Abstract
The trend of globalization and outsourcing makes supply unreliable and companies begin to have supplier diversity embedded into their procurement departments. Traditionally, contract suppliers are a major supply channel for many companies, while the effectiveness of reactive supply sources, such as spot markets, is often ignored. Spot markets have negligible lead times and higher average prices comparing with contract suppliers. In our research, procurement utilizes the combinatorial benefits of proactive supply (a contract supplier) and reactive supply (a spot market). The uncertainties of yields, spot prices, and demand, and the correlations among them are also taken into consideration when designing procurement plans. The objectives of this paper were to evaluate the effectiveness of dealing with uncertain supply using the spot market along with the contract supplier and to model the dependences among all the potential uncertainties. This research also seeks high expected profits without overlooking the associated variances. The analytical expression to determine the optimal order quantity is obtained under the most general situations where commodities can be both bought and sold via the spot market. Some properties are derived to provide useful managerial insights. In addition, reference scenarios, such as pure contract sourcing and the spot market restricted for buying or selling only, are included for comparison purposes.
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References
Agrawal N, Nahmias S (1997) Rationalization of the supplier base in the presence of yield uncertainty. Prod Oper Manag 6(3):291–308
Anderson TW (1958) An introduction to multivariate statistical analysis. Wiley, New York
Bohrnstedt GW, Goldberger AS (1969) On the exact covariance of products of random variables. J Am Stat Assoc 64(328):1439–1442
Chen JF, Yao DD, Zheng SH (2001) Optimal replenishment and rework with multiple unreliable supply sources. Oper Res 49(3):430–443
Chen SL, Liu CL (2007) Procurement strategies in the presence of the spot market and analytical framework. Prod Plan Control 18(4):297–309
Erdem A, Özekici S (2002) Inventory models with random yield in a random environment. Int J Prod Econ 78(3):239–253
Federgruen A, Yang N (2008) Selecting a portfolio of suppliers under demand and supply risks. Oper Res 56(4):916–936
Federgruen A, Yang N (2009) Optimal supply diversification under general supply risks. Oper Res 57(6):1451–1468
Federgruen A, Yang N (2011) Procurement strategies with unreliable suppliers. Oper Res 59(4):1033–1039
Fu Q, Lee CY, Teo CP (2010) Procurement management using options: random spot price and the portfolio effect. IIE Trans 42(11):793–811
Goel A, Gutierrez G (2004) Integrating spot and futures commodity market in the optimal procurement policy of an assemble-to-order manufacturer. Working Paper. School of Business, Univeristy of Texas, Austin
Gurnani H, Akella R, Lehoczky J (2000) Supply management in assembly systems with random yield and random demand. IIE Trans 32(8):701–714
Haksöz Ç, Kadam C, Orhanl T (2008) Supply risk in fragile contracts. Sloan Manag Rev 49(2):6–8
Haksöz Ç, Seshadri S (2007) Supply chain operations in the presence of a spot market: a review with discussion. J Oper Res Soc 58(11):1412–1429
Hazra J, Mahadevan B (2009) A procurement model using capacity reservation. Eur J Oper Res 193(1):303–316
Inderfurth K, Kelle P (2011) Capacity reservation under spot market price uncertainty. Int J Prod Econ 133(1):272–279
Kawai M (1983) Spot and futures prices of nonstorable commodities under rational expectations. Q J Econ 98(2):235–254
Keren B (2009) The single-period inventory problem: extension to random yield from the perspective of the supply chain. Omega 37(4):801–810
