available. But in reality this received quantity could vary based on supplier’s capacity. Though researchers separately addressed the issue of random capacity of suppliers for different inventory model, these two conceptions namely suppliers’ disruption and random capacity have not yet been considered in a single inventory model
4. It is natural for the retailers/ manufacturers to work with more than one supplier and diversify the order among them when all of them are available to reduce the risk associated with the shortages. So far in the literature in case of suppliers’ disruption when both suppliers are available the diversification issue is not yet addressed.
The above-mentioned gaps are addressed in this thesis.
40 4.2 Problem Statement and Assumptions
A manufacturer maintains two suppliers whose supply become randomly disrupted at random times and remain off for random lengths of time. Even if the suppliers are available they will not always be able to deliver the exact quantity ordered. As their capacity is random they will either deliver the quantity ordered or less than that depending on their capacity distribution. The manufacturer is looking for an appropriate ordering policy to minimize his total inventory cost. The manufacturer used to backorder his demands if they cannot be met at time. Some assumptions are made about the situation described above and ordering policy
Manufacturer has no prior idea about the availability and unavailability of either of the suppliers.
Demand from the retailers/end customers generated according to Poisson process
The length of the ON periods (when they are available) and OFF periods (when they are not available) of the suppliers are exponential random variable which means the hazard rates of the ON and OFF period is constant
Capacity of the suppliers are random and exponentially distributed
When both suppliers are available order is diversified between them
The delivery lead time is taken as zero as it happens to be small compared to the average length of the supply disruptions
Per order cost is same for both of the suppliers
Reorder point is a non negative decision variable
41 4.3 Model Development
At first an inventory model is going to be developed for a single supplier. Taking insights from this relatively simple format, the complicated one, an inventory model for two suppliers are going to be developed next. The assumptions made earlier hold for both single supplier and two suppliers’ model.
4.3.1 Single Supplier Case
Here, a stochastic demand inventory system is considered for a supplier whose supply can be randomly disrupted for periods of random duration. According to the assumptions made earlier the demand generated from customer is a Poisson process with parameter 𝛼 which represents the average demand per unit time. At any time t, the state of the system can either be ON or OFF depending on whether the supplier is available or not. To denote the ON and OFF state 0 and 1 is used throughout this thesis. Thus the model can be formulated as two state (0 and 1) CTMC. According to the assumptions, the length of duration of ON and OFF periods are both exponentially distributed random variable denoted respectively as X and Y with parameters 𝜆 and 𝜇. Here, 1/𝜆 represents the average time between supply disruptions and 1/𝜇 represents the average duration of supply disruptions.
When an order is placed and the state is ON, an order cost of $𝑘 /order is incurred.
Holding cost is $h/unit/time, and the shortage (backorder) cost is $𝜋/unit when the state is OFF and the demand cannot be met. For the sake of generality, time dependent part of the backorder cost as $𝜋̂/unit/time is also included. Fig.4.1 depicts a realization of the inventory-level process with the above assumptions.
Based on the assumptions made earlier, supplier has random capacity and it follows exponential distribution with parameter 𝜃 having distribution function F(x) (F(x) = 1- exp (-θx)). Which means every time when he received an order of q units, the amount he delivered will be either q or less than that (whichever applicable) and depends on the values of its distribution function, density function and the amount ordered. From now on, the amount delivered by the supplier is going to be denoted as 𝐸(𝑞) throughout this thesis.
42
The ordering policy used is: When the inventory level drops to r, if the supplier is ON, an order for q units is placed. But the interesting fact is the order received will be 𝐸(𝑞) as the supplier has random capacity following exponential distribution. It will increases the inventory to 𝐸(𝑞)+ r; that is the (q, r) policy is used when the supplier is available. If the supplier is OFF when the inventory drops to r, then the decision maker has to wait until the supplier becomes available (ON) before an order can be placed. In other words, an (s, S)-type order-up-to 𝐸(𝑞)+ r policy is used after the supplier becomes available again. As the demand follows Poisson distribution and its parameter 𝛼 represents average demand per unit time, received 𝐸(𝑞) amount will be last for the period 𝐸(𝑞)/𝛼.
