Supply Chain Management Part 12 doc

20 A Multi-Agent Model for Supply Chain Ordering Management: An Application to the Beer Game Mohammad Hossein Fazel Zarandi1, Mohammad Hassan Anssari2, Milad Avazbeigi3 and Ali Mohagha

Complexity in Supply Chains:

A New Approach to Quantitative Measurement of the Supply-Chain-Complexity 431 Calinescu, A., et al 2001 Classes of complexity in manufacturing 17th national conference

on manufacturing research, 4–6 September, University of Cardiff, Cardiff, 351–356

Deshmukh, A.V., Talavage, J.J and Barash, M.M., 1992 Characteristics of part mix

complexity measure for manufacturing systems IEEE international conference on systems, man and cybernetics, 18–21 October, New York, USA, 1384–1389

Deshmukh, A.V., Talavage, J.J., and Barash, M.M., 1998 Complexity in manufacturing

systems Part 1: analysis of static complexity IIE Transactions, 30 (7), 645–655 Frizelle, G and Woodcock, E., 1995 Measuring complexity as an aid to developing

operational strategy International Journal of Operations and Production Management, 15 (5), 26–39

Gove, P.B., 1986 Complexity Webster’s third new international dictionary of the English

language unabridged Massachusetts, USA: Merriam-Webster-Verlag

Hoyle, D., 2009 ISO 9000 Quality Systems Handbook Butterworth-Heinemann, Oxford,

UK

Isik, F 2010 An entropy-based approach for measuring complexity in supply chains

International Journal of Production Research, 48 (12), 3681 – 3696

Lee, H.L., Padmanabhan, V and Whang, S., 1997a The bullwhip effect in supply chains

Sloan Management Review, 38 (3), 93-102

Lee, H.L., Padmanabhan, V and Whang, S., 1997b Information distortion in a supply chain:

The bullwhip effect Management Science, 43 (4), 546-558

Makui A and Aryanezhad M B., 2003 A new method for measuring the static

complexity in manufacturing, Journal of the Operational Research Society, 54(5), 555-557

Martin, J W (2007) Lean Six Sigma for supply chain management: The 10-step solution

process The McGraw-Hill Companies, Inc New York, NY

Martínez-Olvera, C 2008 Entropy as an assessment tool of supply chain information

sharing European Journal of Operational Research, 185 (1), 405-417

Milgate, M., 2001 Supply chain complexity and delivery performance: an international

exploratory study Supply Chain Management: An International Journal, 6 (3), 106–

118

Montgomery, C.D., Jennings, C.L., Kulahci, M., 2008 Introduction to Time Series Analysis

and Forecasting (Wiley Series in Probability and Statistics) John Wiley & Sons, Inc., Hoboken, NJ, USA

Ronen, B and Karp, R., 1994 An information entropy approach to the small-lot concept

IEEE Transactions on Engineering Management, 41 (1), 89–92

Sivadasan, S., et al., 2002 An information-theoretic methodology for measuring the

operational complexity of supplier–customer systems International Journal of Operations and Production Management, 22 (1), 80–102

Sivadasan, S., et al., 2006 Advances on measuring the operational complexity of supplier–

customer systems European Journal of Operational Research, 171 (1), 208–226 Shannon, C.E., 1948 A mathematical theory of communication Bell System Technical

Journal, 27, 379–423

Shannon, C.E., Weaver, W., 1949 The Mathematical Theory of Communication University

of Illinois Press, Chicago, IL

Wilding, R., 1998 The supply chain complexity triangle: uncertainty generation in the

supply chain International Journal of Physical Distribution and Logistics Management, 28 (8), 599–616

Yao, D.D., 1985 Material and information flows in flexible manufacturing systems Material

Flow, 2, 143–149

20

A Multi-Agent Model for Supply Chain Ordering Management: An Application to the Beer Game

