Purchasing decisions under stochastic prices: Approximate solutions for order time, order quantity and supplier selection

Abstract

This paper analyzes purchasing strategies for retailers regarding the best timing and amount of purchases when operating under combined timing and quantity flexibility contracts in an environment of uncertain prices. To decrease the computational complexity and make the procedure adaptable to the case of multiple suppliers, we introduce, analyze, and compare a “time strategy” and a “target strategy” and then develop a hybrid “adaptive target strategy” to facilitate and improve the purchasing decision for the case of option contracts with generally rising prices. The adaptive target strategy is simpler and more intuitive than the traditional binomial lattice method, while the risk of failing to meet a target profit can also easily be calculated. We then extend the solution procedure to maximize expected profits in an environment of selecting among multiple suppliers with potentially different price processes, and we further provide risk analysis to help determine a good estimate for the number of option contracts from different suppliers to generate in order to create adequate risk protection. Numerical analysis demonstrates how the number of candidate suppliers impacts the expected profit and the risk. Monte Carlo simulation results demonstrate that the developed solution procedures provide satisfying outcomes and that the calculation is fast, even for multiple-dimension and multiple-supplier cases.

Appendix: Proofs of propositions

Proof of Proposition 1

(A1)

Proof of Proposition 2

Let v=[ln(R)−ln(C(0))−θt]/σ, then R=C(0)exp(θt+σv).

Proof of Proposition 3

We use $\pi_{2}^{C}( x )$ to represent the profit for the target strategy with any level x≤0 under the commitment contract. Then

Taking the expected value on both sides:

When $\theta\le - \frac{\sigma^{2}}{2}$, according to (A1),

When $\theta > - \frac{\sigma^{2}}{2}$, according to (A1),

$$E \bigl[ R - C ( T ) \bigr] = R - C ( 0 )\exp \biggl[ T \biggl( \theta + \frac{\sigma^{2}}{2} \biggr) \biggr] \le R - C ( 0 ) = E \bigl[ R - C ( 0 ) \bigr], $$

Therefore, under commitment contracts, the optimal decision using the target strategy never provides a higher expected profit than the optimal decision using the time strategy.

Proof of Proposition 4

Taking the expected value of (9) on both sides:

As x is chosen to maximize (14), the profit at x satisfies R−C(0)e ^x≥0. Therefore:

$$E\bigl\{ R - C\bigl[ \tau_{\theta} (x) \bigr]\bigm|\tau_{\theta} (x) \le T \bigr\} = R - C( 0 )e^{x} \ge 0, $$

$$E\bigl\{ \bigl\{ R - C\bigl[ \tau_{\theta} (x) \bigr] \bigr \}^{ +} \bigm|\tau_{\theta} (x) \le T \bigr\} = E\bigl\{ R - C\bigl[ \tau_{\theta} (x) \bigr]\bigm|\tau_{\theta} (x) \le T \bigr\}, $$

When θ ₀≤0,we get$\theta\le - \frac{\sigma^{2}}{2}$, according to (A1),

Proof of Proposition 5

According to (14),

When θ ₀≥0, we get $\theta\ge - \frac{\sigma^{2}}{2}$, for any t≥0

So $\max_{x}E[ \tilde{\pi}_{2}^{O}( x ) ] \ge E[ \tilde{\pi}_{2}^{O}( 0 ) ] \ge E[ \pi_{1}^{O}( t ) ]$ for any t≥0.