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July  2018, 14(3): 1105-1122. doi: 10.3934/jimo.2018001

## Portfolio procurement policies for budget-constrained supply chains with option contracts and external financing

 1 School of Management and Economics, University of Electronic Science and Technology of China, Chengdu, China 2 Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong 3 Department of Marketing and International Business, Valdosta State University, Valdosta, USA

* Corresponding author: Xu Chen, E-mail: xchenxchen@263.net, Tel: +86-28-83206622

Received  October 2015 Revised  September 2017 Published  July 2018 Early access  January 2018

This study investigates a budget-constrained retailer's optimal financing and portfolio order policies in a supply chain with option contracts. To this end, we develop two analytical models: a basic model with wholesale price contracts as the benchmark and a model with option contracts. Each model considers both the financing scenario and the no-financing scenario. Our analyses show that the retailer uses wholesale price contracts for procurement, instead of option contracts, when its budget is extremely tight. The retailer starts to use a combination of these two types of contracts when the budget constraint is relieved. As the budget increases, the retailer adjusts the procurement ratio through both types until it can implement the optimal ordering policy with an adequate budget. In addition, the condition for seeking external financing is determined by the retailer's initial budget, financing cost, and profit margin.

Citation: Benyong Hu, Xu Chen, Felix T. S. Chan, Chao Meng. Portfolio procurement policies for budget-constrained supply chains with option contracts and external financing. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1105-1122. doi: 10.3934/jimo.2018001
##### References:

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##### References:
The structure of the optimal order policies
The effects of option contracts without financing
The effects of option contracts with financing
Suppliers possible production quantity function
Nomenclature
 Notation Description $D$ Random variable for market demand with $D\geq0$ $f(x)$ Probability density function for market demand $F(x)$ Cumulative distribution function for market demand, which is a continuous, strictly increasing and invertible function of $x$ with $F(x)=0$ $F^{-1}(x)$ Inverse function of $F(x)$ $p$ Product retail price (＄/unit) $c$ Product manufacturing cost (＄/unit) $s$ Product salvage value (＄/unit) $g$ Retailer's shortage penalty (＄/unit) $w$ Product wholesale price under wholesale price contracts (＄/unit) $w_1$ Product wholesale price under option contracts (＄/unit) $b$ Product option price (＄/unit) $w_2$ Option exercise price (＄/unit) $q$ Retailer's order quantity in the basic model $q^1$ Retailer's firm order quantity in the model with option contracts $q^2$ Retailer's option order quantity in the model with option contracts $q^1+q^2$ Retailer's portfolio order quantity in the model with option contracts, denoted as $q^1+q^2=q$ $Y$ Retailer's initial budget $H$ Retailer's financing amount $\lambda_i$ Generalized Lagrange multiplier, $i=1, 2, 3$ $x^+$ $x^+=max(0, x)$ $u$ Mean of market demand, $u=E(D)$ $E(x)$ Expected value of variable $x$ $min(x, y)$ Minimum between $x$ and $y$
 Notation Description $D$ Random variable for market demand with $D\geq0$ $f(x)$ Probability density function for market demand $F(x)$ Cumulative distribution function for market demand, which is a continuous, strictly increasing and invertible function of $x$ with $F(x)=0$ $F^{-1}(x)$ Inverse function of $F(x)$ $p$ Product retail price (＄/unit) $c$ Product manufacturing cost (＄/unit) $s$ Product salvage value (＄/unit) $g$ Retailer's shortage penalty (＄/unit) $w$ Product wholesale price under wholesale price contracts (＄/unit) $w_1$ Product wholesale price under option contracts (＄/unit) $b$ Product option price (＄/unit) $w_2$ Option exercise price (＄/unit) $q$ Retailer's order quantity in the basic model $q^1$ Retailer's firm order quantity in the model with option contracts $q^2$ Retailer's option order quantity in the model with option contracts $q^1+q^2$ Retailer's portfolio order quantity in the model with option contracts, denoted as $q^1+q^2=q$ $Y$ Retailer's initial budget $H$ Retailer's financing amount $\lambda_i$ Generalized Lagrange multiplier, $i=1, 2, 3$ $x^+$ $x^+=max(0, x)$ $u$ Mean of market demand, $u=E(D)$ $E(x)$ Expected value of variable $x$ $min(x, y)$ Minimum between $x$ and $y$
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