Mohammad Hossein Fazel Zarandi1, Mohammad Hassan Anssari2,

Milad Avazbeigi3 and Ali Mohaghar2

1Amirkabir University of Technology

Supply chain ordering management (SCOM), which is the main concern of this book chapter is

an integrated approach to determine the ordering size of each actor of SC to the upstream actor aiming to minimize inventory costs of the whole supply chain SCOM is focused on the demand

of the chain aiming to reduce inventory holding costs, lower slacks, improve customer services, and increase the benefits throughout the entire supply chain (Chaharsooghi et al., 2008)

The observed performance of human beings operating supply chains, whether in the field or

in laboratory settings, is usually far from optimal from a system-wide point of view (Lee & Whang, 1999; Petrovic, 2008) This may be due to lack of incentives for information sharing, bounded rationality, or possibly the consequence of individually rational behaviour that works against the interests of the group In a few cases, the researchers' focus is placed on the coordination and integration of inventory policies between more than three stages (Kimbrough et al., 2002; Mahavedan et al., 1997; Petrovic et al., 1999; Wang & Shu, 2005) When there is no coordination among supply chain partners, each entity makes decision based on its own criteria, which results in local optimization as opposed to global optimum

So called Beer game (Sterman, 1989) is a well-known example of supply chain which has attracted much attention from practitioners as well as academic researchers Optimal parameters of the beer game ordering policy, when customers demand increases, have been analyzed in two different situations It has been shown that minimum cost of the chain (under conditions of the beer game environment) is obtained when the players have

different ordering policies rather than a single ordering policy (Strozzi et al., 2007) Indeed, most of previous works on order policy of beer game use genetic algorithms as optimization technique (Kimbrough et al., 2002; Strozzi et al., 2007)

One ordering policy based on genetic algorithm under conditions of the Beer game environment was introduced (Kimbrough et al., 2002); we call that GA-based algorithm in this chapter GA-based algorithm has some degrees of freedom contrary to 1-1 algorithm; In the GA-based algorithm, each actor of chain can order based on its own rule and learns its own ordering policy in coordination with other members with the aim of minimizing inventory costs of the whole supply chain

One limitation of the GA-based algorithm is the constraint of fixed ordering rule for each member through the time An attempt to mitigate the problem of fixed ordering rules was initiated in (Chaharsooghi et al., 2008), in this study a reinforcement learning model is applied for determining beer game ordering policy The RL model enables agents to have different rules throughout the time In this book chapter we try to extract multiple rules for each echelon in the supply chain using Genetic Algorithm

This book chapter can be viewed as a contribution to the understanding of how to design learning agents to discover insights for complicated systems, such as supply chains, which are intractable when using analytic methods In this chapter, the supply chain is considered as a combination of various multi-agent systems collaborating with each other Thus, SCOM can be viewed as a multi-agent system, consisting of ordering agents Each ordering agent tries to make decisions on ordering size of the relevant echelon by considering the entire supply chain Agents interact and cooperate with each other based on a common goal For example, in a linear supply chain with four echelons (as considered in this chapter), there are four ordering agents in SCOM system, each of which is responsible for ordering decisions in its particular echelon The main objective of ordering agents is to minimize long-term system-wide total inventory cost of ordering from immediate supplier This is a complex task because of the uncertainty embedded in the system parameters (e.g customer demand and lead-times) and demand amplification effect (Forrester, 1961), known as ‘bullwhip effect’(Lee & Wu, 2006; Fazel Zarandi & Avazbeigi ,2008; Fazel Zarandi et al., 2009)

Throughout this study, we use findings from the management science literature to benchmark the performance of our agent-based approach The purpose of the comparison is

to assess the effectiveness of an adaptable or dynamic order policy that is automatically managed by computer programs—artificial agents Also the results of the proposed model are compared with two other existing methods in the literature (Chaharsooghi et al., 2008; Kimbrough et al., 2002)

