A scheme of supplementary production in linear programming scheduling of die casting processes

Abstract

The linear programming model for die casting processes provides the optimal production plan that maximizes the average efficiency of melting furnaces. In this article, we address supplementary production in linear programming scheduling of die casting processes. Since a real die casting campaign is subject to a variety of potential malfunctions, there may exist defective items among final cast products, thus failing to match the quantities of customer orders. The objective of this study is to propose a novel scheme of preventing product shortage by manufacturing additional cast products associated with the result of the linear programming model. In particular, supplementary production is practicable by utilizing the residue of molten alloy and cast scraps or remnants available in each shift. The sequence of casting shifts is also adjusted for facilitating this supplementary production. The proposed scheme does not alter any part of the original scheduling result of the linear programming model. A numerical experiment is conducted using real foundry data to validate integrity and performance of the proposed scheme.

Introduction

It was not until the recent years that die casting processes began to be recognized as a scheduling problem. Intrinsically, die casting procedures have a fixed sequence of operations—ladling and injection of molten alloy, solidification, die opening and ejection of a solidified casting, and spraying and closing of dies. This sequence is executed recursively by the facilities of the foundry, that is, a set of melting furnaces and die casting machines with solid-state ingots, so as to manufacture prescribed quantities of products. ¹ A cycle of the die casting sequence constitutes a casting shift in a foundry. In the shift-based casting process, scheduling usually means determining by a certain criterion the quantities of all the casting types that will be produced in each shift. This allows the problem of die casting scheduling to be solved in the framework of numerical optimization schemes.

The former studies on die casting scheduling are classified according to the applied optimization methods. As a deterministic scheme, linear programming (LP) models have been widely used, especially in the authors’ works.^2–5 LP models are fit for representing die casting processes because the properties of major die casting parameters are described in linear forms. They achieve the maximum efficiency either in raw material utilization^2,3,5 or production time and cost ⁴ of die casting processes.

Compared with deterministic schemes, the stochastic optimization approaches are relatively sparse, among which genetic algorithm (GA) employed in the study by Deb et al.^6,7 is notable. Elegantly designed as their GA operators may be, the main purpose of using GA in the studies by Deb and coworkers^6,7 is just to provide faster solving procedures than an LP model, which for itself generates the optimal solution. As proved in the studies by Yang and Park,^2–5 the LP model can represent general die casting environment with frequent in-process die exchanges, while the computation burden is reasonably managed. Based on these facts, we assert that LP models are more suitable than GA to die casting scheduling. For further qualitative comparison between LP models and GA in die casting scheduling, the readers are referred to the studies by Yang and Park.^2,5

The objective of this article is to propose a scheme of supplementary production in LP scheduling of die casting processes. Our study follows the setting of Yang and Park,^2,5 that is, the LP model that searches for the solution that maximizes the percentage use of molten alloys throughout the entire shifts. Like the scheme of recycling scrap, ³ our scheme is classified into a post-optimization method. Here, we propose a subsequent functionality of die casting processes ensuing from the scheduling result, with no burden of changing the original LP model.

The purpose of the scheme in the study by Park and Yang ³ is to save ingots that are supposed to be expended in the original scheduling result. Meanwhile, in this study, we focus our concern on preventing potential product shortage raised by defective castings after die casting scheduling is actually applied to real casting operations. This is motivated by the fact that in most die casting processes, one cannot guarantee 100% fair quality of cast products. In reality, a great many process parameters—such as inclusions or impurities in the melt, a holding time in the shot chamber, a shot profile, an intensification time and pressure, flow of melt and a temperature profile in the die, a die locking force, and a solidification pattern—contribute to the final quality of a casting.^1,8,9 Commonly observed defects in die castings even constitute a huge list of specifics, such as metallic projections or flash, blow holes, shrinkage cavities, discontinuities, defective surface, incomplete casting, incorrect dimensions or shape, and inclusions. However, previous research works on die casting scheduling provided only an optimal schedule satisfying a predetermined performance criterion and never discussed how to prevent a shortage in the quantity of die cast products upon an actual discovery of defective castings.

The proposed scheme is that whenever defective products are found in a shift, the foundryman attempts to produce additional cast products in the following shifts using the residue of molten alloy and recycling scrap left in the former shifts. As mentioned earlier, neither the calculated amount of molten alloy nor the scheduling result of the LP model have to be altered for applying the proposed method. A heuristic algorithm is presented for determining the quantity and order of the supplementary production according to the priorities of cast products and available resources. The numerical experiments conducted with real foundry data will demonstrate the superiority and applicability of the proposed scheme.

Review: LP model

This work is set in the LP model for die casting scheduling developed by Yang and Park.^2,3 A brief review of the key concept of the model is given in this section. All the materials in this section are drawn from the aforementioned studies.

A die casting foundry consists of $P$ furnaces, each melting a specific kind of alloy, and $Q$ casting machines with multiple die-exchange capabilities. This foundry can make $N$ types of cast products. The customer order is that the foundry should manufacture $d_n$ units of each product type $n=1,2,\dots ,N$ , in total $M$ die casting shifts. Denoting by $x_{n,m}$ the number of casting type $n$ manufactured in the mth shift, the LP model searches all $\mathrm{NM}$ values of $x_{n,m}$ that yield the optimal average efficiency of molten alloy throughout the entire shifts, that is, the values of $x_{n,m}$ that leave the least amount of unused alloy in the entire shifts.

