Capacity-constrained supplier selection model with lost sales under stochastic demand behaviour

Abstract

In today’s market conditions, volume of demand is quite uncertain and thus it is hard to estimate. In many cases, buyer is prone to use supply chain flexibility rather than inventory holding strategy to withstand demand uncertainty. We assume that the buyer releases a replenishment order to the supplier for each cycle (or period) under the contract which is mainly composed of four parameters: (1) supply cost per unit, (2) minimum order quantity, (3) order quantity reduction penalty and (4) maximum capacity of the supplier. Based on these parameters, there are two flexibility options that buyer should evaluate in the order of cycle (1) issue an order smaller than the minimum order quantity and pay the related penalty and (2) place no order and lose the sales. Hence, Q _lost emerges as a critical buyer decision, the order quantity, below which no order is placed. Total expected supply cost plus lost sales, as a function of Q _lost is presented. We derive the optimal Q _lost that minimises the total cost function. Since capacity of each supplier is finite, we then develop a supplier selection model with total cost minimisation over the suppliers subject to capacity constraint that has a stochastic nature stemming from demand behaviour. Linearisation on the model is performed using chance-constrained programming approach. From a given set of supply bids from the potential supply chain partners, the buyer is able to make a quantifiable choice.

Appendices

Appendix 1: Deriving the Ω_qty penalty

$$ \begin{aligned} \Upomega_{\text{qty}} & = \beta \int\limits_{{Q_{\text{lost}} }}^{{Q_{ \hbox{min} } }} {\frac{{Q_{\min} - D}}{{Q_{\min} }}} \Upphi (D){\text{d}}D \\ & = \beta \int\limits_{{Q_{\text{lost}} }}^{{Q_{ \hbox{min} } }} {\frac{{Q_{\min} - D}}{{Q_{\min} }}} \frac{2(D - a)}{(b - a)(m - a)}{\text{d}}D \\ & = \frac{2\beta }{(b - a)(m - a)}\int\limits_{{Q_{\text{lost}} }}^{{Q_{ \hbox{min} } }} {\left( {1 - \frac{D}{{Q_{\min} }}} \right)} (D - a){\text{d}}D = \frac{2\beta }{(b - a)(m - a)}\left[ {\int\limits_{{Q_{\text{lost}} }}^{{Q_{ \hbox{min} } }} {(D - a){\text{d}}D} - \frac{1}{{Q_{\min} }}\int\limits_{{Q_{\text{lost}} }}^{{Q_{ \hbox{min} } }} {\left( {D^{2} - aD} \right){\text{d}}D} } \right] \\ & = \frac{2\beta }{(b - a)(m - a)}\left[ {\frac{{D^{2} }}{2} - aD\left| {\begin{array}{*{20}c} {Q_{\min} } \\ {Q_{\text{lost}} } \\ \end{array} } \right. - \frac{1}{{Q_{\min} }}\left( {\frac{{D^{3} }}{3} - \frac{{aD^{2} }}{2}\left| {\begin{array}{*{20}c} {Q_{\min} } \\ {Q_{\text{lost}} } \\ \end{array} } \right.} \right)} \right] \\ & = \frac{2\beta }{{\left( {b - a} \right)\left( {m - a} \right)}}\left[ {\frac{{Q_{\min}^{2} }}{2} - aQ_{\min} - \frac{{Q_{\text{lost}}^{2} }}{2} + aQ_{\text{lost}} - \frac{1}{{Q_{\min} }}\left( {\frac{{Q_{\min}^{3} }}{3} - a\frac{{Q_{\min}^{2} }}{2} - \frac{{Q_{\text{lost}}^{3} }}{3} + a\frac{{Q_{\text{lost}}^{2} }}{2}} \right)} \right] \\ & = \frac{2\beta }{{\left( {b - a} \right)\left( {m - a} \right)}}\left[ {\frac{{Q_{\min}^{2} }}{2} - aQ_{\min} - \frac{{Q_{\text{lost}}^{2} }}{2} + aQ_{\text{lost}} - \frac{{Q_{\min}^{2} }}{3} + a\frac{{Q_{\min} }}{2} + \frac{{Q_{\text{lost}}^{3} }}{{3Q_{\min} }} - a\frac{{Q_{\text{lost}}^{2} }}{{2Q_{\min} }}} \right] \\ & = \frac{2\beta }{{\left( {b - a} \right)\left( {m - a} \right)}}\left[ {\frac{{Q_{\min}^{2} }}{6} - a\frac{{Q_{\min} }}{2} + aQ_{\text{lost}} - \frac{{Q_{\text{lost}}^{2} }}{2} - a\frac{{Q_{\text{lost}}^{2} }}{{2Q_{\min} }} + \frac{{Q_{\text{lost}}^{3} }}{{3Q_{\min} }}} \right] \\ \end{aligned} $$