The rest of the book chapter is organized as follows In section 2, the proposed GA for agent supply chain is described in detail In section 3, the method is applied on different cases and is compared with other models in the literature Also in this section, the results are discussed Finally in the last section, conclusions are summarized

multi-2 Genetic algorithm with local search for multi-supply chain

2.1 Genetic Algorithm Pseudo Code

Genetic algorithms, originally called genetic plans, were initiated by Holland, his colleagues, and his students at the University of Michigan in the 1970s as stochastic search techniques based on the mechanism of natural selection and natural genetics, have received a great deal

of attention regarding their potential as optimization techniques for solving discrete optimization problems or other hard optimization problems (Masatoshi, 2002)

A Multi-Agent Model for Supply Chain Ordering Management: An Application to the Beer Game 435

2.2 Representation of ordering policies in GA

In the proposed GA, each rules set (ordering policy) is encoded using binary system In Fig

2, the encoding schema is demonstrated Each echelon in the supply chain has w rules All rules are represented in binary system with NumberOfBytes cells which NumberOfBytes is a

parameter of the model The first cell in each echelon rule, stores the sign of the rule 1 is for positive and 0 is for negative These cells are distinguished with grey colour The next

NumberOfBytes-1 bits represent how much to order

1 Initialization A certain number of rules (Ordering Policies) are randomly generated

to form generation 0

2 Pick the first binary rule from the current generation and decode the chosen rule to obtain the decimal ordering rules

3 Agents play the Beer Game according to their current decimal rules

4 Repeat step (3), until the game period (say 35 weeks) is finished

5 Calculate the total cost for the whole team and assign fitness value to the current rule

6 Pick the next rule from the current generation and repeat steps (3), (4) and (5) until the performance of all the rules in the current generation have been evaluated

7 Use GA with local search to generate a new generation of rules and repeat steps (2)

to (6) until the maximum number of generation is reached

Fig 1 The pseudo code of the proposed GA

W rules –instead of one rule– enable each agent to have a more adaptive and dynamic

behaviour The effect of different W’s on system objective function is also studied in next

Rule w-1 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0

Rule w 1 0 0 0 1 0 1 0 1 0 1 1 0 1 0 0 1 0 1 0

Fig 2 Encoding Schema

When it is needed to run a supply chain using a specific ordering policy, first it is mandatory that the chromosome of the ordering policy –similar to that shown in Fig 2– decoded to decimal system Two examples of decoding procedure are shown in Fig 3

Fig 3 Decoding Example

2.3 Objective function

In the MIT Beer Game, each player incurs both inventory holding costs and penalty costs if

the player has a backlog We now derive the total inventory cost function of the whole

supply chain We begin with the needed notation In the MIT Beer Game:

• N is the number of players and is 4

• IN i (t) is the net inventory of player i at the beginning of period t

• C i (t) is the cost of player i at period t

• H i is the inventory holding cost of player i, per unit per period (e.g., in the MIT Beer

Game, US$1 per case per week)

• P i is the penalty/backorder cost of player i, per unit per period (e.g., in the MIT Beer

Game, US$2 per case per week)

• S i (t) is the new shipment player i received in period t

• D i (t) is the demand received from the downstream player in week t (for the Retailer, the

demand from customers)

According to the temporal ordering of the MIT Beer Game, each player’s cost for a given

time period, e.g., a week, can be calculated as following: If IN i (t)≥0, then C i (t)=IN i (t)×H i else

C i (t)=|IN i (t)|×P i , where IN i (t)=IN i (t-1)+S i (t)-D i (t) and S i (t) is a function of both information

lead time and physical lead time The total cost for the supply chain after M periods is

1 1

( )

N M i

1) Selection Operator: In the proposed GA, for selection of the chromosomes from the current

population, the tournament method is chose In this method, at each time two chromosomes

are selected randomly from the current population and then the chromosome with the

minimum cost will be selected as a member of the next population This process continues

until the required chromosomes are chosen for the new population

2) Mutation Operator: Mutation in the proposed GA, includes the replacement of the

zero-cells with one-zero-cells and vice versa The Mutation type indicates that how many zero-cells should

change

3) Crossover Operator: Crossover operator randomly chooses 2*M columns (M: Crossover