Table 1 summarizes the LP model for a general die casting process. The objective function $E$ ( $0<E\le 1$ ) denotes the average efficiency of molten alloy. $w_n$ is the unit weight of product type $n$ and $W_{p,m}$ is the amount of alloy provided by furnace $p$ in the mth shift. Note that though only one furnace is holding the corresponding molten alloy in each shift, practically the foundry is equipped with more than one furnace for a kind of alloy. During the shift, the idle furnace is usually cleaned and recharged with a new batch of ingots, and it melts the ingots while the other furnace is used. Hence, the capacities of the furnaces that melt the identical type of alloy vary with the shift, leading to the notation $W_{p,m}$ . $r_p$ and $u(p,i)$ ( $i=1,\dots ,r_p$ ) denote the number of product types made in molten alloy by furnace $p$ and the indices of such product types, respectively. With these definitions of parameters, a slight reflection of $E$ shows that it expresses the average efficiency of actually used alloy with respect to the provided amount.

Table 1.

LP model for a die casting process.

Maximize	$E=\frac{1}{\mathrm{MP}}\sum _{m=1}^M\sum _{p=1}^P\sum _{i=1}^{r_p}\frac{w_{u(p,i)}{x}_{u(p,i),m}}{W_{p,m}}$
Subject to	$x_{n,m}\in \mathbb{N}$
	(C1) $\sum _{m=1}^M{x}_{n,m}=d_n$
	(C2) $\sum _{i=1}^{r_p}{w}_{u(p,i)}{x}_{u(p,i),m}\le W_{p,m}$
	(C3) $\sum _{j=1}^{s_q}(t_{v(q,j)}{x}_{v(q,j),m}+{\overline{t}}_{v(q,j)})\le T_m$
	$n=1,\dots ,N,m=1,\dots ,M$
	$p=1,\dots ,P,q=1,\dots ,Q$

Aside from the trivial constraint that $x_{n,m}\in \mathbb{N}$ , that is, $x_{n,m}$ is a nonnegative integer, there are three constraint terms given in Table 1. The equality constraint in the third row means that the total manufactured quantity of each product type should be equal to the ordered volume. The inequality in the fourth row is the weight constraint that the total weight of cast products manufactured in a shift should not exceed the provided amount of molten alloy. Finally, the inequality in the fifth row describes the time constraint imposed on casting machines. $t_n$ denotes the cycle time for machine $q$ to make one unit of casting type $n$ , while ${\overline{t}}_n$ is the setup time needed to install a corresponding die for casting type $n$ . $s_q$ and $v(q,j)$ , $j=1,\dots ,s_q$ , are analogous to $r_p$ and $u(p,i)$ , that is, casting machine $q$ can make $s_q$ kinds of product types, of which indices are $v(q,1),\dots ,v(q,s_q)$ . The left side of the inequality is thus the total processing time that machine $q$ expends in the mth shift for producing the scheduling result $x_{n,m}$ . This time should be bounded by $T_m$ , a fixed operation time available in the mth shift.

Defective products in die casting processes

In spite of the optimized die casting scheduling devised by either GA^6,7 or LP^2–5 and its flawless execution, they do not automatically guarantee successful production quality wise and thus quantity wise in a real foundry. Nonconforming castings or defectives with one or multiple defects are sometimes manufactured inevitably in a die casting shop. In case nonconforming units are produced, a die caster is often unable to avoid product shortage, which cannot be predicted a priori.

In reality, a variety of process variables, some being controllable and others uncontrollable, can contribute to the final quality of a casting. Some of them in a long list are nonmetallic inclusions or impurities in the melt, wave formation by an improper shot profile and premature solidification in the shot chamber, an intensification time and pressure, flow of melt and a temperature profile in the die, a solidification pattern, die spray and die locking force, and so on.^10–12

Porosity including blow holes, shrinkage cavities, ¹³ misrun, discontinuities, incomplete casting, flash or metallic projections, ¹⁴ defective surface, incorrect dimensions or shape, and inclusions are only a handful types of commonly observed defects in die castings. Figures 1 and 2 show typical examples of die casting defects: defectives containing porosities due to gas and shrinkage during solidification and misrun due to premature solidification from an insufficient pressure or volume of melt.

Figure 1.

Porosity defects in die castings.

Figure 2.

Misrun in die castings of a mobile phone frame.

Supplementary production

Motivation and terminologies

Our objective is to produce as many supplementary units as possible in order to compensate for product shortage caused by defective units. Note that our scheme does not entail any additional molten alloy nor additional rounds of casting shifts; it is carried out with the original amount of alloy and the fixed number of shifts, that is, $M$ shifts in our setting. Of course, if the number of defective units is found to be excessively large, then no other methods exist except for running an extra shift with additional ingots to produce the lost portion.

In the LP scheduling, the supplementary production is made possible by the following two properties of shift-based die casting processes. The first is that after we acquire the scheduling result, manufacturing the allotted quantity $x_{n,m}$ of each product type leaves some unused amount of molten alloy in a shift, termed alloy residue. ³ By using this amount that would be otherwise wasted, we can manufacture extra units of a product type made of the alloy. The second property is the existence of scrap or remnants of a casting removed in the manufacturing phase. A configuration of every solidified casting product contains accessories, such as biscuit, gate, runner, and overflow. They are to be trimmed off before a solidified casting is shaped into the final product but are indispensable for the die casting process itself. At the end of a casting shift, the accessories of all the produced units are piled as scraps. Since scrap quality is usually high enough to be recycled in the subsequent shifts, we can also apply it to supplementary production.

In the previous study, ³ we used alloy residue and scrap to save the predetermined amount of solid ingots. In this article, instead, we employ them to minimize the product shortage raised by the failures of die casting lines. It can be said that this study also contributes to enhancing the efficiency of the die casting process because without our scheme, higher cost with regard to alloy, electricity, and time would be expended in running extra shifts with retained resources.

Before presenting an algorithm for supplementary production, we postulate the following essential principles that define the production policy.