In inventory models with stochastic elements such as considering here, it is useful to identify the regenerative cycles for the purpose of constructing the (average cost) objective function. Referring to Fig.4.1, it is found that cycles of this process start when the inventory goes up to a level of 𝐸(𝑞)+ r units. Once the cycle is identified, the renewal reward theorem (RRT) [Ross 1983] can be used to construct the average cost objective function as a ratio of the expected cost per cycle to the expected cycle length; that is,
𝐴𝐶(𝑞, 𝑟) =𝐶𝑇00
00= 𝐸(𝑐𝑜𝑠𝑡 𝑝𝑒𝑟 𝑐𝑦𝑐𝑙𝑒)
𝐸(𝑐𝑦𝑐𝑙𝑒 𝑙𝑒𝑛𝑔𝑡ℎ) ………(4.1)
4.3.1.1 Calculation of Transient Probabilities
Supply uncertainties have been modeled in the literature by using stochastic lead times. The model which is going to be developed here bears some similarity to the (q, r) model with stochastic lead times. The difference here is that the order quantity is greater than or equal to q, depending on the status of the supplier when an order must be placed. Before developing the objective function, the probabilities Pij (t) = Pr {being in state j at time t starting in state i at time 0}, i, j = 0, 1, needed to be defined.
The transient probabilities P00 (q) and P01 (q) are well known for this two-state CTMC (Ross, 1983) and they are obtained as
𝑃00(𝑞) = 𝑃0+ 𝑃1𝑒−(𝜆+𝜇)𝑡 ………(4.2)
43
𝑃01(𝑞) = 𝑃1− 𝑃1𝑒−(𝜆+𝜇)𝑡 ………(4.3)
Where,
𝑃0 = 𝜆+𝜇𝜇 ………(4.4)
𝑃1 = 𝜆+𝜇𝜆 ………(4.5)
Here, t represents time when inventory will again drop down to the re order point r after receiving 𝐸(𝑞) amount from the supplier. Now, if X is the time when inventory level hits reorder point and 𝑓𝑥(𝑥) is its probability density function, then