Type) from the randomly chosen chromosome from the current population Then, the

position of two columns changes in the selected chromosome

4) Rearrangement Operator as Local Search of GA: Rearrangement operator, first randomly

choose a chromosome from the chromosomes selected by the Selection method, then choose

two cells randomly and change the positions of those cells randomly If the new

chromosome had a smaller cost function, then the operator adds the new chromosome to the

new population Otherwise, the operator repeats the process until an improvement occurs

3 Results and conclusions

To validate the proposed system, some experiments are designed The experiments and

their results are summarized in Tables 1 and 2 In the following, each experiment is

described in detail

A Multi-Agent Model for Supply Chain Ordering Management: An Application to the Beer Game 437

Experiment Number Of Bytes W Best Ordering Policy Lead Time

4 5 4 [0,2,12,4;4,8,5,8;0,4,4,8;0,9,3,2] 2 for all echelons

16 4 4 [0,1,3,3;1,3,5,6;0,2,6,6;0,0,7,3] 2 for all echelons

Table 1 Best ordering policies achieved by the method

In the first experiment, the performance of the multi-agent system is tested under

deterministic conditions The customer demands four cases of beer in the first 4 weeks, and

then demands eight cases of beer per week starting from week 5 and continuing until the

end of the game (35 weeks) When facing deterministic demand with penalty costs for every

player (The MIT Beer Game), the optimal order for every player is the so-called ‘‘pass

order,’’ or ‘‘one for one’’ (1–1) policy—order whatever is ordered from your own customer

As the result shows ([0, 0, 0, 0]) we found that the artificial agents can learn the 1–1 policy

consistently

In the second experiment, we explored the case of stochastic demand where demand is

randomly generated from a known distribution, uniformly distributed between [0, 15] Lead

time for all echelon is a constant value through the time and is 2 In this case the model is

compared with (Kimbrough et al., 2002) as the result show, the model outperforms

Kimbrough’s model

In experiment 3 and 4, the influence of window basis (w) on the objective function of the

problem is studied As it can be seen, more number of rules leads to smaller values of total

cost This supports the idea that more number of rules enables the agents to be more

adaptive and flexible to the environmental changes

Experiment Demand Total Best

Cost

Worst Total Cost

Avera

ge Total Cost

1-1 Best Total Cost

GA Best Total

RL Best Total Cost

1 All the demands are 8 except

Table 2 Comparison of models with other models in the literature

In experiments 5 to 9, the model is evaluated under more challenging conditions The

demand and lead time are both nondeterministic and have distribution function uniform [0,

15] and [0, 4] respectively The results are compared with 1-1 ordering policy (Chaharsooghi

et al., 2008; Kimbrough et al., 2002) The best objective function achieved by the model is

1979 which is much smaller than (Chaharsooghi et al., 2008) results (2417) Again the

positive effect of window basis can be seen as the number of window basis increases to

some extent the best objective function value decreases A trend stops at window basis equal

to 5 This can be due to the exponential growth in the search space, which makes the search

process so complex for GA (with the current encoding schema 25*5*4 = 2100 possible

solutions exist)

A Multi-Agent Model for Supply Chain Ordering Management: An Application to the Beer Game 439

Fig 4 Customer Demand in comparison with retailer

Fig 5 Customer Demand in comparison with wholeseller

Fig 6 Customer Demand in comparison with Distributer

Fig 7 Customer Demand in comparison with manufacturer

In experiments 10, 11 and 12, the proposed window basis model is again compared with 1-1

ordering policy 1-1 ordering policy is described in (Kimbrough et al., 2002; Sterman, 1989)

In all cases, the model has a better performance The ordering values of four echelons base

on the best ordering policy achieved by the model for experiment 10 are depicted in fig 4, 5,

6 and 7

In the last 4 experiments, the model is applied on a periodic function with the function of