1. All the product types have priorities with respect to the supplementary production. This means that whenever product shortage occurs at the end of a shift, the product type with the highest priority must go through supplementary production first in the following shifts, no matter how much shortage the product may have. Without loss of generality, we assume that the indices of all product types are enumerated according to the priority, that is, product type 1 has the highest priority and product type N the lowest.

2. If defective castings are produced in the present shift, we use alloy residue in the next shift and available scrap at their maximum to execute supplementary production of the lost units. In other words, we admit no delay in supplementary production with respect to shifts. Also, if the supplementary production for a product type is not completed at the end of the next shift, then the remaining shortage will be automatically transferred to the subsequent shifts. The reason for stipulating this policy is that the molten alloy residue left in a shift had better be consumed immediately rather than go to waste. Thus, making as many supplementary cast products as possible at the present shift will optimize the usage of alloy residue.

3. Cast scrap is also fully devoted to manufacturing supplementary products that are generated up to the previous shift. Compared with alloy residue, scrap recycling has an additional time restriction. Since scrap is obtained from a completely cooled casting and quality tests, it goes through some preparation phase before recycled, such as trimming, sorting, cleaning, and remelting. To depict this feature in our setting, we will specify how many shifts a cast scrap must wait at least before being used in supplementary production.

We first introduce the number of defective units that are generated when the foundry produces the scheduling result $x_{n,m}$ in real die casting procedures. Let ${\theta }_{n,m}\in \mathbb{N}$ $(0\le {\theta }_{n,m}\le x_{n,m})$ be the number of defective units found among $x_{n,m}$ products of casting type $n$ manufactured in the mth shift. ${\theta }_{n,m}$ is a random variable whose value is mainly determined by the critical process parameters of die casting, numerous chance causes or outer environment around die casting lines, and the value of $x_{n,m}$ —the occurrences of defective units in a shift will increase as does the quantity of the product type.

Define another variable ${\chi }_{n,m}\in \mathbb{N}$ as the quantity of product type $n$ that is manufactured in the mth shift for counterbalancing defective units. ${\chi }_{n,1}=0$ for any $n$ , since, as derived from Principle 2, supplementary production is not initiated until the second shift. Note that even though we may acquire some alloy residue at the end of the casting procedure in the first shift, we cannot activate the extra die casting procedure again in the same shift.

For further usage, let ${\pi }_n\in \mathbb{N}$ denote the present quantity of product type $n$ that has yet to be manufactured for compensating defective units of product type $n$ accumulated so far. Assume that ${\pi }_n$ is measured at the end of every shift. If new defective units of product type $n$ are found, ${\pi }_n$ will increase by the number of defective units. However, if some supplementary products are manufactured in a shift, the value of ${\pi }_n$ will decrease by the quantity of the supplementary production. At the end of the mth shift, hence, ${\pi }_n$ is reduced to

$${\pi }_n=\sum _{i=1}^m({\theta }_{n,m}-{\chi }_{n,m})$$

(1)

Define $R_{p,m}$ as the residue of alloy $p$ in the mth shift, and define $S_{p,m}$ as the total weight of scrap obtained from alloy $p$ in the mth shift. Since alloy residue is the difference between the amount of molten alloy and the total volume of the manufactured products made of the alloy, $R_{p,m}$ is derived as

$$\begin{array}{c}\hfill R_{p,m}=W_{p,m}-\sum _{i=1}^{r_p}{w}_{u(p,i)}{x}_{u(p,i),m}\hfill \end{array}$$

(2)

As introduced before, $r_p$ is the number of product types made in molten alloy by furnace $p$ and $u(p,i)$ is the index of the ith product type.

To describe the scrap volume $S_{p,m}$ , define ${\beta }_n$ $(0<{\beta }_n\le 1)$ as the yield rate of casting type $n$ and let

$$\begin{array}{l}{\gamma }_n:=1-{\beta }_n\end{array}$$

(3)

The weight of scrap removed from a unit of casting type $n$ is thus ${\gamma }_n{w}_n$ , $w_n$ being the unit weight of product type $n$ . Note that since a defective unit is usually diagnosed before trimming scrap, any defective unit serves as scrap with its whole body. Considering this feature, we induce the formula of $S_{p,m}$ as follows

$$\begin{array}{c}\hfill S_{p,m}=\sum _{i=1}^{r_p}{\gamma }_{u(p,i)}{w}_{u(p,i)}(x_{u(p,i),m}-{\theta }_{u(p,i),m})+\sum _{i=1}^{r_p}{w}_{u(p,i)}{\theta }_{u(p,i),m}\hfill \end{array}$$

(4)

As introduced in Principle 3, we define the delay in scrap use and denote it by $d$ ( $d\ge 2$ ). Scrap $S_{p,m}$ can be recycled only in the $(m+d)\text{th}$ shift or later. This means that any scrap generated in the $(M-d+1)\text{th}$ or later shift is never employed in the supplementary production.

Like ${\pi }_n$ , we define a variable ${\Omega }_p$ as the scrap amount of alloy $p$ that is accumulated so far and is available to recycling for the supplementary production. After completing the mth shift, scrap $S_{p,(m-d)}$ will be added to ${\Omega }_p$ , and some amount that is used for supplementary production will be subtracted from ${\Omega }_p$ . In contrast to ${\pi }_n$ , no deterministic value of ${\Omega }_p$ exists with respect to a shift because the amount of scrap that will be expended in each shift is problem dependent.

