Product costing in the strategic formation of a supply chain

Abstract

This study presents manufacturing and distribution network models for strategic supply chain planning. Where prices of intermediate products in the supply chain are not readily available, we study the influence on the formation of the supply chain of using either direct product costing or full absorption costing for setting internal prices. We develop a generalized Benders decomposition-based approach to solve the models. Problems representing a variety of scenarios that simulates different economic environments are solved. Our computational results show that the product costing method could influence the concentration of production activities at potential manufacturing locations, production and shipment quantities, product prices, and allocation of profits in the supply chain.

Appendix

In this section we present Problem SP$'$. We introduce new variables to formulate this problem. In constraints (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 $Y_{i-1\ell j}$ represent the cost of outbound shipment of products produced by activity $i-1$ in country $\ell$ for country j. We construct problem SP$'$ from Problem SP by replacing constraints (6) with equivalent Constraints (6–1), (6–2), and (6–3) as shown below.

We note that for Problem FAC, Constraints (5’) and (6’) must be rewritten in a similar way.

Next we eliminate the AGP variables by substituting (10) in (1) and (2). We also decompose Constraints (11) into two, i.e. Constraints (11-1) and (11-2). Problem SP$'$ is shown next. For ease of reference, we display the dual variable (represented by subscripted Greek letters) on the rightmost side of each corresponding primal constraint of Problem SP$'$.

Problem SP$'$:	Dual Variables
$Z(X,V)=\mathop {{ Maximize}}\limits _{U,{ COST},{ MARKUP},S,W} \sum \limits _{j=1}^n {e}_j \left( \sum \limits _{i=1}^m \sum \limits _{k=1}^n {MARKUP_{ijk} } -\sum \limits _{i=1}^m {g_{ij} V_{ij}} +(S_j^+ - S_j^{-} )-U_j \right)$
s.t.
${ tax}_j \left[ \sum \limits _{i=1}^m {\sum \limits _{k=1}^n {MARKUP_{ijk} +(S_j^+ - S_j^- )} }\right] -U_j \le tax_j \sum \limits _{i=1}^m {g_{ij} V_{ij} } , \forall j=1\ldots n$	(2)	$\alpha _j$
${ COST}_{1jk} =c_{1j} X_{1jk}, \forall k=1,\ldots ,n, \forall j=1,\ldots ,n$,	(5)	$\beta _{1jk}$
$COST_{ijk} =W_{ijk} +c_{ij} X_{ijk}, \forall k=1,\ldots ,n, \forall j=1,\ldots ,n, \forall i=2,\ldots m$,	(6-1)	$\beta _{ijk}$
$W_{ijk} =\frac{X_{ijk}}{\sum \limits _{s=1}^n {X_{i-1sj}} }\sum \limits _{\ell =1}^n {(COST_{i-1\ell j} +MARKUP_{i-1\ell j} +r_{i-1\ell j} X_{i-1\ell j} )} , \forall k=1,\ldots ,n, \forall j=1,\ldots ,n, \forall i=2,\ldots m$,	(6-2)	$\pi _{ijk}$
$Y_{i-1\ell j} =r_{i-1\ell j} X_{i-1\ell j} , \forall \ell =1,\ldots ,n, \forall j=1,\ldots ,n, \forall i=2,\ldots m$.	(6-3)	$\varsigma _{i\ell j}$

