The simulation for the option contract in the case of one retailer and multiple ... This document explores option contracts in supply chains. .... maintenance, etc.).
19th International Conference on Production Research
SUPPLY CONTRACT WITH OPTIONS a
A. Gomez_Padilla , T. Mishina a
b
Industrial Engineering Department, University of Guadalajara, Av. Revolucion No. 1500, Guadalajara, Jalisco, Mexico b Department of Management Science and Engineering, Akita Prefectural University, 84-4 Ebinokuti, YuriHonjo, Akita, Japan
Abstract The purpose of this paper is to analyze the impact of an option contract for two companies that are both members of a supply chain: a retailer and a supplier. An option contract is one where the retailer orders a quantity of units and has a right to modify his order if necessary. This ordered quantity may be modified in any sense (it may be bigger or smaller than the initial order) without restrictions by buying an option premium in advance from the supplier. This document presents a model to calculate the performance of an option contract in terms of contract value for the two companies engaged. First, the case of multiple suppliers and one retailer, and then the particular case of one supplier and one retailer are considered. Simulation is used since demand is nonstationary and the theoretical model presented is dynamic. The simulation for the case of one retailer and one supplier permits to compare the performance improvement obtained using this kind of contract. The simulation for the option contract in the case of one retailer and multiple suppliers demonstrates that it will be beneficial not only within the retailer or the, but to the supply chain as a whole. Keywords: options, contracts, supply chain, performance
1 INTRODUCTION This document explores option contracts in supply chains. With an option contract the buyer (retailer in this case) may modify its ordered quantity to the seller (supplier in this case) by paying and option premium. It is considered the general case of one retailer and multiple suppliers with different contracts and the particular case of one retailer and one supplier with an option contract. In the general case, the suppliers provide the retailer an identical product and the retailer has to satisfy the final market demand. Neither the retailer nor the suppliers may affect the market behavior. Demand is a nonstationary stochastic process. The retailer will use the available information of the demand from previous periods to establish the growth rate and the upper and lower limits of growth of the demand. With this information the option premium may be calculated. This kind of contract may have applications in a variety of sectors. For example, in the service sector, if a travel agency reserves a number of airplane tickets from a company but then, after observing variations on his expected demand, decide to increase or diminish the number of reserved seats. In production companies, if a retailer passes an order for a new product, but then realizes that this new product is being more (or less) successful than expected, it will then be interesting for him to modify the order. In both cases, the airline company and the supplier of the new product will receive the compensation for the modification on the order by the option premium. The model for the general case of multiple suppliers and one retailer is presented, as well as the particular case of one supplier and one retailer. The decisions criteria for each member of the chain are presented. After presenting the model, the study continues by a simulation approach. It was decided to simulate since demand is nonstationary and the theoretical model presented is dynamic. Simulation is needed in order to test under which conditions this contract will be interesting and to define the proportion of some variables.