Finally, in analogy with the alloy residue $R_{p,m}$ , we define time residue $T_{q,m}$ , which represents the time still available to casting machine $q$ in the mth shift after the machine completes the manufacturing of the assigned product quantities $x_{n,m}$ . From the inequality of time constraint of the LP model (see Table 1), $T_{q,m}$ is derived as

$$\begin{array}{c}\hfill T_{q,m}=T_m-\sum _{j=1}^{s_q}(t_{v(q,j)}{x}_{v(q,j),m}+{\overline{t}}_{v(q,j)})\hfill \end{array}$$

(5)

where $s_q$ is the number of the product types made by casting machine $q$ and $v(q,j)$ is the index of those product types.

Algorithm

By the property of shift-based die casting operations, the solution set $x_{n,m}$ of the LP model is independent of relative orders of shifts. Exchanging the scheduling result of one shift with that of another shift is obviously allowable and does not change the average efficiency of molten alloy. We take advantage of this property to provide the die casting process with a more favorable circumstance for supplementary production. Intuitively speaking, defective units produced in the latter shifts are more difficult to be compensated than those produced in the earlier shifts because fewer shifts are left available for supplementary production. Also, the larger the quantity of a product type is manufactured in a shift, the more likely the defective units are found in the shift. Hence, once the LP model obtains the solution $x_{n,m}$ , it would be desirable to rearrange it so that the scheduling results with large product quantities are placed at the earlier shifts and those with small product quantities at the latter shifts.

A restraint on rearranging the scheduling result is that exchanging the solution subsets in terms of the shift should not change the amount of molten alloy assigned to the original shift. Recall that multiple furnaces melting the same kind of alloy are engaged in the die casting procedure, each furnace having different alloy capacity and working alternately with respect to the shift. If we do not consider the iterative order of furnace usage, the same furnace might be disposed to two adjacent shifts by rearrangement. We should avoid such a situation since the furnace could not have the time needed to be cleaned and recharged.

The product type with the highest priority, or product type 1, is used as the criterion for rearrangement. Assume that $k$ ( $k>1$ ) distinctive furnaces are used for melting the alloy of product type 1. Then, we sort out the solution $x_{n,m}$ with respect to the shift variable $m$ such that it satisfies

$$\begin{array}{l}\begin{array}{cccccc}{x}_{1,1}& \ge & x_{1,(k+1)}& \ge & x_{1,(2k+1)}& \ge \cdots \\ x_{1,2}& \ge & x_{1,(k+2)}& \ge & x_{1,(2k+2)}& \ge \cdots \\ ⋮& & ⋮& & ⋮& \\ x_{1,k}& \ge & x_{1,2k}& \ge & x_{1,3k}& \ge \cdots \end{array}\end{array}$$

(6)

Among $k$ distinctive furnaces, $x_{1,h},x_{1,(k+h)},x_{1,(2k+h)},\dots$ are made from the hth one, where $h=1,\dots ,k$ . Thus, they should be sorted out exclusively with each other. If two shifts have the same quantity of product type 1, one with the larger quantity of product type 2 comes first and so on. With the above order, we give product type 1 more possibility of going through supplementary production as the shift advances forward.

We are now ready to address in formal terms the algorithm for supplementary production, namely, to determine online the additional quantity of each product type in a shift for compensating product shortage caused by occurrences of defective units. Recall that $d$ is the number of shifts for which generated scrap has to wait before recycled and $M$ is the total number of shifts. It then follows that the algorithm is divided into three stages according to the range of the present shift with respect to $d$ and $M$ .

Algorithm 1: supplementary production

• I. $m=1$

  As stated before, no supplementary production is executed in the first shift. Thus, after manufacturing all the assigned scheduling result $x_{1,1},x_{2,1},\dots ,x_{N,1}$ , we just inspect each final product, diagnose any defective unit, and collect generated scrap. Using the defined variables, we record the relevant information, that is, ${\theta }_{n,1}$ , ${\chi }_{n,1}=0$ , ${\pi }_n={\theta }_{n,1}$ , $S_{p,1}$ , and ${\Omega }_p=0$ where $n=1,\dots ,N$ and $p=1,\dots ,P$ .

• II. $2\le m\le d$

  When the die casting process runs shifts 2 to $d$ , supplementary production is made possible, but only with alloy residue; scrap is not ready to be recycled yet. Starting from the product type with the highest priority, that is, product type 1, we determine the quantity ${\chi }_{n,m}$ by assessing and updating the accessible resource as follows

1. Set $m:=2$ .

2. Set $n:=1$ .

3. Find $p$ and $q$ that are the indices of the alloy and casting machine that make product type $n$ , respectively.

4. If $R_{p,m}<w_n$ or $T_{q,m}<t_n$ , then go to Step 6.

5. Else, set ${\chi }_{n,m}$ such that

$$\begin{array}{c}\hfill {\chi }_{n,m}=\underset{a}{max}\{a\in \mathbb{N}|a\le {\pi }_n,R_{p,m}-a{w}_n\ge 0,T_{q,m}-a{t}_n\ge 0\}\hfill \end{array}$$

(7)

Update the involved parameters as

$$\begin{array}{l}{R}_{p\mathrm{,}m}:=R_{p\mathrm{,}m}-w_n{\chi }_{n\mathrm{,}m}\\ T_{q\mathrm{,}m}:=T_{q\mathrm{,}m}-t_n{\chi }_{n\mathrm{,}m}\\ {\pi }_n:={\pi }_n-{\chi }_{n\mathrm{,}m}+{\theta }_{n\mathrm{,}m}\end{array}$$

(8)

where ${\theta }_{n,m}$ is the quantity of units diagnosed as defective casting in the real die casting procedure.

• 6. If $n=N$ , go to Step 7; else, replace $n$ by $n+1$ and return to Step 3.

• 7. If $m=d$ , terminate the algorithm; else, replace $m$ by $m+1$ and return to Step 2.