$\sum \limits _{\ell =1}^n {(COST_{m\ell j} +MARKUP_{m\ell j} +r_{m\ell j} X_{m\ell j} )} +(S_j^+ - S_j^- )=\rho _j d_j , \forall j=1,\ldots ,n$,	(7)	$\xi _j$
$\frac{COST_{ijk} +MARKUP_{ijk} }{X_{ijk}} = \frac{ COST_{ij\ell } +MARKUP_{ij\ell } }{X_{ij\ell }}, \forall j=1,\ldots n, \forall k=1,\ldots ,n, \forall \ell =1,\ldots ,n$, $k\ne \ell ,\forall i=1,\ldots m$	(8)	$\phi _{ijk\ell }$
$\frac{COST_{i-1\ell j} +MARKUP_{i-1\ell j} }{X_{i-1\ell j}} \le \frac{\sum _{k=1}^n {(COST_{ijk} +MARKUP_{ijk} )} }{\sum _{k=1}^n {X_{ijk}} } , \forall \ell =1,\ldots n, \forall j=1,\ldots ,n, \forall i=2,\ldots ,m$	(9)	$\psi _{i\ell j}$
${ MARKUP}_{ijk} \hbox {-rmax}_{ij} COST_{ijk} \le 0, \forall j=1,\ldots n, \forall k=1,\ldots ,n, \forall i=1,\ldots m$,	(11-1)	$\tau _{ijk}$
$MARKUP_{ijk} \hbox {-rmin}_{ij} COST_{ijk} \ge 0, \forall j=1,\ldots n$, $\forall k=1,\ldots ,n, \forall i=1,\ldots m$,	(11-2)	$\varphi _{ijk}$
$COST_{ijk} +MARKUP_{ijk} \le MX_{ijk}, \forall j=1,\ldots n, \forall k=1,\ldots ,n, \forall i=1,\ldots m$,	(12)	$\gamma _{ijk}$
$MARKUP_{ijk} , COST_{ijk} ,U_j ,S_j^+ , S_j^- \ge 0, \forall j=1, 2,\ldots ,n$,	(13)
$W_{ijk} \ge 0, \forall k=1,\ldots ,n, \forall j=1,\ldots ,n, \forall i=1,\ldots m$.	(17)

Let $f(X,V,{ COST}, { MARKUP},U,S,W,Y)$ represent the objective function of Problem SP$'$ and $G(X,V,{ COST}, { MARKUP}, U,S,W,Y)\ge 0$ the constraint set of Problem SP$'$. Denote the LP dual of Problem SP$'$ by Problem DSP$'$ and also let the set of dual variables $\{\alpha ,\beta ,\pi ,\varsigma ,\xi ,\phi ,\psi ,\tau ,\varphi ,\gamma \}$ be represented by $\mu$ for fixed (X, V). The dual problem can be formulated as:

Problem DSP $'$:

$$\begin{aligned} Z(X,V)= & {} \mathop {\hbox {Minimize}}\limits _{\mu \ge 0} L(X,V,\mu ) , \hbox {where}\\ L(X,V,\mu )= & {} \mathop {\hbox {Maximize}}\limits _{{ COST},{ MARKUP},U,S,W,Y\ge 0} f(X,V,{ COST},{ MARKUP},U,S,W,Y)\\&+\, \mu G(X,V,{ COST}, { MARKUP},U,S,W,Y). \end{aligned}$$

Given (X, V), let the primal optimal solution to Problem SP$'$ be represented by (${ COST}^*, \textit{MARKUP}^*, U^*, S^*, W^*, Y^*$) and its dual optimal solution 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 an LP solver. In this study, we used CPLEX (2002)’s LP solver. We substitute (${ COST}^*, \textit{MARKUP}^*, U^*, S^*, W^*, Y^*$) and $\mu *$ in $L(X,V,\mu )$. This yields a linear equation in variables Xand V. We note that $f(X,V,COST,{ MARKUP},U,S,W,Y)$ and $G(X,V,{ COST}, { MARKUP}, U,S,W,Y)$ need not be linearly separable in (X, V) and (COST, MARKUP, U, S, W, Y). Moreover, the dual solution $\mu *$ can be dependent on the values of (X, V) (see Geoffrion 1972). The expansion of $L(X,V,\mu )$ is

