Contingency planning during the formation of a supply chain

Abstract

With today’s growing number of geographically dispersed facilities amplifying the likelihood of supply disruptions, contingency planning has become an important strategic issue for manufacturers and distributors. This paper studies how the addition of reserve capacity in the supply chain—one of the most common strategies in contingency plans against supply disruption—can alleviate the effects of supply disruption on product and income streams and total supply chain profit. We also perform a preliminary study to find the potential implications of utilizing excess production capacity for alternative uses particularly when a firm tries selling some of its intermediate products to an external buyer who in turn could process them into finished products and compete with the firm in the same markets. We formulate a network design optimization model for supply chain contingency planning and present a decomposition procedure which exploits the natural separation between the logistics and pricing decisions in the model. Computational results are presented.

Appendices

Appendix 1

We now present our solution procedure. Consider the formulation of Problem (SP).

Problem (SP):

$$\begin{aligned} \begin{array}{l} \displaystyle Z(X)=\mathop {{\textit{Maximize}}}\limits _{{\textit{AOP}},U,{\textit{COST}},{\textit{MARKUP}},S} \sum \limits _{j=1}^n {e_j ({\textit{AOP}}_j -U_j } ) \\ \displaystyle {\textit{subject to}}:\; (2), (5){-}(12),(14) \hbox { and } (15). \\ \end{array} \end{aligned}$$

Next we develop an alternative formulation that is equivalent to Problem (SP) and call it (SP’). Problem (SP’) has the same optimal solution and optimal value as Problem (SP), but it has the structure we need to generate cuts for the restricted master problem that are linear in variable X as will be shown later. To make this possible, we introduce new variables. In Constraints (5) and (6), denote $W_{ijk}$ to be the cost allocation of each unit produced by activity i in country j which is shipped to country k. Also, let $V_{p(i)\ell j}$ represent the cost of outbound shipment of products produced by activity p(i) in country $\ell$ bound for country j. We construct Problem (SP’) from Problem (SP) by replacing Constraints (5) with equivalent Constraints (5-1) and (5-2) as shown below.

$$\begin{aligned} {\textit{COST}}_{1jk} =W_{1jk} +c_{1j} X_{1jk}, \quad \forall k=1,\ldots ,n, \forall j=1,\ldots ,n, \end{aligned}$$

(5-1)

$$\begin{aligned} W_{1jk} =\frac{g_{1j} X_{1jk}}{\sum \nolimits _{\ell =1}^n {X_{1j\ell }} }, \quad \forall k=1,\ldots ,n, \forall j=1,\ldots ,n \end{aligned}$$

(5-2)

We also replace Constraints (6) with equivalent Constraints (6-1), (6-2) and (6-3).

$$\begin{aligned} {\textit{COST}}_{ijk} =&W_{ijk} +c_{ij} X_{ijk}^ ,\quad \forall k=1,\ldots ,n, \forall j=1,\ldots ,n, \quad \forall i=2,\ldots m, \end{aligned}$$

(6-1)

$$\begin{aligned} W_{ijk} =&\frac{X_{ijk}}{\sum \nolimits _{s=1}^n {X_{p(i)sj}} }\sum \limits _{\ell =1}^n {({\textit{COST}}_{p(i)\ell j} +{\textit{MARKUP}}_{p(i)\ell j} +V_{p(i)\ell j} +g_{ij} )} , \\ \forall k=&1,\ldots ,n, \forall j=1,\ldots ,n, \quad \forall i=2,\ldots m. \end{aligned}$$

(6-2)

$$\begin{aligned} V_{p(i)\ell j}=&r_{p(i)\ell j} X_{p(i)\ell j} , \quad \forall \ell =1,\ldots ,n, \forall j=1,\ldots ,n, \forall i=2,\ldots m. \end{aligned}$$

(6-3)

Notice that when the variable X is fixed, Constraints (5-1) thru (6-3) above are linear in both variables COST, MARKUP, W, and V. In the same token, when the variables COST, MARKUP, W, and V are fixed, Constraints (5-1) thru (6-3) are linear in the variable X. We need this modeling artifice in order to generate linear cuts for the restricted master problem as will be shown later. Next we eliminate AOP by substituting (10) in the Objective Function (1) and Constraints (2). We also break up Constraints (11) into two constraints. Problem (SP’) is shown next. For ease of reference, we display the dual variables (represented by subscripted Greek letters) on the right side of each constraint of Problem (SP’).