F(x)=|MaxDemand*sin(x.Π/Period)| (2) and the impact of different window basis is studied in this function Max Demand is 7 and

period is 8 As table 2 shows, models with window basis with the 2 multiples have a better

performance

It should be noted that in the first 12 experiments, the genetic population is 100, the number

of generation is 400, the mutation, crossover and the rearrangement ration are 0.2 In the last

four experiments, the genetic population is 300, the number of generation is 400, the

crossover and mutation ratio are 0.3 and the rearrangement ratio is 0.2

4 Conclusion

In this a new intelligent multi-agent system is proposed for determination of the best

ordering policy in order to minimize the cost of supply chain

The model is compared with previous models in the literature and as the results show, the

model outperforms all the previous models

The best ordering policy is obtained by a new genetic algorithm which is equipped with

some local searches One limitation of the previous presented GA-based algorithms is the

constraint of fixed ordering rule for each member through the time To resolve this problem

a new concept –window- is introduced in this book chapter Application of the window

basis enables the agents to have different ordering rules throw the time Experiment results

prove that the new multi-agent system is capable of finding patterns in nondeterministic

and periodic data both

A Multi-Agent Model for Supply Chain Ordering Management: An Application to the Beer Game 441

5 References

Chaharsooghi, S.K., Heydari, J & Zegordi, S H (2008) A reinforcement learning model for

supply chain ordering management: An application to the beer game Decision

Support Systems Vol 45, page numbers 949–959

Fazel Zarandi, M.H & Avazbeigi, M (2008) A Fuzzy Logic Approach to prove

Bullwhip Effect in Supply Chains, Proceedings of AMERICA, Toronto, ON.,

Canada

Fazel Zarandi, M.H., Avazbeigi, M & Turksen, I B (2009) An intelligent fuzzy Multi-Agent

System for reduction of bullwhip effect in supply chains, Proceedings of NAFIPS09,

Cincinnati, Ohio, USA

Forrester, J W (1961) Industrial Dynamics, Cambridge, MIT Press, Massachusetts

Giannoccaro, I., Pontrandolfo, P & Scozzi, B (2003) A fuzzy echelon approach for inventory

management in supply chains European Journal of Operational Research, Vol 149,

page numbers 185–196

Kimbrough, S.O., Wu, D J & Zhong, F., 2002 Computers play the beer game: can artificial

agents manage supply chains? Decision Support Systems, Vol 33, page numbers

323–333

Lee, H & Whang, S (1999) Decentralized multi-echelon supply chains: incentives and

information Management Science, Vol 45(5), page numbers 633– 640

Lee, H T & Wu, J C (2006) A study on inventory replenishment policies in a two echelon

supply chain system Computers & Industrial Engineering, Vol 51, page numbers

257–263

Masatoshi, S (2002) Genetic Algorithms and Fuzzy Multiobjective Optimization, (Operations

Research Computer Science Interfaces Series), Massachusetts: Kluwer Academic Publishers

Mahadevan, S., Marchalleck, N., Das, K T & Gosavi, A (1997) Self-improving factory

simulation using continuous-time average-reward reinforcement learning

Proceedings of the 14th International Conference on Machine Learning, page numbers

202–210

Petrovic, D., Roy, R & Petrovic, R (1999) Modeling and simulation of a supply chain in an

uncertain environment European Journal of Operational Research, Vol 109, page

numbers 299–309

Petrovic, D., Xie, Y., Burnham, K & Petrovic, R (2008) Coordinated control of distribution

supply chains in the presence of fuzzy customer demand European Journal of

Operational Research, Vol 185, page numbers 146–158

Sterman, J (1989) Modeling managerial behavior: misperceptions of feedback in a dynamic

decision making experiment Management Science, Vol 35 (3), page numbers 321–

339

Strozzi, F., Bosch, J & Zaldívar, J M (2007) Beer game order policy optimization under

changing customer demand Decision Support Systems, Vol 42, page numbers 2153–

2163

Wang, J & Shu, Y.-F (2005) Fuzzy decision modeling for supply chain management Fuzzy