Maddah B, Salameh MK, Karame GM (2009) Lot sizing with random yield and different qualities. Appl Math Model 33(4):1997–2009
Markowitz H (1952) Portfolio selection. J Fin 7(1):77–91
Martel A, Diaby M, Boctor F (1995) Multiple items procurement under stochastic nonstationary demands. Eur J Oper Res 87(1):74–92
McKinsey & Company and CAPS Research (2000) Coming into focus: using the lens of economic value to clarify the impact of B2B marketplaces. White Paper. McKinsey & Company, New York
Nagali V, Hwang J, Sanghera D, Gaskins M, Pridgen M, Thurston T, Mackenroth P, Branvold D, Scholler P, Shoemaker G (2008) Procurement risk management (PRM) at Hewlett-Packard company. Interfaces 38(1):51–60
Seifert R, Thonemann U, Hausman W (2004) Optimal procurement strategies for online spot markets. Eur J Oper Res 152(3):781–799
Spinler S, Huchzermeier A (2006) The valuation of options on capacity with cost and demand uncertainty. Eur J Oper Res 171(3):915–934
Spinler S, Huchzermeier A, Kleindorfer PR (2003) Risk hedging via options contracts for physical delivery. OR Spectr 25(3):379–395
Tang C (2006) Perspectives in supply chain risk management. Int J Prod Econ 103(2):451–488
Tsay A (1999) The quantity flexibility contract and supplier-customer incentives. Manag Sci 45(10):1339–1358
Wilson EJ (1994) The relative importance of supplier selection criteria: a review and update. Int J Purch Mater Manag 30(3):35–41
Yano C, Lee H (1995) Lot sizing with random yields: a review. Oper Res 43(2):311–334
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Appendices
Appendix 1: Derivation of the expected profit for the BS model
Under the assumption of buying and selling, the buyer’s profit can be expressed as follows:
So, the expected value of the buyer’s profit can be computed as follows:
Appendix 2: Derivation of the profit variance for the BS model
The variance of the buyer’s profit can be computed as follows:
As for \(E [{( {r-s})^2d^2}]\), we have that
The calculation process of \(E[( {s-w})^2Y^2]\) is similar to that of \(E[ {( {r-s})^2d^2}]\), and the result is as follows:
As for \(E [{{d}Y( {r-s})( {s-w})}]\), defining \(u=s-r\) and \(v=s-w\), we have that \(E [{dY( {r-s})( {s-w})} ]=-E( {dYuv})\). Moreover, \(E( {dYuv})=E( {dY\cdot uv})=E( {dY})E( {uv})+\mathrm{Cov}(dY,uv)\), where \(E( {dY})=\mu _d \mu _Y +\rho _{d,y} \sigma _d \sigma _Y \) and \(E( {uv})=E( {s^2-sw-rs+rw})=( {\mu _s^2 +\sigma _s^2 })-(w+r)\mu _s +rw \quad =( {\mu _s -w})( {\mu _s -r})+\sigma _s^2 \).
Before we compute \(\mathrm{Cov}(dY,uv)\), let us introduce an important result from Bohrnstedt and Goldberger (1969). Let \(x_1 ,x_2 ,x_3,\) and \(x_4 \) be jointly distributed random variables. By definition, the covariance between products \(x_1 x_2 \) and \(x_3 x_4 \) is as follows:
where \(C\) is short for covariance. Let \(\Delta x_1 =x_1 -E(x_1 )\), \(\Delta x_2 =x_2 -E(x_2 )\), \(\Delta x_3 =x_3 -E( {x_3 }),\) and \(\Delta x_4 =x_4 -E(x_4 )\). We have that
Based on the result from Anderson (1958), under multivariate normality, all third moments vanish and
Therefore,
Hence,
Because \(C( {Y,u})=C( {Y,s-r})=E \{ {[{Y-E( Y)}] [{( {s-r})-E( {s-r})}]}\}= E \{ {[{Y-E( Y)}] [{s-E( s)}]}\}=C(Y,s)\) and \(C( {Y,v})= C( {Y,s})\), we have that \(C( {Y,u})=C( {Y,v})=C( {Y,s})\). Thus,
Therefore, the profit variance is computed as follows:
The square of the buyer’s expected profit is as follows:
After substituting \([E( {\Pi _\mathrm{BS} })]^2\) into \(\mathrm{Var}( {\Pi _\mathrm{BS} })\), we have that
Appendix 3: Proof of Proposition 1
Since \(\mu _Y =\mu _y Q\) and \(\sigma _Y =\sigma _y Q\), substituting them into \(\mathrm{Var}( {\Pi _\mathrm{BS} })\), we obtain that
Then, we find that \(\mathrm{Var}( {\Pi _\mathrm{BS} })\) is a quadratic function with the form of \(f( Q)=aQ^2+bQ+c\), where
Since \(0\le \rho _{y,s}^2 \le 1\), we have that \(a>0\). Therefore, the parabola opens upward.