Problem (SP’):	Dual Variables
$Z(X)=\mathop {{\textit{Maximize}}}\limits _{{\textit{COST}},{\textit{MARKUP}},U,S,W,V} \sum \limits _{j=1}^n {e_j \left( \sum \limits _{i=1}^m {\sum \limits _{k=1}^n {{\textit{MARKUP}}_{ijk} } } +S_j^+ - S_j^- -U_j \right) }$
s.t.
${\textit{tax}}_j \left[ \sum \limits _{i=1}^m {\sum \limits _{k=1}^n {\textit{MARKUP}}_{ijk} +(S_j^+ - S_j^- )}\right] -U_j \le 0,\quad \forall j=1\ldots n$	(2)	$\alpha _j$
${\textit{COST}}_{1jk} =W_{1jk} +c_{1j} X_{1jk}\quad \forall k=1,\ldots ,n, \forall j=1,\ldots ,n,$	(5-1)	$\beta _{1jk}$
$W_{1jk} =\frac{g_{1j} X_{1jk}}{\sum \limits _{\ell =1}^n {X_{1j\ell }} },\qquad \forall k=1,\ldots ,n,\quad \forall j=1,\ldots ,n$	(5-2)	$\pi _{1jk}$
${\textit{COST}}_{ijk} =W_{ijk} +c_{ij} X_{ijk} \quad \forall k=1,\ldots ,n,\quad \forall j=1,\ldots ,n,\quad \forall i=2,\ldots m,$	(6-1)	$\beta _{ijk}$
$W_{ijk} =\frac{X_{ijk}}{\sum \limits _{s=1}^n {X_{p(i)sj}} }\sum \limits _{\ell =1}^n {({\textit{COST}}_{p(i)\ell j} +{\textit{MARKUP}}_{p(i)\ell j} +V_{p(i)\ell j} +g_{ij} )} ,\quad \forall k=1,\ldots ,n,\quad \forall j=1,\ldots ,n,\quad \forall i=2,\ldots m,$	(6-2)	$\pi _{ijk}$
$V_{p(i)\ell j} =r_{p(i)\ell j} X_{p(i)\ell j} , \quad \forall \ell =1,\ldots ,n, \quad \forall j=1,\ldots ,n, \quad \forall i=2,\ldots m.$	(6-3)	$\varsigma _{i\ell j}$
$\sum \limits _{\ell =1}^n {({\textit{COST}}_{m\ell k} +{\textit{MARKUP}}_{m\ell j} +r_{m\ell j} X_{m\ell j} )} +(S_j^+ - S_j^- )=\rho _j d_j , \quad \forall j=1,\ldots ,n,$	(7)	$\xi _j$
$\frac{{\textit{COST}}_{ijk} +{\textit{MARKUP}}_{ijk} }{X_{ijk}} = \frac{ {\textit{COST}}_{ij\ell } +{\textit{MARKUP}}_{ij\ell } }{X_{ij\ell }}, \quad \forall j=1,\ldots n, \quad \forall k=1,\ldots ,n, \quad \forall \ell =1,\ldots ,n,$
$k\ne \ell ,\quad \forall i=1,\ldots m$	(8)	$\varphi _{ijk\ell }$
$\frac{{\textit{COST}}_{p(i)\ell j} +{\textit{MARKUP}}_{p(i)\ell j} }{X_{p(i)\ell j}} \le \frac{\sum \nolimits _{k=1}^n {({\textit{COST}}_{ijk} +{\textit{MARKUP}}_{ijk} )} }{\sum \limits _{k=1}^n {X_{ijk}} } , \quad \forall \ell =1,\ldots n, \quad \forall j=1,\ldots ,n, \quad \forall i=2,\ldots ,m$	(9)	$\psi _{i\ell j}$
${\textit{MARKUP}}_{ijk} \hbox {-rmax}_{ij} {\textit{COST}}_{ijk} \le 0, \quad \forall j=1,\ldots n, \quad \forall k=1,\ldots ,n, \quad \forall i=1,\ldots m,$	(11-1)	$\tau _{ijk}$
${\textit{MARKUP}}_{ijk} \hbox {-rmin}_{ij} {\textit{COST}}_{ijk} \ge 0, \quad \forall j=1,\ldots n, \quad \forall k=1,\ldots ,n, \quad \forall i=1,\ldots m,$	(11-2)	$\phi _{ijk}$
${\textit{COST}}_{ijk} +{\textit{MARKUP}}_{ijk} \le MX_{ijk}^ , \quad \forall j=1,\ldots n, \quad \forall k=1,\ldots ,n, \quad \forall i=1,\ldots m,$	(12)	$\gamma _{ijk}$
$U_j ,S_j^+ , S_j^- \ge 0, \quad \forall j=1, 2,\ldots ,n,$	(14)
${\textit{MARKUP}}_{ijk} , {\textit{COST}}_{ijk} ,W_{ijk} ,V_{ijk} \ge 0, \quad \forall k=1,\ldots ,n, \quad \forall j=1,\ldots ,n, \quad \forall i=1,\ldots m.$	(15)