Sets and Systems, Vol 150, page numbers 107–127

Wang, J & Shu, Y.-F (2007) A possibilistic decision model for new product supply chain

design, Journal of Operational Research, Vol 177, page numbers 1044–1061

Nita H Shah1 and Kunal T Shukla2

1Department of Mathematics, Gujarat University, Ahmedabad – 380009, Gujarat,

2JG College Of Computer Application, Drive – in road, Ahmedabad – 380054, Gujarat,

India

1 Introduction

The classical economic order quantity model of Wilson’s was developed with the assumption that the buyer must pay off immediately on arrival of the goods in the inventory system In fact, offering buyers to delays payment for goods received is considered as a sales promotional tool in the business world With offer of trade credit, vendor increases sales, attracts more buyers and reduces on – hand stock level Under this marketing strategy, the time of the buyer’s capital tied up in stock reduced which eventually reduces the buyer’s holding cost of finance In addition, during this allowable credit period, the buyer can earn interest on the generated revenue For the small – scale industries having a limited finance, the trade credit acts as a source of short – term funds Goyal (1985) developed an economic order quantity model with a constant demand rate under the condition of permissible delay in payments After that numbers of variants of the trade credit problem have been analyzed For example Shah (1993a, 1993b), Aggarwal and Jaggi (1995), Kim et al (1995), Jamal et al (1997), Shinn (1997), Chu et al (1998), Chen and Chung (1999), Chang and Dye (2001), Teng (2002), Chung and Huang (2003), Shinn and Hwang (2003), Chung and Liao (2004, 2006), Chung et al (2005), Teng et al (2005), Ouyang et al (2005) and their cited references For up – to day available literature on permissible delay period, refer to the article by Shah et al (2010)

The above cited references assume that the vendor offer the buyer a “one – part” trade credit, i.e the vendor offers a permissible delay period If the account is settled within this period, no interest is charged to the buyer As a result, with no incentive for making early payments, and earning interest through generated revenue during the credit period, the buyer postpones payment up to the last day of the permissible period offered by the vendor

As an outset, from the vendor’s end, offering trade credit leads to delayed cash inflow and increases the risk of cash flow shortage and bad debt To increase cash inflow and reduce the risk of a cash crisis and bad debt, the vendor may offer a cash discount to attract the buyer to pay for goods earlier i.e the vendor offers a “two – part” trade credit to the buyer

to balance the trade off between delayed payment and cash discount For example, under an agreement, the vendor agrees to a 2% discount to the buyer’s purchase price if payment is made within 10 days Otherwise, full payment is to be settled within 30 days after the

delivery In financial management, this credit is denoted as “2|10 net 30” If the vendor only offers the buyer a 30 days credit period, i.e “one – part” trade credit, then this is denoted as

“net 30” (Brigham, 1995) The papers related to this credit policy are by Lieber and Orgler (1975), Hill and Riener (1979), Kim and Chung (1990), Arcelus and Srinivasan (1993), Arcelus et al (2001, 2003) Ouyang et al (2002), Chang (2002) and Huang and Chung (2003) developed inventory models when two – credit policy is offered by the vendor to the buyer The above cited model’s are derived either from the vendor’s or the buyer’s end However, the two players may have their own goals The decision taken from the buyer’s end may not be agreeable to vendor and vice versa Lee et al (1997) argued that without coordinated inventory management in the supply chain may result in excessive inventory investment, revenue reduction and delays in response to customer satisfaction Therefore, the joint discussion is more beneficial as compared to the individual decision Goyal (1976) first developed a single vendor – single buyer integrated inventory model Banerjee (1986) extended Goyal’s (1976) model under assumption of a lot – for – lot production for the vendor Later, Goyal (1988) established that if vendor produces an integer multiple of the buyer’s purchase quantity then the inventory cost can be reduced Lu (1995) generalized Goyal’s (1988) model by relaxing the assumption that the vendor can supply to the buyer only after finishing the entire lot size Bhatnagar et al (1993), Goyal (1995), Viswanathan (1998), Hill (1997, 1999), Kim and Ha (2003), Kalle et al (2003), Li and Liu (2006) developed more batching and shipping policies for an integrated inventory model However, these articles did not incorporate the effect of trade credit on the integrated optimal decision Abad and Jaggi (2003) developed a vendor – buyer integrated model assuming lot – for – lot production under a permissible delay in payments Later, Shah (2009) extended Abad and Jaggi’s(2003) model for deteriorating items In both the articles, the vendor offered a “one – part” trade credit to the buyer