The purpose is to first test different conversion rates to calculate the option premium. The retailer will prefer a low conversion rate while the supplier will prefer a high one. The option premium also considers the growth rate and the upper and lower limits of growth. First it is simulated the case when there is one retailer and one supplier exchanging products with an option contract. In this part it was obtained information on the importance of an option contract with one supplier. Then it was simulated the case of one retailer and i suppliers, where each supplier has different contract. In this part results are compared to determine if this contract will be interesting for the case of multiple suppliers. In this document, as in [1], the retailer will be refered as “he” and each supplier as “she” to simplify and facilitate comprehension. The literature survey is presented in next section, and in section 3 the general model is introduced. The document continues with the model for the particular case of one supplier and one retailer in section 4. The simulation results ere presented on section 5. Section 6 concludes and present perspectives for further work. 2 LITERATURE SURVEY The fact of get engaged to provide and order quantities and to pay financial amounts implies decision making, decisions that will be expressed and established in the contract. These decisions are made in order to achieve certain goals fixed by each company. Strategic decisions are those that influence the long term evolution of the company. Regarding a contract, this concerns first of all, the decision of establishing a contractual relation or not. Non exhaustive examples of strategic decisions would be: future buying options, management of transaction specific investments, cost analysis of transactions, intellectual propriety, reselling licenses, commercial agreements, cooperation dynamics, technological evolution, change rate fluctuation, legal
instances and relational exchanges. All this is defined during the negotiation process. Important decisions stablished in contracts are prices and volumes. The prices are fixed depending on the type of contract, and the contract determines the financial flow or transfer between the companies. Volumes normaly depend on demand. A complete study of prices and volumes for different kinds of contracts is presented in [1] under a newsvendor perspective, i.e., no stock is possible. The author analysed five contracts (wholesale, buy back, quantity flexibility, revenue sharing and rebate) and identified under which conditions the chain will be coordinated. Concerning quantity commitments and the option to modify this, [2] distinguish two approaches. In the first one the contract fixes the total physical flow to be ordered on N periods, and it is necessary to distribute this quantity the best it is possible in the periods ("flexibility of the quantity"). In the second approach the contract fixes total financial flow on N periods and the objective is to distribute the better the orders per period. In [2] The "flexibility of the quantity" refers to the variability that the ordered quantity can present from one period to another. If the quantity ordered on one period can be different from the quantity ordered at the previous period then quantity is flexible. The quantity to be ordered on a fixed horizon is established by the contract. The quantities to be ordered on each period remain to be fixed, and they can be different from one period to another. In their study two types of are distinguished: wholesale price and capacity reservation. [3] analyse and compare six types of contacts in a situation where stock is held. The relations between two companies vertically related are studied: a supplier and a client, who coordinate by logistic and economic decisions. The objectif is to model and evaluate the decisions of each actor, for each studied contract: wholesale price, rebate, quantity flexibility, buy back, revenue sharing and capacity reservation. The author studied if the contract allows to effectively coordinate the chain. The study of the contracts by [3] was made in order to improve the benefit of the upstream company (supplier) compared to the benefit obtained with a wholesale price contract. The research highlights the importance of the contract for the individual benefits, for the replenishment policy of the downstream company (retailer) and in the contractual strategy of the upstream company. The obtained results where that the quantity flexibility and buy back contracts make it possible to effectively coordinate the chain, i.e., they simultaneously maximizes the profits of each actor and of the whole chain. This means that with these two contracts it is possible to fix the contract parameters that will optimise the benefit for the two companies and for the chain formed by them. An option contract was modelled in [4] for a two periods horizon. The upstream compay engages to buy a fixed quantity per period plus a certain number of options (or additional units of the product) which it "buys" at each period for the following period, at a higher price. The upstream compay orders the quantities for the first and second period, and it reservs a quantity of options. The options represent the quantity of units the downstream company can order that the upstream compay engages to provide if they are required. The upstream compay offers the ordered quantities for a wholesale price for the two periods. The options are reserved by the payment of a price. The downstream company must pay an additional price for the actually ordered options (exerted options). The authors modelled the problem for a normally distributed demand over each period and with nonnull correlation between the two periods. Their objective is to maximize the profit and to determine the number of options to reserve.