We note that for any X,  Problem (SP’) is a concave program with a finite optimal solution. This implies that for any given X, an optimal solution to the dual of Problem (SP’) exists. Next we show how we employ the optimal primal and dual solutions of Problem (SP’) to construct a cut. Let f(COST, MARKUP, U, S, W, V) represent the objective function of Problem (SP’) and $G(COST,MARKUP,U,S,W,V)\ge 0$ the constraint set of problem (SP’). Denote the LP dual of problem (SP’) by Problem (DSP’). Also let the set of dual variables $\{\alpha ,\beta ,\pi ,\varsigma ,\xi ,\psi ,\tau ,\phi ,\varphi ,\gamma \}$be represented by $\mu$ for any given X. The dual problem can be formulated as:

Problem (DSP’):

$$\begin{aligned} Z(X)=\mathop {\hbox {Minimize}}\limits _{\mu \ge 0} L(X,\mu ) \end{aligned}$$

where $L(X,\mu )=\mathop {\hbox {Maximize}}\limits _{{\textit{COST}},{\textit{MARKUP}},U,S,W,V\ge 0} f({\textit{COST}},{\textit{MARKUP}},U,S,W,V)+ \mu G({\textit{COST}},{\textit{MARKUP}},U,S,W,V)$. Expanding $L(X,\mu )$ yields:

$$\begin{aligned}&L(X,\mu )\\&\quad =\mathop {\hbox {Maximize}}\limits _{{\textit{COST}},{\textit{MARKUP}},U,S,W,V\ge 0} \left\{ \sum \limits _{j=1}^n e_j \left( \sum \limits _{i=1}^m {\sum \limits _{k=1}^n {{\textit{MARKUP}}_{ijk} } } -\sum \limits _{i=1}^m {g_{ij} V_{ij}} +(S_j^+ - S_j^- )-U_j \right) \right. \\&\qquad +\sum \limits _{j=1}^n {\alpha _j \left( tax_j \left[ \sum \limits _{i=1}^m {\sum \limits _{k=1}^n {{\textit{MARKUP}}_{ijk} } } +(S_j^+ -S_j^- )\right] -U_j\right) } \\&\qquad +\sum \limits _{k=1}^n {\sum \limits _{j=1}^n {\beta _{1jk} ({\textit{COST}}_{1jk} -W_{1jk} -c_{1j} X_{1jk} )} }\\&\qquad + \sum \limits _{k=1}^n {\sum \limits _{j=1}^n {\pi _{1jk} \left( W_{1jk} \sum \limits _{\ell =1}^n {X_{1j\ell } } -g_{1j} X_{1jk} \right) } } \\&\qquad +\sum \limits _{k=1}^n {\sum \limits _{j=1}^n {\sum \limits _{i=2}^m {\beta _{ijk} ({\textit{COST}}_{ijk} -W_{ijk} -c_{ij} X_{ijk} )} } } \\&\qquad +\sum \limits _{k=1}^n \sum \limits _{j=1}^n \sum \limits _{i=2}^m \pi _{ijk} \left( W_{ijk} \sum \limits _{s=1}^n {X_{p(i)sj}} -X_{ijk} \sum \limits _{\ell =1}^n \left( COST_{p(i)\ell j}\right. \right. \\&\qquad +\left. \left. {\textit{MARKUP}}_{p(i)\ell j} +V_{p(i)\ell j} +g_{ij} \right) \right) \\&\qquad +\sum \limits _{i=2}^m {\sum \limits _{\ell =1}^n {\sum \limits _{j=1}^n {\varsigma _{p(i)\ell j} (V_{p(i)\ell j} -r_{p(i)\ell j} } } } X_{p(i)\ell j} )\\ \end{aligned}$$

$$\begin{aligned}&\qquad +\sum \limits _{j=1}^n { \xi _j \left[ \sum \limits _{\ell =1}^n {({\textit{COST}}_{m\ell k} +{\textit{MARKUP}}_{m\ell j} +r_{m\ell j} X_{m\ell j} )} +(S_j^+ - S_j^- )-\rho _j d_j \right] } \\&\qquad +\sum \limits _{i=1}^m \sum \limits _{j=1}^n \sum \limits _{k=1}^n \sum \limits _{ \ell =1, \ell \ne k}^n \phi _{ijk\ell } (({\textit{COST}}_{ijk} +{\textit{MARKUP}}_{ijk} )X_{ij\ell }\\&\qquad - ({\textit{COST}}_{ij\ell } +{\textit{MARKUP}}_{ij\ell } )X_{ijk} ) \\&\qquad +\sum \limits _{i=2}^m \sum \limits _{j=1}^n \sum \limits _{\ell =1}^n \psi _{i\ell j} \left[ \sum \limits _{k=1}^n X_{ijk} ({\textit{COST}}_{p(i)\ell j} +{\textit{MARKUP}}_{p(i)\ell j} )\right. \\&\qquad -\left. (X_{p(i)\ell j} \sum \limits _{k=1}^n ({\textit{COST}}_{ijk} + {\textit{MARKUP}}_{ijk}) \right] \\&\qquad +\sum \limits _{i=1}^m \sum \limits _{j=1}^n \sum \limits _{k=1}^n \tau _{ijk} ({\textit{MARKUP}}_{ijk} -\hbox {rmax}_{ij} {\textit{COST}}_{ijk}) \\&\qquad +\sum \limits _{i=1}^m {\sum \limits _{j=1}^n {\sum \limits _{k=1}^n {\varphi _{ijk} (\hbox {rmin}_{ij} {\textit{COST}}_{ijk} -{\textit{MARKUP}}_{ijk} )} } }\\&\qquad +\left. \sum \limits _{i=1}^m {\sum \limits _{j=1}^n {\sum \limits _{k=1}^n {\gamma _{ijk} ({\textit{COST}}_{ijk} +{\textit{MARKUP}}_{ijk} -MX_{ijk} )} } } \right\} \\ \end{aligned}$$