Appendix 4: Derivation of the optimal order quantity \(Q^*\)
The objective function \(Z( Q)\) equals to \(E(\Pi _\mathrm{BS} )-k\mathrm{Var}(\Pi _\mathrm{BS} )\). Since \(Y=yQ\), \(\mu _Y =\mu _y Q\), and \(\sigma _Y =\sigma _y Q\), substituting them into \(Z(Q)\), we obtain that
Thus,
Since \(\rho _{y,s} \) is a correlation coefficient, we have that \(1-\rho _{y,s}^2 \ge 0\). Therefore, \(\frac{d^2Z(Q)}{dQ^2}\le 0\). So, \(Z( Q)\) is concave and the optimal order quantity \(Q^*\) can be obtained through \(\frac{dZ( Q)}{dQ}=0\). Based on the expression of \(Z( Q)\), we have that
Let \(\frac{dZ( Q)}{dQ}=0\). Then, the optimal order quantity is as follows:
Appendix 5: Proof of Proposition 2
Based on the analytical expression of the optimal order quantity \(Q^*\), \(\frac{dQ^*}{d\rho _{d,y} }\) is computed as follows:
Since \(\rho _{d,y} <0\), in order to investigate the effect of \(\rho _{d,y} \) on the optimal order quantity \(Q^*\), we need to calculate \(\frac{dQ^*}{d(-\rho _{d,y} )}\), which is computed as follows:
Let \(\frac{dQ^*}{d(-\rho _{d,y} )}>0\), we have that \(\frac{-\sigma _d \sigma _y [{( {\mu _s -r})( {\mu _s -w})+\sigma _s^2 }]}{( {w-\mu _s })^2\sigma _y^2 +( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 +\rho _{y,s}^2 \sigma _y^2 \sigma _s^2 +2\rho _{y,s} \mu _y \sigma _s \sigma _y ( {\mu _s -w})}>0.\) According to a conclusion in Appendix 3, \(( {w-\mu _s })^2\sigma _y^2 +( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 +\rho _{y,s}^2 \sigma _y^2 \sigma _s^2 +2\rho _{y,s} \mu _y \sigma _s \sigma _y ( {\mu _s -w})>0\). So, to guarantee that \(\frac{dQ^*}{d(-\rho _{d,y} )}>0\), we should have that \(-\sigma _d \sigma _y [{( {\mu _s -r})( {\mu _s -w})+\sigma _s^2 }]>0\).
Appendix 6: Proof of Proposition 3
Based on the analytical expression of the optimal order quantity \(Q^*\), \(\frac{dQ^*}{d\rho _{d,s} }\) is computed as follows:
Since \(( {w-\mu _s })^2\sigma _y^2 +( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 +\rho _{y,s}^2 \sigma _y^2 \sigma _s^2 +2\rho _{y,s} \mu _y \sigma _s \sigma _y ( {\mu _s -w})>0\), to guarantee that \(\frac{dQ^*}{d\rho _{d,s} }>0,\,-\sigma _d \sigma _s [{\mu _y ( {r-\mu _s })-\rho _{y,s} \sigma _s \sigma _y }]\) must be greater than 0.
Appendix 7: Proof of Proposition 4
When all the risks are not correlated, we have that
Thus, \(\frac{dQ^*}{d\mu _d }=\frac{\mu _y \sigma _s^2 }{( {w-\mu _s })^2\sigma _y^2 +( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 }>0\).
Appendix 8: Proof of Proposition 5
When all the risks are not correlated, we have that
To guarantee that \(\frac{dQ^*}{d\mu _y }>0\), we must have that \(( {\frac{\mu _s -w}{2k}+\mu _d \sigma _s^2 })[ ( {w-\mu _s })^2\sigma _y^2 +( \sigma _y^2-\mu _y^2 )\sigma _s^2 ]>0\).
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Hong, Z., Lee, C.K.M. & Nie, X. Proactive and reactive purchasing planning under dependent demand, price, and yield risks. OR Spectrum 36, 1055–1076 (2014). https://doi.org/10.1007/s00291-013-0344-5
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DOI: https://doi.org/10.1007/s00291-013-0344-5