${\chi }_{n,m}$ defined in equation (7) is the maximum quantity of product type $n$ that can be additionally manufactured in the mth shift with the given $R_{p,m}$ and $T_{q,m}$ . While scrap is not used in this phase, it is accumulated for later usage. Since scrap recycling requires the delay in $d$ shifts, none of $S_{p,m}$ , $m=2,\dots ,d$ , are added to the parameter ${\Omega }_p$ at this phase.

• III. $d+1\le m\le M$

When $m>d$ , the supplementary production can use scrap accumulated up to the $(m-d)\text{th}$ shift. For instance, at the start of the $(d+1)\text{th}$ shift, that is, the first shift in which scrap can be employed, the scrap amount of alloy $p$ is ${\Omega }_{d+1}=S_{p,1}$ . The amount expended for supplementary production at the present shift will be deducted from ${\Omega }_p$ , and as the shift moves onward, new scrap $S_{p,i}$ generated in the $d$ previous shift will be added to ${\Omega }_p$ and so on.

1. Set $m:=d+1$ .

2. Set $n:=1$ and update the scrap amounts as

   $$\begin{array}{l}{\Omega }_p:={\Omega }_p+S_{p,(m-d)},\forall p=1,\dots \mathrm{,}P\end{array}$$
   (9)

3. Find $p$ and $q$ that are the indices of the alloy and casting machine that make product type $n$ , respectively.

4. If $R_{p,m}+{\Omega }_p<w_n$ or $T_{q,m}<t_n$ , then go to Step 6.

5. Else, set ${\chi }_{n,m}$ such that

   $$\begin{array}{c}\hfill {\chi }_{n,m}=\underset{a}{max}\{a\in \mathbb{N}|a\le {\pi }_n,R_{p,m}+{\Omega }_p-a{w}_n\ge 0,T_{q,m}-a{t}_n\ge 0\}\hfill \end{array}$$
   (10)

   Define $\chi {\prime }_{n,m}\in \mathbb{N}$ as

   $$\begin{array}{l}{{\chi }^{\prime }}_{n\mathrm{,}m}:=\underset{b}{{\displaystyle \max }}\left\{b\in \mathbb{N}|b\le {\chi }_{n\mathrm{,}m},R_{p\mathrm{,}m}-w_{n}b\ge 0\right\}\end{array}$$
   (11)

   Update $R_{p,m}$ , $T_{q,m}$ , and ${\pi }_n$ as

   $$\begin{array}{l}{R}_{p\mathrm{,}m}:=R_{p\mathrm{,}m}-w_n{{\chi }^{\prime }}_{n\mathrm{,}m}\\ T_{q\mathrm{,}m}:=T_{q\mathrm{,}m}-t_n{\chi }_{n\mathrm{,}m}\\ {\pi }_n:={\pi }_n-{\chi }_{n\mathrm{,}m}+{\theta }_{n\mathrm{,}m}\end{array}$$
   (12)

   where ${\theta }_{n,m}$ is the quantity of defective units that are detected in the real die casting procedure. Furthermore, update the scrap amount ${\Omega }_p$ as

   $$\begin{array}{l}{\Omega }_p:={\Omega }_p-w_n({\chi }_{n\mathrm{,}m}-{{\chi }^{\prime }}_{n\mathrm{,}m})\end{array}$$
   (13)

6. If $n=N$ , go to Step 7; else, replace $n$ by $n+1$ and return to Step 3.

7. If $m=M$ , terminate the algorithm; else, replace $m$ by $m+1$ and return to Step 2.▪

Part III is the main stage of Algorithm 1. When $d+1\le m\le M$ , we may use both alloy residue and scrap in the supplementary production. At the onset of the mth shift, the scrap amount $S_{p,(m-d)}$ is added to ${\Omega }_p$ as it is now eligible for supplementary production (equation (9)). Among the alloy residue $R_{p,m}$ and the scrap ${\Omega }_p$ , we should fully utilize $R_{p,m}$ because it would be otherwise discarded at the end of the mth shift. $\chi {\prime }_{n,m}$ defined in equation (11) realizes this scheme. Once the quantity of supplementary production ${\chi }_{n,m}$ is derived by equation (10), we produce as many units as possible, that is, $\chi {\prime }_{n,m}$ units using $R_{p,m}$ , and the remaining ${\chi }_{n,m}-\chi {\prime }_{n,m}$ units using ${\Omega }_p$ . Figure 3 illustrates Part III of the proposed algorithm for supplementary production.

Figure 3.

Algorithm for supplementary production.

As mentioned before, the superiority of the proposed method is the fact that it utilizes the redundant amount of alloy residue and scrap with the maximum efficiency for preventing the production shortage in the online casting procedure. After applying Algorithm 1 up to the last shift, we will obtain total $\sum _{m=1}^M{\chi }_{n,m}$ units of product type $n$ as the supplementary production for compensating $\sum _{m=1}^M{\theta }_{n,m}$ defective units that have occurred during the real casting operation. Hence, the proposed scheme has the following rate of supplementary production, termed $\phi$ ( $0\le \phi \le 1$ )

$$\begin{array}{l}\phi :=\frac{{\displaystyle \sum _{n=1}^N{\displaystyle \sum _{m=1}^M{\chi }_{n\mathrm{,}m}}}}{{\displaystyle \sum _{n=1}^N{\displaystyle \sum _{m=1}^M{\theta }_{n\mathrm{,}m}}}}\end{array}$$

(14)

If $\sum _{m=1}^M{\chi }_{n,m}<\sum _{m=1}^M{\theta }_{n,m}$ for some product type $n$ , we have no other choice but to run extra shifts for producing the shortage of $\sum _{m=1}^M{\theta }_{n,m}-\sum _{m=1}^M{\chi }_{n,m}$ units.