Given X, let the primal optimal solution of Problem (SP’) be represented by (COST*, MARKUP*, U*, S*, W*, V*) and the optimal solution of Problem (DSP’) by $\mu ^{*}=\{\alpha ^{*}, \beta ^{*},\pi ^{*},\varsigma ^{*},\xi ^{*},\phi ^{*}, \psi ^{*},\tau ^{*},\varphi ^{*},\gamma ^{*}\}$. These optimal solutions can be obtained easily by solving problem (SP’) directly using CPLEX (IBM ILOG CPLEX 2012). We substitute (COST*, MARKUP*, U*, S*, W*,V*) and $\mu ^{*}$ in$L(X,\mu )$. This yields a linear equation in variable X. We note that with Geoffrion’s generalized Benders decomposition (Geoffrion 1972), $f({\textit{COST}},{\textit{MARKUP}},U,S,W,V)$ and $G(X,{\textit{COST}},{\textit{MARKUP}},U,S,W,V)$ need not be linearly separable in X and (${\textit{COST}}, {\textit{MARKUP}}, U, S, W,V$). Furthermore, the dual solution in the generalized Benders decomposition is dependent on the values of X. Next we discuss the master problem.

Define the set $A=\{X$such that X satisfies constraints (3), (4), and (13)}. In other words set A represents the feasible set of product flows on our supply chain network. We can reformulate our original Problem (P) by introducing in the objective function a variable $\theta$ such that $\theta \le L(X,\mu )$ for every $X\in A$(and $X's$ associated $\mu ^{*})$. The new model is called the master problem. Because the number of $X's$ in set A can be enormous, we shall solve a “restricted” master problem instead. Define $w = {\vert }\hbox {B}{\vert }$ where B $\subset A$. Let $L_t (X,\mu )$ be the linear cut generated at iteration t. The restricted master problem is formulated as a network flow problem with linear side constraints. It is given by:

Problem ($R_w$):

$$\begin{aligned} \begin{array}{l} \mathop {{\textit{Maximize}}}\limits _{\theta \ge 0} \theta \\ {\textit{subject to}}:\; (3)-(4),(13) \hbox { and } \\ \theta _ \le L_t (X,\mu ), \quad \forall t=1,\ldots ,w. \\ \end{array} \end{aligned}$$

The restricted master problem’s objective function value is an upper bound (UB) on the optimal value of Problem (P), whereas the sub-problem’s optimal value is a lower bound (LB) on the optimal value of Problem (P).

Appendix 2

This section presents the model formulations of the firm’s and external buyer’s pricing decisions in the Stakelberg game. For tractability, we assume linear demand. We start first with the model formulation of the external buyer’s problem. We determine the external buyer’s optimal price given the firm’s wholesale and market prices. Then we present the firm’s problem which is Problem (P) with modifications in the technological constraints.

1.1 A.1 The external Buyer’s problem

Let w represent the wholesale price per unit that the firm charges the external buyer for the intermediate product. This is the landed cost of the intermediate product delivered from echelon B. Define $\hat{{c}}$ to be the unit cost incurred by the external buyer to process the intermediate product, which it buys from the firm, into a finished product. Let $d_{ej} (d_{\textit{fj}} )$ be the external buyer’s (firm’s) market demand in country j. Also, let $p_{ej}\; (p_{\textit{fj}})$ be the external buyer’s (firm’s) price in market j. Finally, define $\pi _e (w,p_f )$ to be the external buyer’s profit given the firm’s wholesale and market prices. Assume the parameters $a_2 , b, \hbox {and} \gamma$ of the linear demand function to be the same for all countries. Also, assume that the firm charges its affiliates and the external buyer the same wholesale price for the intermediate product.¹⁵ The external buyer’s problem is as follows.

Problem (EB):

$$\begin{aligned} \mathop {{\textit{Max}}}\limits _{p_{ej} ,\quad \forall j} \pi _e (w,p_f )= & {} \sum \limits _{j=1}^n {(p_{ej} -(w+\hat{{c}}))d_{ej} } \nonumber \\= & {} \sum \limits _{j=1}^n {(p_{ej} -(w+\hat{{c}}))(d_j -d_{\textit{fj}} (p_{ej} ,p_{\textit{fj}} ))}\nonumber \\= & {} \sum \limits _{j=1}^n {(p_{ej} -(w+\hat{{c}}))(d_j -a_2 d_j +bp_{\textit{fj}} -\gamma p_{ej} )} \end{aligned}$$

(22)