Ho et al (2008) studied impact of a “two – part” trade credit policy in the integrated inventory model This model assumed that units in inventory remain of 100% utility during the cycle time However, the products like medicines and drugs, food products, vegetables and fruits, fashion goods, x – ray films etc loose its 100% utility in due course of time In this chapter, we analyze effect of a “two – part” trade credit policy in the integrated inventory model when units are subject to constant deterioration and demand is retail price sensitive The supplier offers the buyer a cash discount if payment is made before an allowable period, and if the buyer does not pay within the allowable period, the full account against purchases made before the delay payment due date The joint profit is maximized with respect to the optimal payment policy, selling price, lot – size and the number of shipments from vendor

to buyer in one production run An algorithm is developed to determine the optimal policy Numerical examples are given to validate the theoretical results The sensitivity analysis of the optimal solutions with respect to model parameters is also carried out

2 Assumptions and notations

The proposed model is formulated using the following assumptions and notations

1 The integrated inventory system comprises of a single – vendor and single buyer for a single item

2 Shortages are not allowed

3 The inventory holding cost rates excluding interest charges for the vendor is Iv and for the buyer is Ib

4 To accelerate the cash inflow and reduce the risk of bad debt, the vendor offers a

discount β (0 < β < 1) off the purchase price, if the buyer settles the account within time

M1 Otherwise, the full account is due within time M2, where M2>M1≥0

A Collaborative Vendor – Buyer Deteriorating Inventory Model

for Optimal Pricing, Shipment and Payment Policy with Two – Part Trade Credit 445

5 The vendor’s unit production cost is $ Cv and unit sale price is $ Cb The buyer’s unit retail price is $ P HereP C> b>(1−β)C b>C v

6 During the allowable credit period to the buyer, the vendor opts to give up an immediate cash inflow until a later date Thus, the vendor endures a capital opportunity cost at a rate Ivo during the time between delivery and payment of the item

7 During period [M1, M2], a cash flexibility rate f is available to quantize the advantage vc

of early cash income for the vendor

8 During the credit period (i.e M1 or M2), the buyer earns interest at a rate of Ibe on the revenue generated by selling the product

9 The demand rate for the item is a decreasing function of the sale price and is given

by ( )R P =aP−η, where a > 0 is scaling demand, and η > 1 is a price – elasticity

coefficient

10 The capacity utilization “ρ” is defined as the ratio of the demand rate, R(P) to the production rate p(P), i.e ρ = R(P)/p(P) where ρ < 1 and is fixed

11 The buyer’s cycle time is T, order quantity is Q per order

12 The buyer’s ordering cost per order is Ab

13 During the production period, the vendor produces in batches of size nQ (where n is a positive integer) and incurs a batch set up cost Av After the production of first Q units, the vendor ships them to the buyer and then makes continuous shipping at every T- units of time until the vendor’s inventory level depletes to zero

14 The units in inventory deteriorate at a constant rate, θ (0 < θ < 1) The deteriorated can

neither be repaired nor replaced during the cycle time T

3 Mathematical model

The inventory on hand depletes due to price – sensitive demand and deterioration of units The rate of change of inventory at any instant of time ‘t’ is governed by the differential equation,