$$\begin{aligned}&L(X,V,\mu )\\&\quad =\mathop {\hbox {Maximize}}\limits _{{ COST},{ MARKUP},U,S,W\ge 0} \left\{ \sum _{j=1}^n {e}_j \left( \sum _{i=1}^m {\sum _{k=1}^n {{ MARKUP}_{ijk} } } -\sum _{i=1}^m {g_{ij} V_{ij}} +(S_j^+ - S_j^- )-U_j \right) \right. \\&\qquad +\sum _{j=1}^n \alpha _j \left( { tax}_j \left[ \sum _{i=1}^m {\sum _{k=1}^n {{ MARKUP}_{ijk} } } +S_j^+ -S_j^- \right] -U_j\right. \\&\left. \qquad -\,{ tax}_j \left( \sum _{i=1}^m {g_{ij} V_{ij} } +\sum _{i=1}^m {\sum _{k=1}^n {r_{ijk} X_{ijk} } } \right) \right) \\&\qquad +\sum _{k=1}^n {\sum _{j=1}^n {\beta _{1jk} ({ COST}_{1jk} -c_{1j} X_{1jk} )} } + \sum _{k=1}^n {\sum _{j=1}^n {\sum _{i=2}^m {\beta _{ijk} ({ COST}_{ijk} -W_{ijk} -c_{ij} X_{ijk} )} } } \\&\qquad +\sum _{k=1}^n {\sum _{j=1}^n {\sum _{i=2}^m {\pi _{ijk} \left( W_{ijk} \sum _{s=1}^n {X_{i-1sj}} -X_{ijk} \sum _{\ell =1}^n ({ COST}_{i-1\ell j} +{ MARKUP}_{i-1\ell j} +Y_{_{i-1\ell j} })\right) } } } \\&\qquad +\sum _{i=2}^m {\sum _{\ell =1}^n {\sum _{j=1}^n {\varsigma _{i-1\ell j} (Y_{i-1\ell j} -r_{i-1\ell j} } } } X_{i-1\ell j} ) \\&\qquad +\sum _{j=1}^n { \xi _j \left[ \sum _{\ell =1}^n {({ COST}_{m\ell j} +{ MARKUP}_{m\ell j} +r_{m\ell j} X_{m\ell j} )} +(S_j^+ - S_j^- )-\rho _j d_j \right] } \\&\qquad +\sum _{i=1}^m \sum _{j=1}^n \sum _{k=1}^n \sum _{ \ell =1, \ell \ne k}^n {\phi }_{{ ijk}\ell } (({ COST}_{ijk} +{ MARKUP}_{ijk} )X_{ij\ell }\\&\qquad - \,({ COST}_{ij\ell } +{ MARKUP}_{ij\ell } )X_{ijk} ) \\&\qquad +\sum \limits _{i=2}^m \sum \limits _{j=1}^n {\sum }_{\ell =1}^n {\psi }_{i\ell j} \left[ \sum _{k=1}^n X_{ijk} ( { COST}_{i-1\ell j} +{ MARKUP}_{i-1\ell j} )\right. \\&\left. \qquad -\,X_{i-1\ell j} \sum _{k=1}^n {({ COST}_{ijk} +} { MARKUP}_{ijk} ) \right] \\&\qquad +\sum _{i=1}^m {\sum _{j=1}^n {\sum _{k=1}^n {\tau _{ijk} \hbox {(}{} { MARKUP}_{ijk} -\hbox {rmax}_{ij} { COST}_{ijk} )} } }\\&\qquad + \sum _{i=1}^m {\sum _{j=1}^n {\sum _{k=1}^n {\varphi _{ijk} (\hbox {rmin}_{ij} { COST}_{ijk} -{ MARKUP}_{ijk} )} } } \\&\qquad \left. +\sum _{i=1}^m {\sum _{j=1}^n {\sum _{k=1}^n {\gamma _{ijk} ({ COST}_{ijk} +{ MARKUP}_{ijk} -MX_{ijk} )} } } \right\} . \\ \end{aligned}$$