Next we find the external buyer’s optimal price as follows. Taking the first partial derivative of $\pi _e$ with respect to $p_e$ yields

$\frac{\partial \pi _e }{\partial p_{ej} }=(d_j -a_2 d_j +bp_{\textit{fj}} -\gamma p_{ej} )-\gamma (p_{ej} -(w+\hat{{c}}))\quad .$ Since $\frac{\partial ^{2}\pi _e }{\partial p_{ej} ^{2}}=-2\gamma <0$, setting $\frac{\partial \pi _e }{\partial p_{ej} }=0$ yields the external buyer’s optimal price

$$\begin{aligned} p_{ej}^*=\frac{1}{2}\left[ {\frac{(1-a_2 )d_j +bp_{\textit{fj}} }{\gamma }+(w+\hat{{c}})} \right] \end{aligned}$$

(23)

For a given b,  as $\gamma$ increases, $(b- \gamma )$ decreases (which implies a strong competition) and this causes $p_{ej}^*$ to decrease in turn. Also, $p_{ej}^*$ is increasing in $p_{\textit{fj}}$ and $(w+\hat{{c}})$.

1.2 A.2 The firm’s problem

The firm’s problem is to find the optimal wholesale price $w*$and the market price $p_{\textit{fj}}^*$. It is basically Problem (P) with some modifications. Let $\hat{{\delta }} , 0<\hat{{\delta }}<1$, be the firm’s percentage gross profit (after deducting distribution and sales expenses). It follows $\rho _j =\hat{{\delta }}p_{\textit{fj}} , \quad \forall j.$ The firm’s problem is given below.

Problem (P’):

$$\begin{aligned} {{\textit{Maximize}}} \; Z= \sum \limits _{j=1}^n {e_j (AOP_j -U_j } ) \end{aligned}$$

(1)

subject to: (2)–(6), (8)–(16), and

$$\begin{aligned}&\sum \limits _{\ell =2}^n {({\textit{COST}}_{m\ell j} +{\textit{MARKUP}}_{m\ell j} +r_{m\ell j} X_{m\ell j} )} +\left( S_j^+ -S_j^- \right) \\&\quad =\rho _j d_{\textit{fj}} (p_{\textit{fj}} ,p_{ej}^*), \forall j=1,\ldots ,n,\,\, \,\, p_{\textit{fj}} \ge 0,\quad \forall j. \end{aligned}$$

(7')

We note that the index $\ell$ in (7’) starts at 2 since the plant in country 1 has been designated as the external buyer. Also note that $p_{\textit{fj}}$ is a decision variable in Problem (P’).

The right hand side of (7’) can be expressed as:

$$\begin{aligned} \rho _j d_{\textit{fj}} (p_{\textit{fj}} ,p_e^*)= & {} \delta p_{\textit{fj}} (a_2 d_j -bp_{\textit{fj}} +\gamma p_{ej}^*)\nonumber \\= & {} \frac{\delta }{2}\left[ {p_{\textit{fj}} d_j \left( {a_2 +1+\frac{\gamma (w+\hat{{c}})}{d_j }} \right) -bp_{\textit{fj}}^2 } \right] \end{aligned}$$

(24)

Notice that (24) is quadratic in $p_{\textit{fj}}$. To obtain a solution to Problem (P’), we can approximate (24) using a piecewise linear function (see Wagner 1969). Finally, the optimal wholesale price is the landed cost of the intermediate product which is given by

$$\begin{aligned} w^*= \quad \sum \limits _{\ell =1}^n {\left( {\frac{X_{p(2)\ell 1} }{\sum \limits _{s=1}^n {X_{p(2)s1} } }} \right) \left( {\frac{{\textit{COST}}_{p(2)\ell 1} +{\textit{MARKUP}}_{p(2)\ell 1} }{X_{p(2)\ell 1} }+r_{p(2)\ell 1} } \right) } , \end{aligned}$$

(25)

where the subscript p(2) represents echelon B and the subscripts $\ell$ and s represent source countries of the intermediate product.