Thesis submitted to the Department of Industrial and Manufacturing Engineering, Bangladesh University of Engineering and Technology in partial fulfillment. Thesis titled “Development of an Inventory Model for Two Suppliers with Random Capacity Considering Supply Disruptions” submitted by Imtiaz Ahmed, Student no. P, was accepted as satisfactory in partial fulfillment of the requirements for the degree of Master of Science in Industrial and Production Engineering on May 4, 2014. Figure 6.2 Cost minimization (Best cost and Population . average) with the number of generations in GA 67 Figure 6.3 Relationship between average cost and in order.
Fig.6.4 Relationship between average cost and order quantity used for supplier 2 to create policy when both suppliers are available. Fig.6.5 Relationship between average cost and order quantity used for supplier 1 to arrive at a policy when only supplier 1 is available. Fig.6.6 Relationship between average cost and order quantity used for supplier 2 for policy when only supplier 2 is available.
LIST OF TABLES
NOMENCLATURE
Rationale of the Study
Again, suppliers may not always supply the exact quantity as ordered, so it is a fact that it is safe for companies to work with more than one supplier and spread the order among the available suppliers. Thus, considering the uncertainty of future supply, the uncertain return of suppliers and the issue of diversification in a single inventory model can not only reflect the actual scenario, but also help the manager to reduce costs. Although quite a bit of research has already been done in this area of disruption management, most of it deals with only one supplier.
So far, very few papers have considered the possibility of more than one provider in the field of disruption management. Although consideration of random capacity of suppliers has been added when determining the optimal ordering policy for a particular inventory model, but almost no one has considered the possibility of both supply interruption and random capacity of suppliers. Moreover, the diversification issue that arises when more than one supplier is available has not been addressed or overlooked so far in this field of disruption management.
Objectives of the study
Otherwise, the company may face a situation where the supplier delivers a much smaller quantity than the ordered quantity, which can cause not only serious damage to its reputation, but also a large economic loss. Again, almost all considered unit demand to avoid complexity in model design. But in reality, the demand generation process is not as simple as it is believed in the existing literature.
Considering these factors, the consideration of random capacity together with random supply disruption in the case of two suppliers when the demand is not unitary is still an open problem and constitutes the object of the proposed thesis. To optimize the final equation of the final objective of the formulated model using a suitable non-linear optimization technique to obtain the value of the decision variables which is mainly the state of the relevant order quantities and the reorder point. So, in short, the proposed research will help manufacturers develop an optimal inventory model to address the problem of future supply disruptions and random capacity of suppliers in response to random demand.
Outline of the Methodology
Nevertheless, using the concavity property of the expected profit function, they proposed an approximation to the optimal solution. They proved that the optimal inventory policy is of the type (s, S), where s depends on the state of the supplier in the last period, but S does not. For a given i, the size of the reorder point sni increases with the probability that the current order would be successful.
In addition, they studied the sensitivity of the optimal order quantity to different values of the model parameters. The duration of the ON period was assumed to be Ek (i.e. k-stage Erlangian) and the OFF period was common. They assumed that the duration of the ON periods for the two providers were distributed as Erlang random variables.
Note that supplier unavailability in our model only affected the acceptance of new replenishment orders in the future. For the linear cost model, they derived the structural properties and bounds of the optimal policy. He found that a supplier's percentage of uptime and the nature of the disruptions (frequent but short versus infrequent but long) were important determinants of the optimal strategy.
In Mohebbi and Hao (2006), they assumed that the processing of the open order 'resumes' when the supplier switches back to the on status. But in their work they assume that the processing of the open order will be 'restarted'. They obtained the form of the optimal policy and analyzed how supply disruption affected the profit function and the optimal policy.
The resulting rules were shown to be independent of the duration and frequency of breakdown. It was shown that the optimal repair schedule was dependent on the shortage cost parameters, as well as the magnitude of the disruption. Therefore, very little is known about the convergence properties of the method (Singer and Nelder, 2009).