𝑃01 = ∫ {(0∞ 𝜆+𝜇𝜆 )[1 − 𝑒−(𝜆+𝜇)𝑥]𝑓𝑥(𝑥)}𝑑𝑥 ………(4.6)
Because demand is a poisson process, X is distributed Erlang with parameter 𝛼 and 𝐸(𝑞) where
𝑓𝑥(𝑥) = 𝛼(𝛼𝑥)𝛤𝐸(𝑞)𝐸(𝑞)−1𝑒−𝛼𝑥, 𝑥 > 0 ………(4.7) We thus have
𝑃01 = (𝜆+𝜇𝜆 )𝛤𝐸(𝑞)𝛼 ∫ {[1 − 𝑒0∞ −(𝜆+𝜇)𝑥](𝛼𝑥)𝐸(𝑞)−1𝑒−𝛼𝑥}𝑑𝑥 ………(4.8)
𝑃01 = (𝜆+𝜇𝜆 )𝛤𝐸(𝑞)𝛼 [∫ (𝛼𝑥)0∞ 𝐸(𝑞)−1𝑒−𝛼𝑥𝑑𝑥 − ∫ {[𝑒0∞ −(𝜆+𝜇)𝑥](𝛼𝑥)𝐸(𝑞)−1𝑒−𝛼𝑥}𝑑𝑥] (4.9)
For the first part of integration Let 𝛼𝑥 = 𝑦 So, 𝑑𝑥 =𝑑𝑦𝛼
44 𝑃01 = ( 𝜆
𝜆 + 𝜇) 1
𝛤𝐸(𝑞)[∫ 𝑦∞ 𝐸(𝑞)−1𝑒−𝑦
0
𝑑𝑥
− ( 𝜆
𝜆 + 𝜇) 𝛼
𝛤𝐸(𝑞)∫ {[𝑒−(𝜆+𝜇)𝑥](𝛼𝑥)𝐸(𝑞)−1𝑒−𝛼𝑥
∞
0
} 𝑑𝑥
We know, 𝛤𝑡 = ∫ 𝑥0∞ 𝑡−1𝑒−𝑥𝑑𝑥; so, ∫ 𝑦0∞ 𝐸(𝑞)−1𝑒−𝑦𝑑𝑥 = 𝛤𝐸(𝑞) Now, 𝑃01= ( 𝜆
𝜆 + 𝜇) − ( 𝜆
𝜆 + 𝜇) 𝛼
𝛤𝐸(𝑞)∫ {(𝛼𝑥)∞ 𝐸(𝑞)−1𝑒−(𝜆+𝜇+𝛼)𝑥
0
}𝑑𝑥
𝑃01 = ( 𝜆
𝜆 + 𝜇) − ( 𝜆
𝜆 + 𝜇) 𝛼𝐸(𝑞)
𝛤𝐸(𝑞)∫ {𝑥∞ 𝐸(𝑞)−1𝑒−(𝜆+𝜇+𝛼)𝑥
0
}𝑑𝑥
𝑃01 = ( 𝜆
𝜆 + 𝜇) − ( 𝜆
𝜆 + 𝜇) 𝛼𝐸(𝑞) 𝛤𝐸(𝑞)
1
(𝜆 + 𝜇 + 𝛼)𝐸(𝑞)−1∫ {[(𝜆 + 𝜇∞
0
+ 𝛼)𝑥]𝐸(𝑞)−1𝑒−(𝜆+𝜇+𝛼)𝑥}𝑑𝑥
Let (𝜆 + 𝜇 + 𝛼)𝑥 = 𝑧
So, 𝑑𝑥 =(𝜆+𝜇+𝛼)1 𝑑𝑧 𝑃01 = ( 𝜆
𝜆 + 𝜇) − ( 𝜆
𝜆 + 𝜇) 𝛼𝐸(𝑞) 𝛤𝐸(𝑞)
1
(𝜆 + 𝜇 + 𝛼)𝐸(𝑞)∫ {𝑧∞ 𝐸(𝑞)−1𝑒−𝑧
0
}𝑑𝑧
𝛤𝐸(𝑞) = ∫ {𝑧𝐸(𝑞)−1𝑒−𝑧
∞
0
}𝑑𝑧
𝑃01 = ( 𝜆
𝜆 + 𝜇) − ( 𝜆
𝜆 + 𝜇) 𝛼𝐸(𝑞) 𝛤𝐸(𝑞)
1
(𝜆 + 𝜇 + 𝛼)𝐸(𝑞)𝛤𝐸(𝑞) 𝑃01 = ( 𝜆
𝜆 + 𝜇) − ( 𝜆
𝜆 + 𝜇) 𝛼𝐸(𝑞) (𝜆 + 𝜇 + 𝛼)𝐸(𝑞)
𝑃01 =𝜆+𝜇𝜆 [1 − ( 𝛼+𝜆+𝜇𝛼 )𝐸(𝑞)] ………(4.10)
𝑃00 = 1 − 𝑃01 ………(4.11)
45 4.3.1.2 Calculation of Cycle Cost
The ordering and holding cost for a single interval which starts with an inventory of 𝐸(𝑞)+ r and ends with r units is
𝐴(𝐸 (𝑞), 𝑟) = 𝑘 +ℎ𝐸(𝑞)𝛼 [𝐸(𝑞)+𝑟+𝑟2 ]
= 𝑘 +ℎ𝐸(𝑞2𝛼2)+ℎ𝐸(𝑞)𝑟𝛼 ………(4.12)
𝐸(𝑞) = 1/𝜃(1 − 𝑒−𝜃𝑞) [Erdem et al. (2006)] ………(4.13) 𝐸(𝑞2) = 2/𝜃2(1 − (1 + 𝜃𝑞)𝑒−𝜃𝑞)[Erdem et al. (2006] ………(4.14)
So, the total cost of the cycle (starts with state 0 and end with state 0) is
𝐶00= 𝑃00(𝐸(𝑞))𝐴(𝐸 (𝑞), 𝑟)+𝑃01(𝐸(𝑞))[𝐴(𝐸 (𝑞), 𝑟) + 𝐶10(𝑟)] ………(4.15)
