The effect of yield variability on the decision to improve the performance of multistage production systems

Abstract

The paper presents a modeling framework that assists management to allocate capital resources to improve the yield and the yield variability in a multistage production system. The results indicate that production stages with low yield and/or high yield variability receive more capital resources for improvement. The applicability of the model is demonstrated in a real-life manufacturing firm in the food-packing industry. The solution of the model provides directions of improving the stages of a multistage production system.

Appendices

Appendix A

Proof of Proposition 1

Using Eqs. 3 and 6 and introducing the Lagrange multiplier λ we obtain

$$ \begin{array}{rll} L&=&\sum\limits_{i=1}^{K}\sum\limits_{j=1}^{N}\big(1-Y_{ij}\big)\epsilon_{ij}n_{ij}c_{ij}\big(\xi\big(\epsilon_{ij}\big)\sigma_{ij}\big) \\ &&-\;\lambda \left\{B-\sum\limits_{i=1}^{K}\sum\limits_{j=1}^{N} \big[m_{ij}-\gamma_{ij}\epsilon_{ij}\big] \right\}. \end{array} $$

(9)

From Eq. 9 we obtain the following first order conditions:

$$ \begin{array}{rll} \partial L/\partial \epsilon_{ij} & = & \big(1-Y_{ij}\big)n_{ij}\beta\rho^{\delta}\sigma_{ij}^{\delta}(1+\delta)\epsilon_{ij}^{\delta} - \lambda \gamma_{ij}=0,\\ i&=&1, \ldots, N, \end{array} $$

(10)

$$ \partial L/\partial \lambda = B-\sum\limits_{i=1}^{K}\sum\limits_{j=1}^{N} \big[m_{ij}-\gamma_{ij}\epsilon_{ij}=0\big]. $$

(11)

Solving Eq. 10 for ϵ _ij we obtain

$$ \epsilon_{ij}=\lambda^{1/\delta}\alpha_{ij}, $$

(12)

where a _ij is defined as in Eq. 6. Substituting Eqs. 12 in 11 and after simplifying, we obtain the solution in Eqs. 5 and 6.□

Proof of Proposition 2

Taking the derivative of the objective function in Eq. 7, with respect to ϵ _ij , equating to zero and simplifying, we obtain the solution in Proposition 2.□

Proof of Proposition 3

Condition 1 is implied from Eq. 5 since $\epsilon^\ast_{ij} \geq 0.$ Condition 2 stems from Eq. 5 where $\epsilon^\ast_{ij} \leq 1.$ Condition 3 stems from Eqs. 8 and 6 since $\epsilon^{\ast\ast}_{ij} \leq 1.$□

Appendix B

The model presented in HSD uses the following notation:

y _j :

Mean yield of production stage j, j = 1, ..., N.

c _j :

Expected cost incurred from a defect observed at production stage j.

n _j :

Annual number of products processed at production stage j.

N :

Number of production stages in the multistage production system.

AC :

Annual expected cost incurred from defects observed in all production stages.

B :

Budget available for improving the yield of the production stages.

h _j :

A parameter that characterizes the investment needed to improve the yield loss, 1 − y _j , at production stage j.

ε _j :

The remaining percentage of the initial yield loss (1 − y _j ) after improvement in yield has been established. In other words, 1 − ε _j represents the percentage improvement or reduction in the initial yield loss at production stage j.

The mathematical program in HSD is given by:

$$ \min_{\epsilon_j,\ j=1, \ldots, N}\ \ \ \sum\limits_{j=1}^{N} \epsilon_j\big(1-y_j\big)n_jc_j $$

$${\rm s.t.}\ \ \ B=\sum\limits_{j=1}^{N} h_j\big[\big(1/\epsilon_j\big)-1\big]. $$

The objective function represents the annual expected cost incurred from defects after improvement has been implemented and the constraint represents the budget availability.

The solution to the HSD model is given by:

$$ \epsilon_j^\ast=\sqrt{\alpha_j}\frac{\sum_{q=1}^{N} \big(h_q/\sqrt{\alpha_q}\big)}{B+\sum_{q=1}^{N} h_q},\, \, \, where \, \, \, \alpha_j=h_j\big[\big(1\!-\!y_j\big)n_jc_j\big]. \label{optimal-invest} $$

The solution results in an optimal budget allocation that minimizes the annual expected cost incurred from defects observed in all production stages, by reducing the yield loss (1 − y _j ) at production stage j to $(1-y_j)\epsilon_j^\ast.$