1. (jointly work with Maria Panta and George Kaimakamis
Exploring the impact of variables revenues and cost on the profit function with 3 members manufacturer ,wholesaler, retailer
Vassilios Vrysagotis,
(jointly work with Maria Panta and George Kaimakamis
ISBEFA 2012, KEFALONIA

2. Agenda Introduction Literature Review Solution Process Findings
Further research
References

3. Introduction
On our presentation we deal with a supply chain which is modeled as queuing network.
The parts of the queuing network (echelons of the supply chain) have stochastic leadtimes which are distributed according phase type distributions.
The supply chain is evaluated by two performance measures wip (mean inventory) and fill rate
We aim to explore the impact of variables , revenues and cost on the profit function for a distribution channel.

4. Distribution Channel
Manufacturer
Wholesaler
Retailer
Demand

5. Literature Review
Phase type distributions in multiechelon inventory system: From our research one paper [1] stresses the application of phase type distributions in inventory management .
Queuing networks: concerning supply queuing networks we can mention the paper of Kerbache and Smith [2] where types of supply queuing networks are presented.
Lost sales : several researchers have dealt with this feature of inventory system. Some of them are
Anderson and Melchiors [3] who examined a two-echelon inventory model with lost sales and one outstanding order
Zipkin [4] present exact performance measures for systems with lost sales and (S,s) inventory policy
Johansen and Thorstenson [5] consider a continuous review (r, Q) inventory system with Poisson demands and at most one order outstanding. The replenishment lead time is gamma distributed. Demands not covered immediately from inventory are lost

6. Literature Review (2)
Concerning Markovian chains and processes(or stochastic programming) : we discern the paper of Ollson & Hill[6] where steady distribution is derived and other performance measures such as WIP are calculated. On the paper of Dekker et al [7] a stochastic inventory model is presented and inventory measures as well as cost expressions are calculated. Further, an inventory model with Poisson demand and Erlangian lead times is analyzed on Johansen ‘s paper [8]. Also , on the paper of Chao and Zhou a stochastic inventory model is described and many cost expressions are derived[9]. On the paper of Liu and Lee [10] a markovian model is analyzed and performance measures are derived for a multi item base stock policy. On the paper of Shang and Shong [11] an analysis of a markovian model is presented in relation with constraints concerning fill rate.

7. Linear Regression Model
Solution Process
Input
Data for WIP
Data for fill rate
Profit function
Process
Linear Regression Model
Output R2
Conclusions

8. Input (1) Profit Function :
PROFIT (annual) =300*fillrate*lamda*Cp - Ch*WIP- 300*Cs*(1-fillrate
Where :
WIP : it refers to the average inventory which it is hold in the whole supply chain
Fill Rate: The percentage of the demand orders covered by the inventory
lamda : demand rate
Cp : selling price
Ch : holding cost of a unit
Cs : Cost of shortage (no having products to offer to customers) of a unit
We assume 300 working days of a year.

9. Input (2) Variable Revenue = 300*fillrate*lamda*Cp
Variable Cost = Ch*WIP-300*Cs*(1-fillrate)
We proceed with a regression analysis concerning the above mentioned variables

10. Output – Findings From our analysis , we can argue
The distribution cost has no such important role for a profit despite the fact all the efforts in businesses target at cost reduction
The revenues have vital role for a distribution channel.
Thus, the majority of business efforts must aim to revenues and especially to increase revenues.

11. References
On the use of phase type distributions for inventory management with supply disruptions. Baris Balcioglu, Ulku Gurler. Applied Stochastic Models in Business and Industry, 2011
Queuing networks and the topological design of supply chain systems. L. Kerbache , J. Mac Gregor. International Journal of Productions Economics, 2004, pp
A two-echelon inventory model with lost sales. Andersson, Jonas, Melchiors, Philip International Journal of Production Economics, Vol. 69, 2001, pp. 307–315.
Foundations of Inventory Management, Zipkin H. P. (2000), McGraw Hill, pp.279
Optimal and approximate (Q, r) inventory policies with lost sales and gamma-distributed lead time, Johansen Søren Glud, and Thorstenson Anders International Journal of Production Economics, vol , ,(1993), pp

12. References
A two echelon base stock inventory model with Poisson demand and sequential processing of orders at the upper echelon . R. Olsson , R.Hill. European Journal of Operational Research 177 (2007)
On the S-1, S Lost Sales Inventory Model with Priority Demand Classes. R.Dekker, R.M. Hill, M..J. Kleign , R.H. Teunter. Naval Research Logistics
Base stock policies for the lost sales inventory system with Poisson demand and Erlangian lead times. S. G. JOhansen , International Journal of Productions Economics, 93-94, 2005, pp
Probabilistic solution and Bounds for serial inventory systems with discounts and average costs. X. Chao ,S. Zhou Naval Research Logistics
Evaluation of inventory policies with unidirectional substitutions J.Liu, C.G.Lee European Journal of Operational Research 94 (1996) 527—547
A closed form approximation for serial inventory systems and its application to system design. K. Shang, .J.S. Song . Manufacturing and Service Operations Management vol. 8 No. 4 Fall pp
Exploring the impact of variables revenues and cost on the profit function with 3 members manufacturer ,wholesaler, retailer Maria Panta, Vassilios Vrysagotis, George Kaimakamis (under print)