The reason behind the above is when inventory level drops to r, the state will be ON (0) with probability P00 (E(q)) and 1(OFF) with probability P01 (E(q)). If the state is ON, the cost incurred will only be holding and inventory cost which is weighed by the probability of the event P00 (E(q)). On the other hand, if the state is OFF and inventory drops to r, the cost will be the summation of inventory and holding cost and the cost that incurred from the time when inventory drops to r and state is OFF to the beginning of the next cycle (again ON state), this cost is termed as C10. Equation 4 can be written as following
𝐶00= 𝐴(𝐸 (𝑞), 𝑟) + 𝑃01(𝐸(𝑞))𝐶10(𝑟) ………(4.16)
As, ordering and holding cost A (E(q), r), probabilities P00 (E(q)) and P01 (E(q)) are already defined; now C10 have to be defined. Referring to Fig. 4.1, which is equal to
46
C10 = ℎ𝑦 (𝑟+𝑟−𝑦𝛼2 ) 𝑦𝛼 < 𝑟 ………(4.17)
=ℎ𝑟2𝛼2+ 𝜋(𝑦𝛼 − 𝑟) + 𝜋̂ ∫0𝑦−𝑟/𝛼(𝑦𝛼 − 𝑟)𝑑𝑦 𝑦𝛼 ≥ 𝑟
=ℎ𝑟2𝛼2+ 𝜋(𝑦𝛼 − 𝑟) + 𝜋̂(𝑦22𝛼+3𝑟2𝛼2− 2𝑦𝑟) ………(4.18) So that
𝐶10(𝑟) = ∫ ℎ𝑦 (0𝛼𝑟 𝑟+𝑟−𝑦𝛼2 ) 𝜇𝑒−𝜇𝑦𝑑𝑦 + ∫ [𝑟∞ ℎ𝑟2𝛼2+ 𝜋(𝑦𝛼 − 𝑟) +
𝛼
𝜋̂(𝑦22𝛼+3𝑟2𝛼2−
2𝑦𝑟)]𝜇𝑒−𝜇𝑦𝑑𝑦 ………(4.19)
𝐶10(𝑟) = ℎ𝑟𝜇 ∫ 𝑦𝑒0𝑟𝛼 −𝜇𝑦𝑑𝑦−ℎ𝛼𝜇2 ∫ 𝑦0𝛼𝑟 2𝑒−𝜇𝑦𝑑𝑦+ℎ𝑟2𝛼2𝜇∫ 𝑒𝑟∞ −𝜇𝑦𝑑𝑦
𝛼
+ 𝜋𝛼𝜇 ∫ 𝑦𝑒𝑟∞ −𝜇𝑦𝑑𝑦
𝛼 − 𝜋𝑟𝜇 ∫ 𝑒−𝜇𝑦𝑑𝑦 + 𝜋̂𝛼𝜇/2 ∫ 𝑦𝑟∞ 2𝑒−𝜇𝑦𝑑𝑦
𝛼 𝑟∞
𝛼 − 2𝜋̂𝑟𝜇 ∫ 𝑦𝑒𝑟∞ −𝜇𝑦𝑑𝑦
𝛼 +
3𝜋̂𝑟2𝜇/2𝛼 ∫ 𝑒𝑟∞ −𝜇𝑦𝑑𝑦
𝛼 ………(4.20)
𝐶10(𝑟) = ℎ𝑟𝜇[𝑒−𝜇−𝜇𝑦]0
𝑟
𝛼−ℎ𝛼𝜇2 [𝑦2−𝜇𝑒−𝜇𝑦−2𝑦𝑒𝜇−𝜇𝑦2 −2𝑒𝜇−𝜇𝑦3 ]0
𝑟
𝛼+ℎ𝑟2𝛼2𝜇[𝑒−𝜇−𝜇𝑦]𝑟
𝛼
∞+ 𝜋𝛼𝜇[−𝑦𝑒𝜇−𝜇𝑦−𝑒−𝜇𝑦𝜇2 ]𝑟
𝛼
∞− 𝜋𝑟𝜇[𝑒−𝜇−𝜇𝑦]𝑟
𝛼
∞+𝜋̂𝛼𝜇2 [𝑦2−𝜇𝑒−𝜇𝑦−2𝑦𝑒𝜇−𝜇𝑦2 −2𝑒𝜇−𝜇𝑦3 ]𝑟
𝛼
∞− 2𝜋̂𝑟𝜇 [−𝑦𝑒𝜇−𝜇𝑦−𝑒−𝜇𝑦𝜇2 ]𝑟
𝛼
∞+ 3𝜋̂𝑟2𝜇/2𝛼[𝑒−𝜇−𝜇𝑦]𝑟
𝛼
∞ ………(4.21)
𝐶10(𝑟) = 𝑒−𝜇𝑟𝛼 [−𝜋̂𝑟𝜇 +ℎ𝛼𝜇2+𝜋𝛼𝜇 +𝜋̂𝛼𝜇2] +ℎ𝑟𝜇 −ℎ𝛼𝜇2 ………(4.22) Now C00 can be easily calculated from equation 4.15.