Appendix 2: Deriving the Ω_lost penalty

$$ \begin{aligned} \Upomega_{\text{lost}} & = \gamma \int\limits_{a}^{{Q_{\text{lost}} }} {\frac{D}{{Q_{\min} }}} \Upphi (D){\text{d}}D \\ & = \gamma \int\limits_{a}^{{Q_{\text{lost}} }} {\frac{D}{{Q_{\min} }}\frac{2(D - a)}{(b - a)(m - a)}{\text{d}}D} \\ & = \frac{2\gamma }{{(b - a)(m - a)Q_{\min} }}\int\limits_{a}^{{Q_{\text{lost}} }} {\left( {D^{2} - aD} \right){\text{d}}D} \\ & = \frac{2\gamma }{{(b - a)(m - a)Q_{\min} }}\left[ {\frac{{D^{3} }}{3} - a\frac{{D^{2} }}{2}\left| {\begin{array}{*{20}c} {Q_{\text{lost}} } \\ a \\ \end{array} } \right.} \right] = \frac{2\gamma }{{(b - a)(m - a)Q_{\min} }}\left[ {\frac{{Q_{\text{lost}}^{3} }}{3} - a\frac{{Q_{\text{lost}}^{2} }}{2} + \frac{{a^{3} }}{6}} \right] \\ \end{aligned} $$

Appendix 3: Deriving optimal Q_lost

$$ \begin{aligned} \frac{\partial \Upomega }{{\partial Q_{\text{lost}} }} = & 0 \\ & \Rightarrow \frac{2\beta }{(b - a)(m - a)}\left[ {a - Q_{\text{lost}} - \frac{{aQ_{\text{lost}} }}{{Q_{\min} }} + \frac{{Q_{\text{lost}}^{2} }}{{Q_{\min} }}} \right] + \frac{2\gamma }{{(b - a)(m - a)Q_{\min} }}\left[ {Q_{\text{lost}}^{2} - aQ_{\text{lost}} } \right] = 0 \\ & \underbrace {{\left[ {\frac{2(\beta + \gamma )}{{(b - a)(m - a)Q_{\min} }}} \right]}}_{A}Q_{\text{lost}}^{2} + \underbrace {{\left[ { - \frac{{2a(\beta + \gamma ) + 2\beta Q_{\min} }}{{(b - a)(m - a)Q_{\min} }}} \right]}}_{B}Q_{\text{lost}} + \underbrace {{\frac{2a\beta }{(b - a)(m - a)}}}_{C} = 0 \\ \end{aligned} $$

$$ \Updelta = B^{2} - 4{\text{AC}}\;{\text{and}}\;Q_{\text{lost}}^{[1,2]} = \frac{ - B \pm \sqrt \Updelta }{2A} $$

$$ \begin{aligned} Q_{{{\text{lost}}}}^{{[1,2]}} & = \frac{{a(\beta + \gamma ) + \beta Q_{{{\text{min}}}} \pm \sqrt {a^{2} + \left( {\beta + \gamma } \right)^{2} + \beta ^{2} Q_{{{\text{min}}}}^{2} - 2a\beta (\beta + \gamma )Q_{{{\text{min}}}} } }}{{2\left( {\beta + \gamma } \right)}} \\ & = \frac{{a(\beta + \gamma ) + \beta Q_{{{\text{min}}}} \pm \left( {a(\beta + \gamma ) - \beta Q_{{\min }} } \right)}}{{2(\beta + \gamma )}} \to Q_{{{\text{lost}}}}^{{[1]}} = a\;{\text{and}}\;Q_{{{\text{lost}}}}^{{[2]}} = \frac{{\beta Q_{{\min }} }}{{(\beta + \gamma )}} \\ \end{aligned} $$