Case study

Data for an example problem

A data set is obtained from a real foundry ¹⁵ and is adjusted to a manageable size for the considered scheduling problem. We implement the LP model with Risk Solver Platform^® v9.5, ¹⁶ a standard optimization software package based on Microsoft Excel that can perform linear/quadratic programming with various search methods.

The foundry receives an order of producing five die casting types ( $N=5$ ) with different alloys and sizes altogether in a period consisting of 15 consecutive shifts ( $M=15$ ; 8 day shifts and 7 night shifts) in total. In each shift, the die casting plant employs two casting machines ( $Q=2$ ) and two melting furnaces ( $P=2$ ), each of which is exclusively assigned to an alloy type: furnace $p=1$ melts KS D6008-AC4D (T6) alloy and furnace $p=2$ melts ASTM B26-A356 alloy. Note that though the number of employed furnaces is two, another furnace melting the same alloy is reserved for each furnace; one furnace of each pair works in every day-shift, that is, odd-numbered shift, while the other one is idle for cleaning and remelting of a new batch of ingots, and vice versa in every night-shift, that is, even-numbered shift. Table 2 shows indices and parameters of the alloys and furnaces used in the case study.

Table 2.

Specification of alloy and melting furnace with $P=2$ and $M=15$ .

Molten alloy	KS D6008-AC4D (T6)		ASTM B26-A356
Furnace index $p$	$p=1$		$p=2$
Capacity $W_{p,m}$ (kg)	700	800	800	900
Assigned shift $m$	Odd	Even	Odd	Even

The demanded quantity $d_n$ and other related parameters of each product type $n$ are summarized in Table 3. The foundry must cast five types of castings denoted by A–E in Table 3 to meet the order. Product A ( $n=1$ ) and product C ( $n=3$ ) are two different types of crank housing whose alloy is identical with each other (see Figure 4). Since both castings are made of the same D6008-AC4D (T6) alloy, they share the same furnace $p=1$ . Due to different weights, however, the former is cast by the 2500-ton high-capacity die casting machine $q=1$ and the latter by the standard machine $q=2$ . The other three products are cylinder head (product B, $n=2$ ), knuckle arm (product D, $n=4$ ), and turbo impeller (product E, $n=5$ ). All of them are made of B26-A356 alloy and thus share the same furnace $p=2$ . However, since the cylinder head requires a high-capacity machine, it is cast by machine $q=1$ along with crank housing A. Meanwhile, knuckle arm and turbo impeller share the standard machine $q=2$ with crank housing C.

Table 3.

Product and machine specification with $N=5$ and $Q=2$ .

Product	A	B	C	D	E
Type index $n$	$n=1$	$n=2$	$n=3$	$n=4$	$n=5$
Quantity $d_n$	350	350	430	300	380
Gross weight $w_n$ (kg)	14.7	14.2	12.5	9.7	10.4
Yield rate ${\beta }_n$ (%)	86	82	74	84	89
Assigned furnace $p$	1	2	1	2	2
Machine index $q$	1	1	2	2	2
Die type	1-A	1-B	2-C	2-D	2-E
Cycle time $t_n$ (min)	9.4	9.2	6.7	5.5	5.6
Setup time ${\overline{t}}_n$ (min)	10	10	8	7	8

Figure 4.

Crank housing A and C.

In order to produce different types of castings in a die casting machine, the machine has to possess the capability of conducting quick die exchange. For example, crank housing (product A) and cylinder head (product B) are cast by machine $q=1$ in die 1-A and die 1-B, respectively. As long as product quantities of both types are positive in a shift, the machine should change the loaded die from 1-A to 1-B or vice versa in the shift. Moreover, the die-exchange procedure has to be executed fast enough that the sequence of completing the production of one product type and initiating the production of another type can be regarded seamless. Otherwise, the setup time for die exchange would become excessive, which makes it impossible to cast different product types consecutively in a single shift. Recently, automatic tie bar pulls and quick die exchange using auto clamping and unclamping devices, as shown in Figure 5 for instance, have made short runs practically feasible, and high-mix, low-volume production employing frequent die exchange is popular in the industry. ⁵ The cycle time and setup time for each casting are also shown in Table 3. Finally, $T_m$ , the available time of each shift $m$ , is set to be 480 min for all $m$ .

$$T_m=480min,\forall m=1,\dots ,15$$

(15)

Figure 5.

Quick die change device.

Table 4 shows the scheduling result of the LP model for the problem represented by Tables 2 and 3. All the product quantities $x_{1,m},\dots ,x_{5,m}$ per the mth shift, $m=1,\dots ,15$ , are displayed along with $E_{1,m}$ and $E_{2,m}$ , namely, the efficiency of alloys 1 and 2 in the mth shift, respectively. As a high average efficiency $E=93.9\%$ is obtained as the objective function, we assert that the LP model has found the optimal solution.

Table 4.

Scheduling result.

Shift ( $m$ )	$x_{1,m}$	$x_{2,m}$	$x_{3,m}$	$x_{4,m}$	$x_{5,m}$	$E_{1,m}$ (%)	$E_{2,m}$ (%)
1	47	0	0	0	76	98.7	98.8
2	45	4	1	81	0	84.3	93.6
3	45	0	3	0	76	99.9	98.8
4	43	6	0	82	1	79.0	99.0
5	43	2	5	1	73	99.2	99.7
6	43	5	0	83	0	79.0	97.3
7	39	5	10	0	69	99.8	98.6
8	24	6	23	0	54	80.0	71.9
9	2	47	53	6	7	98.8	99.8
10	19	30	40	29	0	97.4	78.6
11	0	50	56	9	0	100.0	99.7
12	0	50	64	3	2	100.0	84.4
13	0	50	56	0	8	100.0	99.2
14	0	49	63	5	1	98.4	83.9
15	0	46	56	1	13	100.0	99.8
$\sum x_{n,m}$	350	350	430	300	380	$E$ (%)	93.9

In general, not all the molten alloy in a furnace is poured into a casting die. The oxidized fraction of the amount and inclusions, which would deteriorate the casting quality, should be eliminated before injection. In our case study, we assume that every value of $W_{p,m}$ is the net amount of molten alloy, containing neither the oxidized portion nor inclusions. Hence, all the product quantities $x_{n,m}$ in Table 4 and the supplementary product units that will be induced later are made of the “good-quality” alloy.