In [5] a single period contract is studied at wholesale price where options can be excercised as calls and puts (bidirectional options). When it is exercised as a call option, exercise price is higher than wholesale price. In [5] exists a constant fixed wholesale price. The variation in the amount paid to the supplier comes from the variation in quantity. The case of option contracts design for a single product under an agency perspective is presented in [6]. In their model the supplier is the principal and the retailer is the agent. Their model uses option contracts and renegotiation to prevent the principal to hold up the agent. In [7], the authors explore options in supply contracts where order levels change as a response to exchange rate fluctuations.[8] study portfolios of contracts. They divide contracts in three levels according to the demand tier they are going to satisfy, nature of relationship with the supplier and the purchasing objectives. [8] focuses on inventory holding costs and purchase cost of parts. A special case where it is the supplier who has the option to not to satisfy the quantity commitment is presented in [9], where the retailer will pay a bonus to the supplier if she meets the order quantity. In this paper it is studied the case of an option contract (bidirectional option in the sense of [5]), where prices remain constant but the retailer has to pay an option premium to the supplier. In the simulation of one supplier and one retailer an option contract is compared with a wholesale price and a buy back contract. It was decided to consider a wholesale price because of its widespread use and the buy back contract because in [3] it was demonstrated that it could effectively coordinate the chain. The approach presented in this document is different from [5] since the variation in the amount paid to the supplier comes from the variation in quantity and also from the option premium. In both cases the variation is calculated from expected demand variation. [5] show that a bidirectional option contract provides better results in terms of benefit for the retailer, but it is not considering the impact of this contract over the supplier and over the chain. In this paper the results for the supplier and the chain are also considered. The objective of this document is not to find the better order distribution along the periods of contract validity as in [2], but to analyze how an option contract affects the benefit of the chain and its members. The presented model is not considered a renegotiation possibility, and it is not taking into account fluctuation rates. 3
MODEL
3.1 Variables One retailer has i suppliers. He is engaged with each one of them by a contract lasting a time T. The variables related to demand are: • r =average growth rate of demand • u = upper limit growth of demand • d =lower limit growth of demand • Du(t) = higher expected demand at time t • Dd(t) = lower expected demand at time t • X(t) = expected demand at time t • D(t) = demand at time t The variables related to the contracts are: • p(t) = price of the product on the final market at time t Si
• w (t) =price at which supplier i sells the product to the retailer at time t
19th International Conference on Production Research Si
• b (t) =buy back amount paid by supplier i to the retailer at time t per unit in stock Si
• OP (t)= option premium price paid to supplier i at time t; it may be a call or a put option, and each type of option is calculated in a different way • CR = conversion rate to calculate the option premium price The variables related to the retailer decisions are: R
• S (t) =units sold by the retailer at time t Si
• Q (t) =ordered quantity from the retailer to supplier i at time t R
• J (t) = units available by the retailer at time t
3.3 Sequence of events At a period t, where t Є [0,T], the retailer decides how much to order Q(t). The retailer define this quantity according to the expected variations of demand D(t). In order to determine Q(t) the retailer will analyze the market conditions. To decide Q(t) he will then observe what happened on period (t – 1). He will consider the average growth rate r, the upper limit growth u and the lower limit growth d. From the actual demand at that period, D(t – 1), he will be able to determine the upper and lower expected demands (Du(t) and Dd(t) respectively) for the next period, as well as the expected demand X(t). Figure 1 represents this schematically.
R
• c (t) =cost in which the retailer incurs to make the product available to the client at time t R
• h (t) =inventory holding cost of the retailer at time t
Du
R
• I (t) =inventory hold by the retailer at time t RSi
• I (t) =inventory from supplier i hold by the retailer at time t
u X
R
• β (t) =cost for the retailer of not satisfying demand at time t
r
D(t-1)
R
• B (t) = unsatisfied demand of the retailer at time t
d
• FCR = fix costs of the retailer (production, stock, maintenance, etc.)