$$ \frac{{\partial^{2} \Upomega }}{{\partial Q_{\text{lost}}^{2} }} = \left[ {\frac{4(\beta + \gamma )}{{(b - a)(m - a)Q_{\min} }}} \right]Q_{\text{lost}} - \left[ {\frac{{2a(\beta + \gamma ) + 2\beta Q_{\min} }}{{(b - a)(m - a)Q_{\min} }}} \right] $$

$$ \Upomega^{''} (a) = \frac{{2\beta \left( {a - Q_{\min} } \right) + 2\gamma a}}{{(b - a)(m - a)Q_{\min} }} $$

Since $\Upomega^{''} (a) < 0$ for all problem setting, $Q_{lost}^{[1]} = a$ is the maximum point for the total cost function Ω.

$$ \Upomega^{''} \left( {\frac{{\beta Q_{\min} }}{\beta + \gamma }} \right) = - \frac{{2\beta (a - Q_{\min} ) + 2\gamma a}}{{(b - a)(m - a)Q_{\min} }} $$

Since $\Upomega^{''} \left( {\frac{{\beta Q_{{\min} } }}{\beta + \gamma }} \right) > 0$ for all problem setting, $Q_{{\rm lost}}^{[2]} = \frac{{\beta Q_{{\min} } }}{\beta + \gamma }$ is the minimum point for the total cost function Ω.

Appendix 4: GAMS/Cplex codes for supplier selection model

 

sets

i supplier/I-1*I-5/;

scalar a lower bound of distribution/700/

c mode of distribution/1600/

b upper bound of distribution/3000/

SP sale price      /8/

GD confidence level      /0.05/ ;

parameter Mu mean;

Mu = (a + b + c)/3;

parameter StdSap Standard deviation;

StdSap = sqrt((sqr(a) + sqr(b) + sqr(c) − a*b − a*c − b*c)/18);

parameter Z;

Z = (b − sqrt(GD*(b − a)*(b − c)) − Mu)/StdSap;

parameters Qmin(i) minimum order quantity/

I-1 1300

I-2 1220

I-3 1240

I-4 1205

I-5 1270/;

parameters Ca(i) maximum capacity of each supplier/

I-1 2000

I-2 1800

I-3 2100

I-4 1770

I-5 1800/;

parameters Beta(i) maximum quantity reduction penalty/

I-1 35000

I-2 38000

I-3 33000

I-4 43000

I-5 30000/;

parameter Alfa(i);

Alfa(i) = SP*Qmin(i);

parameter Qlost(i) optimum lost sales;

Qlost(i) = (a*(Beta(i) + Alfa(i)) + Beta(i)*Qmin(i) + sqrt(sqr(a)*sqr(Beta(i) + Alfa(i)) + sqr(Beta(i)*Qmin(i)) − 2*a*Beta(i)*(Beta(i) + Alfa(i))*Qmin(i)))/(2*(Beta(i) + Alfa(i)));

parameter QRP(i) quantity reduction penalty;

QRP(i) = (2*Beta(i)/((b − a)*(c − a)))*(sqr(Qmin(i))/6 − a*Qmin(i)/2 + a*Qlost(i) − sqr(Qlost(i))/2 − a*sqr(Qlost(i))/(2*Qmin(i)) + power(Qlost(i),3)/(3*Qmin(i)));

parameter LSP(i) lost sales penalty;

LSP(i) = (2*Alfa(i)/((b − a)*(c − a)*Qmin(i)))*(power(Qlost(i),3)/3 − a*sqr(Qlost(i))/2 + power(a,3)/6);

parameter TC(i) total cost;

TC(i) = QRP(i) + LSP(i);

variables

af objective function

X(i) proportion of demand that is assigned to each supplier;

positive variable

X(i);

Equations

ObjFun objective function value which is minimised

Eq1

Eq2;

ObjFun.. af = e = sum(i,TC(i)*X(i));

Eq1.. sum(i,X(i)) = e = 1;

Eq2(i).. X(i)*Mu+Z*X(i)*StdSap = l = Ca(i);

model SSM/all/;

solve SSM using lp minimizing af;

display Qlost, Z, Mu, StdSap, QRP, LSP, TC;