Obviously, the solution set $x_{n,m}$ satisfies all the constraints of the LP model described in Table 1. First, comparing the last row of Table 4 and the third row of Table 3, $\sum x_{n,m}$ , the total manufactured volume of product type $n$ , is equal to the ordered quantity $d_n$ for all $n=1,\dots ,5$ , which implies that constraint (C1) of Table 1 is valid. Second, since all the values of $E_{1,m}$ and $E_{2,m}$ do not exceed 100%, the weight constraint (C2) is guaranteed. Finally, a slight examination reveals that the time constraint (C3) is also satisfied throughout all the shifts and casting machines.

Note that the solution set $x_{n,m}$ is sorted out according to equation (6). Out of all types of castings, crank housing A costs most, and its shortage in quantity is most detrimental to the business. Therefore, crank housing A has the highest priority, and thus, we assign it to product type 1. Also, since crank housing A is melted by furnace $p=1$ that has two distinctive capacities of 700 kg in odd-numbered shifts and 800 kg in even-numbered shifts, we can rearrange the solution set to acquire the relation $x_{1,1}\ge x_{1,3}\ge \cdots \ge x_{1,15}$ and $x_{1,2}\ge x_{1,4}\ge \cdots \ge x_{1,14}$ .

Supplementary production

After obtaining the optimal solution from the LP model, we handed the scheduling result to a foundryman for actual casting in the manufacturing plant. To apply the proposed algorithm of supplementary production, we envisage a fault scenario that the casting procedure of each shift generates the defective units ${\theta }_{n,m}$ as shown in Table 5. Based on previous data, it is known that the average fraction of nonconforming or defective castings in the shop ranges 2%–3%, with a little variation depending on the characteristics of the casting type and numerous uncontrollable process parameters. In addition, the number of defective castings increases as does the production quantity of the casting. The synthetic data of Table 5 reflect these properties. We observe that crank housing (product type $n=3$ ), the ordered quantity of which is the maximum among the five types, also has the greatest number of defective units (10 out of 430). However, due to the tight dimensional tolerance and specification on the fatigue strength, the defective rate of knuckle arm (product type $n=4$ ) records the highest rate of 2.66% (8 out of 300).

Table 5.

Defective units.

Shift ( $m$ )	${\theta }_{1,m}$	${\theta }_{2,m}$	${\theta }_{3,m}$	${\theta }_{4,m}$	${\theta }_{5,m}$
1	2	0	0	0	1
2	1	0	0	3	0
3	1	0	0	0	2
4	1	0	0	2	0
5	1	0	0	0	2
6	0	0	0	2	0
7	0	0	1	0	2
8	1	1	0	0	1
9	0	1	2	0	0
10	0	1	0	1	0
11	0	2	1	0	0
12	0	1	2	0	0
13	0	1	1	0	0
14	0	0	2	0	0
15	0	1	1	0	0
Sum	7	8	10	8	8

Table 6 is the result of supplementary production conducted by the proposed algorithm. Here, the delay in scrap use $d$ is set to be 2, that is, a scrap obtained from the present shift can be recycled, if possible, two or more shifts later. Remind that ${\chi }_{n,m}$ is the number of additional units of product type $n$ that are manufactured in the mth shift for compensating the defective units, and ${\pi }_n$ is the remaining number of defective units that has yet to be produced for meeting the ordered quantity. Note that among three residue parameters $R_{p,m}$ , $S_{p,m}$ , and $T_{q,m}$ , only time residue $T_{q,m}$ is displayed in Table 6. The reason is that the volume of alloy residue $R_{p,m}$ and scrap $S_{p,m}$ is ample enough in every shift to produce all the required quantities caused by the defective units of Table 5. In other words, the possibility of accomplishing supplementary production in our case study depends on how much time is available in the casting machines after completing the production of the assigned quantity $x_{n,m}$ in each shift.

Table 6.

Supplementary production result.

$m$	$T_{1,m}$	$T_{2,m}$	${\chi }_{1,m}$	${\chi }_{2,m}$	${\chi }_{3,m}$	${\chi }_{4,m}$	${\chi }_{5,m}$	${\pi }_1$	${\pi }_2$	${\pi }_3$	${\pi }_4$	${\pi }_5$
1	18.2	31.4	0	0	0	0	0	2	0	0	0	1
2	0.2	4.8	0	0	0	0	0	3	0	0	3	1
3	37	11.3	3	0	0	2	0	1	0	0	1	3
4	0.6	0.4	0	0	0	0	0	2	0	0	3	3
5	37.4	9.2	2	0	0	1	0	1	0	0	2	5
6	9.8	0.5	1	0	0	0	0	0	0	0	4	5
7	47.4	3.6	0	0	0	0	0	0	0	1	4	7
8	179.2	0.5	0	0	0	0	0	1	1	1	4	8
9	8.8	29.7	0	0	1	4	0	1	2	2	0	8
10	5.4	29.5	0	0	2	0	2	1	3	0	1	6
11	0	32.3	0	0	0	1	4	1	5	1	0	2
12	0	0.5	0	0	0	0	0	1	6	3	0	2
13	0	37	0	0	3	0	2	1	7	1	0	0
14	9.2	1.8	0	1	0	0	0	1	6	3	0	0
15	36.8	3.5	1	3	0	0	0	0	4	4	0	0
Sum			7	4	6	8	8

We now analyze in detail the result presented in Table 6 shift by shift.