Dd t
• TS(t) = transfer from the retailer to the suppliers at time t Si
• TS
(t) = transfer from the retailer to supplier i at time t
R
• π (t) = profit of the retailer at time t The variables related to the supplier i decisions are: Si
• S (t) =units sold from supplier i to the retailer at time t
t-1
t
Figure 1: Expected demand at time t Q(t) will then be the expected demand X(t) plus the necessary security stock. After deciding Q(t) he will then decide how much to order Si to each supplier, Q (t), where:
Si
• c (t) =production cost of supplier i at time t Si
• M (t) = units produced by supplier i at time t Si
• J (t) = units available by supplier i at time t Si
• h (t) = holding cost of supplier i at time t Si
• I (t) = inventory hold by supplier i at time t Si
• β (t) = cost for supplier i of not satisfying the demand from the retailer at time t Si
• B (t) = demand not satisfied by supplier i at time t Si
• FC = fix costs of supplier i (production, stock, maintenance, etc.) Si
• π (t) = profit of supplier i at time t • ∏(t) = profit of the chain at time t 3.2 Relation between economic variables The price over the final market p(t) is bigger than the sale Si price w (t) from supplier i to the retailer and the production R Si R cost c (t), so that there is a revenue: p(t) > w (t) + c (t). Si This sale price w (t) is bigger than the holding cost of the R retailer h (t), so that stock is interesting. In the case of the buy back contract, to prevent the retailer from making profit out of inventory the holding cost of the R Si retailer h (t) is bigger than the buy back price b (t). The Si R Si relations are then: w (t) > h (t) > b (t). Si The sale price w (t) offered to the retailer is bigger than Si the production cost c (t), which is also bigger than the Si holding cost of the supplier h (t). The holding cost of the Si supplier is bigger than the buy back price b (t), so that this contract represent a benefit for the supplier, otherwise it would be more interesting for him to hold a bigger Si Si Si inventory. The relation will then be: w (t) > c (t) > h (t) > Si b (t).
I
Q(t ) = ∑ Q Si (t )
(1)
i =1
Si
Si
The retailer orders Q (t) units, and he will receive S (t) units from supplier i. It is important to recall that the supplier will receive or not the whole ordered quantities depending on the units available by each supplier. The S units received from the suppliers S (t) is then: I
S S (t ) = ∑ S Si (t )
(2)
i =1
The units available by the retailer will be the units received from the suppliers plus the units hold in stock from the R S R previous period so J (t)= S (t) + I (t-1). After receiving the S units S (t) from suppliers, demand D(t) occurs. If demand is smaller or equal to the units available by the R retailer J (t), he will satisfy all the demand, which will be R his sales S (t)=D(t) and he will have left the unsold units R R R as stock I (t) he will carry for next period, so I (t)= J (t) D(t). If demand is bigger than the units available, he will not be able to satisfy all the demand, and his sales will be for the R R available amount of units S (t) = J (t). The unsatisfied R R demand will be B (t) = D(t)- J (t). The retailer will then decide his order for the next period. Retailer decides how much to order according to the expectance of demand as it was previously explained. Si Each i supplier receives an order for Q (t) units. If the order received is smaller or equal to the available Si Si units J (t), her sales will be for the ordered quantity S Si Si (t)=Q (t) and she will have left the unsold units as stock I Si Si Si (t) she will carry for next period, so I (t)=J (t)-Q (t).
If the ordered quantity is bigger than the units available, she will not be able to satisfy all the demand, in this case Si Si sales will be S (t)=J (t). The unsatisfied demand will be Si Si Si B (t) = Q (t) - J (t). 3.4 Performance evaluation At each period, the retailer and the supplier will evaluate their benefit. The benefit for the retailer is given from sales revenue, minus the cost associated to preparing the units to be sold, the cost associated to the unsold units (holdig cost increased by buy back cost), the opportunity lost cost for unsatisfied demand, his fix costs and minus the expenses of buying to the suppliers. R
R
R
π (t) = p(t) S (t) – c (t)
I
∑S
Si
R
R
(t) – h (t)I (t)
i =1
R
R
(3) – β (t) B (t) – FCR – TS(t) The total transfer will be the sum of transfers to all the suppliers:
The option premium for a put option is:
⎡r − d ⎤ Max( X (t ) − Du(t ),0) ⎥ ⎢ 1 u d − OPSi (t ) = CR ⎢ ⎥ r ⎢ u−r Max( X (t ) − Dd (t ),0)⎥ + ⎣⎢ u − d ⎦⎥
(9)
These formulations come from the Cox-Ross-Rubinstain pricing model. In this approach, it is necessary to calculate the conversion rate CR, which will give the economical dimension to the result. The option premium is interpreted as the probability of demand being higher than the lower bound multiplied by the values it can take, and the probability that demand is bellow the lower bound multiplied by the values it can take. Then it is calculated its proportional value according to the growth rate, and finally multiplied by the conversion rate. The transfer to a supplier with an option contract will be: TS
Si
Si
Si
Si
4
SINGLE SUPPLIER PARTICULAR CASE
(t)= w (t) S (t) +OP (t)
(10)
I
TS (t ) = ∑ TS (t ) Si
(4)
i =1
The benefit of each supplier is calculated from the revenue from the transfer from the retailer diminished by production cost, the holding cost for the unsold units, the opportunity cost for unsatisfied demand and her fix costs. Si
Si
Si
Si
Si
Si
π (t) = TS (t) – c (t)M (t) – h (t)I (t) Si Si Si – β (t) B (t) – FC (5) Si The next step is to define the transfer TS (t) to each supplier. The transfer will depend on the contract. In this document there are considered three types of contracts: wholesale price, buy back and option. The wholesale price contract consists in that the supplier Si asks the retailer a fixed price per unit w (t). Supposing that supplier i has this kind of contract, the transfer from the retailer to this supplier will be: Si Si Si (6) TS (t) = w (t) S (t) In a buy back contract, the supplier asks a unitary price Si w (t) but she will take in charge part of the holding cost of Si the retailer by paying him a sum b (t) per unit in stock. If supplier i has a buyback contract, the transfer will be: Si
Si
Si
Si
RSi
(7) TS (t)= w (t) S (t) – b (t) I (t) RSi Where I (t) is the stock held by the retailer at time t composed by units provided by supplier i. In the two contracts previously described, once the retailer passes an order he has to buy the units ordered. In an option contract, the retailer may change the ordered quantity. Two types of options exist: call and put. When demand is higher than the initial estimation, then the retailer will excercise a call option. When demand is smaller than the initial estimation a put option will be excercised by the retailer. If the retailer decides to modify the ordered quantity, he will have to pay an option Si (t), which is calculated premium to supplier i, OP differently for a call option than for a put option. The option premium for a call option is:
OP Si
⎡r −d ⎤ Max(Du(t ) − X (t ),0) ⎥ 1 ⎢u − d (t ) = C R ⎢ ⎥ r ⎢ u−r + Max(Dd (t ) − X (t ),0)⎥ ⎣⎢ u − d ⎦⎥
(8)
4.1 Description This is a relatively simple case since the retailer does not have to split his order quantity in multiple suppliers. The contract between the retailer and the supplier is an option contract. The transfer is then as shown in equation (10). The benefit for the retailer and the supplier will be, respectively: R
R
R
S
R
R
R
R
π (t)= p(t) S (t) – c (t) S (t) – h (t)I (t) – β (t) B (t) R
S
S
S
– FC – w (t) S (t) – OP (t) S
S
S
S
S
(11)
S
π (t)= w (t) S (t) +OP (t) – c (t)M (t) S
S
S
S
S
(12) – h (t)I (t) – β (t) B (t) – FC The benefit of the chain as a whole will be given by the sum of benefits of both retailer and supplier: R
S
∏= π (t) + π (t) R
R
S
R
R
R
R
R
∏= p(t) S (t) – c (t) S (t) – h (t)I (t) – β (t) B (t) – FC S
S
S
S
S
S
S
– c (t)M (t) – h (t)I (t) – β (t) B (t) – FC
(13)
4.2 Sequence of events The sequence of events is as follows: the retailer will decide the initial order quantity, which will be transmitted to the supplier. After having more accurate information on demand, the retailer will decide if he exercises his option to modify the ordered quantity. He will modify the ordered quantity as to maximize his expected benefit. The retailer will calculate if his benefit increases by doing so. He will then modify the ordered quantity if the benefit of this is bigger than the benefit of not modifying his decision. N O The initial order quantity will be Q (t), and Q (t) is the order quantity if the retailer exercises his option after having a more accurate information of demand. He will R N R O exercise if π (t, Q (t))