Genetic Algorithm
Most articles addressing the issue of random supplier availability concern only one supplier. The manufacturer has no idea in advance of the availability and unavailability of any of the suppliers. The delivery lead time is considered zero because it is small compared to the average duration of the supply interruptions.
For generality, the time-dependent part of the cost of the unfulfilled order is also included as $𝜋̂/unit/time. So the total cost of the cycle is (starts with state 0 and ends with state 0). As in the one-supplier model, in this model with parameters θ1 and θ2, the random capacities of both suppliers are exponentially distributed.
In the case of state 3, when both suppliers are off, he will not order anything and wait for either one or both suppliers to become available again. Here, the time (t) spent in the diagonal entry of the above matrix depends on both the distribution followed by the consumption and the received order quantity that will be consumed. Since the consumption of received order quantity follows the erlang distribution, in the case of the order quantity when both suppliers are available (𝐸(𝑞0)), 𝐷11 will be derived from the following equation.
The value of the matrix 𝑫(𝑡) will be any of the three matrices defined above depending on the order quantity received depending on the condition to be consumed. As before, some of the timing components will be collected from the associated matrix adjustments and the total cycle length T00 will be calculated as follows. So, since the cost per cycle, C00 and the cycle length T00 are determined, the average cost objective function can now be written as.
The goal is to minimize the above equation and obtain the optimal values of the decision variables. In this section, a numerical example is presented to illustrate the solution process of the proposed inventory model. From now on, these parameter values will be used to understand the solution methodology.
Result Discussion
A computer programming code of the model was generated in Matlab R2009a and two of the solution methods were used to get the optimal values. From Fig 6.1 it can be observed that although the initial parameters are set differently and therefore the simplex of the Nelder Mead graphs begins to form a different domain, but the destinations of the paths or the convergence points are almost the same and found at the similar places of the graphs shown. Furthermore, to validate the effectiveness of the result obtained in this approach, another technique genetic algorithm approach is also used to have the optimal values of decision variables that minimize the total cost of inventory model per time unit.
Stall generation (G) and mutation rate (m) have been changed to get a better overview of the result. To observe the effect of the decision variables on total cost, 30 data points for each independent variable are plotted on the X-axis, holding all other variables constant, and the resulting average cost values are plotted on the Y-axis. These graphs provide a clear understanding of the relationship or effect of the variable change on the average cost.
All these graphs reveal the fact that initially the average cost decreases as the value of the decision variables increases. But after reaching the optimal value, the cost started to increase with the increasing value of the decision variables. Figure 6.3 Relationship between average cost and order quantity used for supplier 1 under the policy when both suppliers are available.
In order to study the effects of the model parameters, a sensitivity analysis is performed with the illustrative example described in chapter 5. The optimum value of the decision variables and the corresponding values of the objective function at different levels of the model parameters are shown in Table.6.4. Final average cost is highly sensitive to the increase in holding costs (11% and 22% increase against 20% and 40% increase in holding costs).
To better understand the result, two additional sensitivity analyzes were performed on the effects of 𝜆1/μ1 and ℎ/𝜋̂ on the decision variables and average costs.
Conclusions
This research shows that the traditional assumptions that were usually made in the past years during the design of the inventory model should be relaxed to make the model as close as possible to the real scenario. Therefore, manufacturers/retailers should always be prepared in advance for these adverse situations and include precautions in their inventory model that will minimize the impacts of such situations. The author expects that in the future some of the assumptions of the traditional inventory model will be relaxed and one day it will become just a reflection of the real world.
Recommendations
Joint optimization of inventory policies on a multi-product multi-echelon pharmaceutical system with batching and ordering constraints, European Journal of Operational Research. When supplier availability affects delivery time – an extension of the supply disruption problem. Step response optimization method of switched reluctance motor, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering.