3.1 Vendor’s total profit per unit time

During each production run, the vendor produces in batches of the size nQ with a batch set

up cost Av The cycle length of the vendor is nT- units Therefore, the vendor’s set up cost per unit time is (Av/nT) Using method given by Joglekar (1988), with the unit production

cost Cv, the inventory holding cost rate excluding interest charges Iv and capital opportunity

cost per $ per unit time Ivo, the vendor’s carrying cost per unit time is

For each unit of item, the vendor charges ( (1−K jβ)C b)if the buyer pays at time Mj,j=1,2,

K1 = 1 and K2 = 0 The opportunity cost at the finance rate Ivo per unit time for offering trade

credit is( (1−K jβ)C I b⋅ vo⋅M j⋅Q T) However, if the buyer pays at M1 - time, during M2 – M1

the vendor can use the revenue ((1 – β)Cb) to avoid a cash flow crisis The advantage gain

per unit time from early payment at a cash flexibility rate fvc is

(K j⋅ −(1 β)C f b⋅ vc⋅(M2−M Q T1) / )

Thus, the vendor’s total profit per unit time is the revenue generated plus the advantage

from early payment minus production cost, set up cost, inventory holding cost and

opportunity cost for offering trade credit

3.2 Buyer’s total profit per unit time

The buyer’s ordering cost is Ab for each order of Q – units, so the ordering cost per unit time

is (Ab/T) The inventory holding cost excluding interest charges per unit time is

On the basis of length of the payment time, two cases arise: (i) T< Mj and (ii) T ≥ Mj ; j=1,2

These two cases are shown in Figure 1

Case: (i) T < Mj; j = 1, 2

Here, the buyer’s cycle time ends before the payment time So buyer does not pay

opportunity cost for the items kept in stock The buyer earns interest at the rate of Ibe on the

revenue generated; hence, the interest earned per unit time is,

2 0

A Collaborative Vendor – Buyer Deteriorating Inventory Model

for Optimal Pricing, Shipment and Payment Policy with Two – Part Trade Credit 447

Fig 1 Inventory and interest earned for the buyer under trade credit

Case: (ii) T ≥ Mj; j = 1, 2

In this case, the buyer’s allowable payment time ends on or before the inventory is depleted

to zero The interest earned per unit time is

2 0

( )( )

After the due date Mj, the buyer pays interest charges at the rate of Ibc Therefore, the interest

charges payable per unit time is,

−

∫

The buyer purchase cost per unit time is ((1 – Kj β) Cb Q/T) and revenue generated per unit

time is (PQ/T) Therefore, the buyer’s total profit per unit time is revenue generated plus

interest earned minus the total cost comprises of the purchase cost, ordering cost, inventory

holding cost excluding interest charges and interest charges payable, i.e

1 2

T be

j

C Q A PQ

ββ

θθ

K C I R P

T

θ θ

ββ

θθ

β

θθ

3.3 The joint total profit per unit time

When the buyer and vendor opt for the joint decision, the joint total profit per unit time is,

1 2

θβ

θθ

A Collaborative Vendor – Buyer Deteriorating Inventory Model

for Optimal Pricing, Shipment and Payment Policy with Two – Part Trade Credit 449

θ

θ θ

β

θββ

θθ

βθ

TP P c R P

M T A T

T n T

βθ

θβ

θβ

β

ββ

∂ for j = 1, 2 suggest that TPj (n, P, T) is a concave function in ‘n’

This guarantees that the search for the optimal shipment number n* is reduced to find a

local optimal solution

4.1 Determination of the optimal cycle time ‘T’ for any given ‘n’ and ‘P’

For given n and P, the partial derivative of TPj1 (n, P, T) in (6 – a) with respect to T,

= − ⎜ + ⎟<

∂ ⎝ ⎠ suggests that TPj1 (n, P, T) is a concave function in T

Hence, there exists unique value of T = Tj1(n, P) (say) which maximizes TPj1(n, P, T) Tj1(n, P)

can be obtained by setting TP n P T j1( , , ) 0

A A n T

j v

R P M A

j v

R P M A