47 4.3.1.3 Calculation of Cycle Length
The formulation of objective function requires another thing which is the length of one cycle termed as T00. It can be calculated using the following equation
𝑇00= 𝑃00(𝑞)𝐸(𝑞)𝛼 + 𝑃01(𝑞)[𝐸(𝑞)𝛼 + 𝑇10] ………(4.23) Here T10 = E [time to reach to beginning of the next cycle when inventory drops to r and the state is OFF], the above equation can be also written as
𝑇00= 𝐸(𝑞)𝛼 +𝑃𝜇01 ………(4.24)
So, the average cost objective function now can be written as 𝐴𝐶(𝐸 (𝑞), 𝑟) =𝐶𝑇00
00= 𝐴(𝐸(𝑞),𝑟)+ 𝑃01(𝐸(𝑞))𝐶10(𝑟) 𝐸(𝑞)
𝛼 +𝑃01𝜇 ………(4.25)
Here A (E(q), r) will come from equation 4.12, C10 will come from equation 4.22, P01
will come from equation 4.10
Fig.4.1 Inventory level and status process with a single supplier r
E (q) +r
0
Cycle Cycle Cycle Cycle
Shortage/backorder Inventory level
Time
Status of supplier
ON OFF ON OFF ON OFF ON
48 4.3.2 Two Supplier Case
From now on, it is going to be assumed that the model consists of two suppliers rather than one. By considering a generalization of the previous single supplier model it can be easily developed. These two suppliers might be ON or OFF at different times. Like the previous model, their length of duration of ON and OFF periods are now also exponentially distributed with parameters 𝜆1 and μ1 corresponding to supplier 1 and 𝜆2 and μ2 corresponding to supplier 2. The status of the system can be any of the following mentioned in table 4.1 at any time but at time t=0 it will must in state 0.
Table 4.1 Possible states of the suppliers
State Status of supplier 1 Status of supplier 2
0 ON ON
1 ON OFF
2 OFF ON
3 OFF OFF
As there are four states now it is logical to have state respective three order quantities for state 0, 1 and 2. Similarly it is logical to have three reorder quantities. But if state dependent reorder quantities are considered the regenerative cycle will be very complex. So for simplicity the model is going to be developed for four decision variables which are qi = (q0, q1, q2) as state dependent order quantities and only one common reorder level, r. A related question is why one should go for state dependent order quantities rather than single quantity. The result lies in the values of the exponential distribution parameters of the suppliers. If they have different 𝜆i and μi
values, it will be wise to use different order quantities. For example, the case where the first supplier has a very low 𝜆1 which means a long expected duration for his ON period, if an order must be placed and this supplier is available it would be preferable to place an order q1 with him which should not be too different from the qEOQ of the standard EOQ model. Because he is likely to be available for a long time to come.
Similar comments would apply to the case where 𝜆2 may be very low.
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Another important factor is what to do when both suppliers are available at state 0.
There are three possible solutions. The entire order quantity for state 0 which is q0
could be given to either one of the two available suppliers or it could be diversified into both of the two suppliers. As suppliers have random capacities it will be more justified for the retailer to diversify the order between available suppliers to reduce the risk associated with shortages. Hence two more variable will come into the model namely, 𝑞0′ (quantity ordered from supplier 1) and 𝑞0′′ (quantity ordered from supplier 2) when both suppliers are available. The consideration of random capacities used in single supplier model is also applicable here. This means if any quantity q is ordered, received quantity will be E (q) rather than q. The value of E (q) depends on the capacity distribution. Just like the single supplier model random capacities of the two suppliers are also exponentially distributed in this model with parameters θ1 and θ2. The cost parameters k, h, 𝜋 and 𝜋̂ have the same interpretation as they have in single supplier model. Demand from customer end follows Poisson distribution with parameter α just as previous. But the calculation of transient probabilities, Pij (t) = Pr {being in state j at time t starting in state i at time 0}, i, j = 0, 1, 2, 3 is now a bit difficult as now the problem resembles four state CTMC. The calculations of these probabilities are going to be provided in the next page.
So, if the problem is summarized, it could be somewhat like this: a retailer replenishes its inventory from two of its supplier whose supply faces random disruptions and remain unavailable for random length. Suppliers have random capacities following exponential distribution. The policy that retailer is going to follow is denoted by (q (3), r) = (q0 (𝑞0′, 𝑞0′′) q1, q2, r) which means he will order q0 units when inventory drops to r and both suppliers are available (state 0) but diversify his order into supplier1 (𝑞0′) and supplier 2(𝑞0′′), in case of state i (i=1,2) he will order qi units when inventory drops to r. In case of state 3 when both suppliers are off he will order nothing and wait either any or both of the suppliers become available again. As soon as that happens he will order to make the inventory E (qi) + r; i= 0, 1, 2.
So, just like the previous model, the average cost objective function is 𝐴𝐶(𝑞0(𝑞0′, 𝑞0′′), 𝑞1, 𝑞2, 𝑟) =𝐶𝑇00
00= 𝐸[𝐶𝑜𝑠𝑡 𝑝𝑒𝑟 𝑐𝑦𝑐𝑙𝑒]
𝐸[𝑙𝑒𝑛𝑔𝑡ℎ 𝑝𝑒𝑟 𝑐𝑦𝑐𝑙𝑒] ………(4.26)