1. $m=1$ . In the first shift, all the values ${\chi }_{n,m}$ are zero because no supplementary production is possible due to the lack of available resource. Hence, ${\pi }_n$ is equal to ${\theta }_n$ , the number of defective units generated in the first shift (see also Table 5).

2. $m=2$ . The second shift falls into part (II) of Algorithm 1, where the scrap volume cannot be used yet for supplementary production. The alloy and time residues are derived as $R_{1,2}=126\text{kg}$ , $R_{2,2}=57.5\text{kg}$ , $T_{1,2}=0.2min$ , and $T_{2,2}=4.8min$ . Applying assignment (7), we obtain ${\chi }_{n,2}=0$ for all $n=1,\dots ,5$ . This result can also be deduced by just comparing the cycle time $t_n$ of product types shown in Table 3 and the time residue $T_{1,2}$ and $T_{2,2}$ . Clearly, $T_{1,2}$ and $T_{2,2}$ are less than the cycle time of any product type, implying that no product unit can be additionally cast by either machine 1 or machine 2. Since defective units ${\theta }_{1,2}=1$ and ${\theta }_{4,2}=3$ are generated in this shift (the third row of Table 5), the cumulative numbers of defective units that must be compensated are ${\pi }_1=3$ , ${\pi }_2={\pi }_3=0$ , ${\pi }_4=3$ , and ${\pi }_5=1$ at the end of the second shift.

3. $m=3$ . From the third shift onward, we can use the scrap volume obtained in the two or more shifts earlier for supplementary production. In the third shift, the alloy and time residues are derived as $R_{1,3}=1\text{kg}$ , $R_{2,3}=9.6\text{kg}$ , $T_{1,3}=37min$ , and $T_{2,3}=11.3min$ . For obtaining the available scrap amount ${\Omega }_p$ , we calculate the scrap volume generated in two shifts earlier, that is, in the first shift: $S_{1,1}=96.7\text{kg}$ and $S_{2,1}=86.9\text{kg}$ . Then, by equation (9) in part (III) of Algorithm 1, we have ${\Omega }_1=S_{1,1}=96.7$ and ${\Omega }_2=S_{2,1}=86.9$ . Let us now apply assignment (10) to each product type in order of priority. First, recalling that ${\pi }_1=3$ and product type 1 is made of the alloy from furnace 1 and by casting machine 1, it follows

$$\begin{array}{c}\hfill R_{1,3}+{\Omega }_1=1+96.7=97.7\ge 3w_1=3\times 14.7=44.1\hfill \\ \multicolumn{1}{c}{T_{1,3}=37\ge 3t_1=3\times 9.4=28.2}\end{array}$$

(16)

This leads to ${\chi }_{1,3}=3$ , that is, all the three defective units of product type 1 generated up to the second shift can be compensated by supplementary production in the third shift. Since the amount of alloy residue $R_{1,3}$ is very small, $\chi {\prime }_{1,3}=0$ by definition (11), which means that all the supplementary products are made of the available scrap ${\Omega }_1$ . Once ${\chi }_{1,3}$ and $\chi {\prime }_{1,3}$ are determined, the residue parameters are updated according to equations (12) and (13) as follows

$$\begin{array}{l}{R}_{1,3}:=R_{1,3}-w_1{{\chi }^{\prime }}_{1,3}=1-14.7\times 0=1\\ T_{1,3}:=T_{1,3}-t_1{\chi }_{1,3}=37-9.4\times 3=8.8\\ {\Omega }_1:={\Omega }_1-w_1({\chi }_{1,3}-{{\chi }^{\prime }}_{1,3})=96.7-14.7\times (3-0)=52.6\end{array}$$

(17)

In this way, we continue to execute the algorithm of the supplementary production until the last shift. Referring to the last two rows of Table 6, all the defective units of product types 1, 4, and 5 could be compensated online throughout the casting shifts, whereas some of the product types 2 and 3 are left unfinished ( ${\pi }_2=4$ and ${\pi }_3=4$ ). Hence, an extra shift should be run to manufacture this shortage. Substituting the data of the last rows of Tables 5 and 6, we derive the rate of supplementary production $\phi$ as

$\begin{array}{l}\phi =\frac{7+4+6+8+8}{7+8+10+8+8}=0.80\end{array}$

(18)

Therefore, by the proposed algorithm, the die casting process could supplement 80% of all the generated defective units. Note that the result of our case study demonstrates the virtue of the proposed scheme of supplementary production that it does not require any additional shift or resource aside from the prescribed amount for the LP-based schedule.

Conclusion

The ability to cope with unexpected occurrences of defective units is a requisite for guaranteeing the reliability of modern manufacturing systems. As exposed to various uncertainties of industrial processes, the die casting operation is not an exception. In this article, we have proposed a novel supplementary production of die casting based on the LP model. A major accomplishment of the proposed scheme is to fully utilize surplus resources—molten alloy, scrap, and manufacturing time—of a current die casting shift for additional production of defective units generated in the previous shifts. Since the amount of surplus resources may not be enough to produce all the defective units, each product type is given the priority of supplementary production, and an algorithm is presented to assign surplus resources to producing supplementary product units according to their priorities. The proposed scheme can be applied to any scheduling result obtained by the LP model since it preserves the original solution set; only the order of execution between the shifts is adjusted for acquiring the highest resource potential fit for the algorithm. The numerical experiment with a synthetic outcome of defective units validates the applicability and performance of the proposed scheme.