### Abstract

This paper considers a supply contract in a two-stage supply chain consisting of one supplier and one buyer. Market demand is stochastic, and there are two opportunities for demand forecasts. With the first forecast at the beginning of production, the buyer puts a firm fixed order Q to the supplier, purchasing a quantity q_{o} of options at the same time. With the second forecast at the beginning of the sales season, the buyer determines execution quantity q_{e} of the option within the purchased quantity of options (q_{ϵ} ≤ q_{o}). The supplier determines option price p_{o} and option execution price p_{e}, while the buyer determines Q, q_{o}, q_{ϵ} under the given p_{o} and p_{ϵ}. Both buyer and supplier try to maximize their own profit; however, the buyer's decision variables Q, q_{o}, q_{e} are dependent on the supplier's decision variables p_{o} and p_{ϵ}. The purpose of this research is to determine the optimal p_{o} and p_{ϵ} for the supplier. The solution process is two-fold. In the first step, we solve the buyer's optimization problem to determine optimal Q, q_{o} and q_{ϵ}, which can maximize the buyer's profit. The optimal order quantity Q∗, option quantity q_{o}∗ and execution quantity q_{ϵ}∗ are functions of p_{o} and p_{ϵ}. Therefore, in the second step, we solve the supplier's optimization problem to determine the optimal p_{o} and p_{ϵ} so as to maximize the supplier's profit. We construct the supplier's profit model and derive an analytical function to determine the optimal option price p∗_{o} and option execution price p∗_{ϵ}. Since the analytical function consists of seven order implicit system equations, we apply a numerical solution procedure using Mathematica instead of deriving the closed analytical form of p∗_{o} and p∗_{ϵ}. Through numerical experiments, we show that the optimal option price p∗_{o} and option execution price p∗_{e} always provide maximum profit to the supplier, while the buyer keeps maximum profit at the same time.

Original language | English |
---|---|

Pages (from-to) | 23-32 |

Number of pages | 10 |

Journal | Journal of Japan Industrial Management Association |

Volume | 68 |

Issue number | 1 |

Publication status | Published - 2017 |

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### Keywords

- Execution price
- Optimization
- Option price
- Supply chain management
- Supply contract

### ASJC Scopus subject areas

- Strategy and Management
- Management Science and Operations Research
- Industrial and Manufacturing Engineering
- Applied Mathematics

### Cite this

*Journal of Japan Industrial Management Association*,

*68*(1), 23-32.

**A research on optimal option prices in supply contract.** / Huang, Liqun; Shimizu, Tomoya; Matsukawa, Hiroaki.

Research output: Contribution to journal › Article

*Journal of Japan Industrial Management Association*, vol. 68, no. 1, pp. 23-32.

}

TY - JOUR

T1 - A research on optimal option prices in supply contract

AU - Huang, Liqun

AU - Shimizu, Tomoya

AU - Matsukawa, Hiroaki

PY - 2017

Y1 - 2017

N2 - This paper considers a supply contract in a two-stage supply chain consisting of one supplier and one buyer. Market demand is stochastic, and there are two opportunities for demand forecasts. With the first forecast at the beginning of production, the buyer puts a firm fixed order Q to the supplier, purchasing a quantity qo of options at the same time. With the second forecast at the beginning of the sales season, the buyer determines execution quantity qe of the option within the purchased quantity of options (qϵ ≤ qo). The supplier determines option price po and option execution price pe, while the buyer determines Q, qo, qϵ under the given po and pϵ. Both buyer and supplier try to maximize their own profit; however, the buyer's decision variables Q, qo, qe are dependent on the supplier's decision variables po and pϵ. The purpose of this research is to determine the optimal po and pϵ for the supplier. The solution process is two-fold. In the first step, we solve the buyer's optimization problem to determine optimal Q, qo and qϵ, which can maximize the buyer's profit. The optimal order quantity Q∗, option quantity qo∗ and execution quantity qϵ∗ are functions of po and pϵ. Therefore, in the second step, we solve the supplier's optimization problem to determine the optimal po and pϵ so as to maximize the supplier's profit. We construct the supplier's profit model and derive an analytical function to determine the optimal option price p∗o and option execution price p∗ϵ. Since the analytical function consists of seven order implicit system equations, we apply a numerical solution procedure using Mathematica instead of deriving the closed analytical form of p∗o and p∗ϵ. Through numerical experiments, we show that the optimal option price p∗o and option execution price p∗e always provide maximum profit to the supplier, while the buyer keeps maximum profit at the same time.

AB - This paper considers a supply contract in a two-stage supply chain consisting of one supplier and one buyer. Market demand is stochastic, and there are two opportunities for demand forecasts. With the first forecast at the beginning of production, the buyer puts a firm fixed order Q to the supplier, purchasing a quantity qo of options at the same time. With the second forecast at the beginning of the sales season, the buyer determines execution quantity qe of the option within the purchased quantity of options (qϵ ≤ qo). The supplier determines option price po and option execution price pe, while the buyer determines Q, qo, qϵ under the given po and pϵ. Both buyer and supplier try to maximize their own profit; however, the buyer's decision variables Q, qo, qe are dependent on the supplier's decision variables po and pϵ. The purpose of this research is to determine the optimal po and pϵ for the supplier. The solution process is two-fold. In the first step, we solve the buyer's optimization problem to determine optimal Q, qo and qϵ, which can maximize the buyer's profit. The optimal order quantity Q∗, option quantity qo∗ and execution quantity qϵ∗ are functions of po and pϵ. Therefore, in the second step, we solve the supplier's optimization problem to determine the optimal po and pϵ so as to maximize the supplier's profit. We construct the supplier's profit model and derive an analytical function to determine the optimal option price p∗o and option execution price p∗ϵ. Since the analytical function consists of seven order implicit system equations, we apply a numerical solution procedure using Mathematica instead of deriving the closed analytical form of p∗o and p∗ϵ. Through numerical experiments, we show that the optimal option price p∗o and option execution price p∗e always provide maximum profit to the supplier, while the buyer keeps maximum profit at the same time.

KW - Execution price

KW - Optimization

KW - Option price

KW - Supply chain management

KW - Supply contract

UR - http://www.scopus.com/inward/record.url?scp=85019205654&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85019205654&partnerID=8YFLogxK

M3 - Article

VL - 68

SP - 23

EP - 32

JO - Journal of Japan Industrial Management Association

JF - Journal of Japan Industrial Management Association

SN - 0386-4812

IS - 1

ER -