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Cutting tool management in manufacturing firms constitutes an essential element in production cost optimization. In order to optimize the cutting tool stock level while concurrently minimizing production costs, a cost optimization model which considers machining parameters is required. This inclusive modeling consideration is a major step towards achieving effectiveness of cutting tool management policy in manufacturing systems with stochastic driven policies for tool demand. This paper presents a cost optimization model for cutting tools whose utilization level is assumed to be optimized in respect of the machining parameters. The proposed cost model in this research incorporated the effects of diversified machining costs ranging from operational through machining, shortage, holding, material and ordering costs. The machining of parts was assumed to be a single cutting operation. Holt-Winters forecasting technique was used to create a stochastic demand dataset for a test scenario in the production of a high-end automotive part. Some numerical examples used to validate the developed model were implemented to illustrate the optimal machining and tool inventory conditions. Furthermore, a sensitivity analysis was carried out to study the influence of varying production parameters such as: machine uptime, demand and cutting parameters on the overall production cost. The results showed that a desired low level of tool storage and holding costs were obtained at the optimal stock levels. The machining uptime had a significant influence on the total cost while tool life and cutting feed rate were both identified as the most influential cutting variables on the total cost. Furthermore, the cutting speed rate had a marginal effect on both costs and tool life. Other cost variables such as shortage and tool costs had significantly low effect on the overall cost. The output trend showed that the feed rate is the most significant cutting parameter in the machining operation, hence influencing the cost the most. Also, machine uptime and demand significantly influenced the total production cost.

Research in the field of cost optimization modelling for stochastic inventory management and control has significantly intensified over the past few decades [

Companies that focus on industrial metal works are often faced with predictive challenges even though there are several quantitative techniques that can be used to improve production processes such as forecasting, probability distributions and optimization related methods amongst others [

In most companies, the behavior of product sales is random, since it is difficult to absolutely predict demand to determine sales. Due to this challenge, the focus is often placed on alternate areas where the lack of a sound process design can be mitigated and controlled. Some of these areas include:

• Production: The lack of a future programming and being left with the expectation of the needs of the clients, manufacturing companies are forced to fulfill the demand of their clients by incurring additional time in the productive processes.

• Inventories. To the poor management of inventory levels, such as the case of high inventory level, this causes high costs and prevents the expansion of the business.

• Finances: Given the high costs of inventories, companies limit their growth in the purchase of new equipment to make production processes more efficient.

To mitigate these and other problems caused by poor planning of the production, it is necessary to design a methodology that provides accurate predictions of the sales of industrial metal mechanical products or mitigate alternate costs generated elsewhere in production [

Existing machining models more often than not, are seen to be analyzing the combination of different cutting materials by focusing on their conditions while conducting analytical, numerical, empirical and artificial intelligence based observations. Robust predictive models are often required to accommodate the complex interaction that exists amongst the trio of a workpiece, the cutting tool and the machine tool [

Modeling a cutting process often constitute integrating a system’s planning process to improve productivity and enhance product quality. However, to have a significant progress in the modelling of machining processes, especially on metallic work pieces, some notable variables capable of improving the quality of output should be taken into cognizance. Some of these include the effect of stress, strain rate and temperature amongst others. From the point of view of demand, when a product with intricate properties such as high strength (e.g. aerospace superalloys) is seasonally or randomly produced, it often becomes difficult for industries machining these materials to adequately estimate the quantity of cutting tools needed to meet the demand. On the other hand, working with high strength materials increases the tendency of tool penalty cost due to short tool life. These effects are central to the increase in the cost of products and a reduction in the overall profit margin. These costs could be associated with storage costs, ordering costs, stock-out or shortage costs.

Traditional procurement policies are subjective and often premised on periodic supply decisions. These policies are usually based on simplified and idealistic assumptions, as well as on the expected cost criterion, without considering machining factors as well as the finished goods throughput. Some research works [

The rate of tool depletion was found to be a function of tool life and machining conditions. Finding the optimal machining conditions related to tool life and obtaining the optimal order quantity could stabilize cycle length and ordering cycle. A study into production cost in the milling of titanium alloys by considering machining conditions, was conducted by Conradie, Dimitrov and Oosthuizen [

Li, Sarker and Yi [

In their research work on cost modelling in milling operations, Parent, Songmene and Kenné [

The above mentioned works, as most resources from the literature, do not consider the attributes that influences the productive length of a tool-life while solving the general optimization problem. It is however important to recognize that the tool-life productive length has a great impact on tool procurement and management of its inventory systems as well as indirectly influence total operating cost.

Based on the reality of market variability, which has got so much impact on production processes, modern day manufacturing firms are faced with a rising level of supply uncertainty and demand variability. These fluctuations are characterized with challenges capable of impacting negatively on the optimum performance level and general sustainability of the manufacturing sector. Industrial firms involved in the production of high-valued parts such as the automotive or aerospace industry amongst others are often posed with these challenges [

Other factors of significant importance in manufacturing processes include machine unit costs, machine uptime costs, ordering costs and lead-time due to their impact on procurement, tool inventory policy and influences on tool lifespan. These considerations if well managed, can reduce the costs associated with production activities, decrease the levels of inventory of tools with low demand and increase the level of service for products considering their seasonality and cyclicality.

This paper presents a combined tool acquisition and cost optimization model capable of predicting and reducing the degree of uncertainty in demand. With this collective information, this will assist in determining the lot-sizing and cycle-time of cutting tools for a better manipulation of the supply chain system. This paper develops a nonlinear cost optimization model based on an inventory policy for cutting tools at optimum machining parameters for the production of high-value mechanical parts. Finally, the study inclusively offers information to a production manager regarding the life of his cutting tools in a production process and permits the flexibility to adapt a production process to suit demands consumer requirements.

In this section, the model development is introduced with the mathematical formulations, assumptions and constraints. Prior to the model formulation, an illustration of the interaction between main cost components and effect of cost modelling is presented as seen in

From

This sub-section presents the assumptions considered in the model development process:

1) Tool orders are made on request and not in predefined cyclic batches.

2) The cutting operations tools are mainly divided into two groups, namely: tools for roughing and finishing operations. However, this research will focus on the use a single cutting operation.

3) Once a cutting tool has reached its life span, it is no longer used to avoid tool breakage and the occurrence of a defective process. This also eliminates the penalty cost for tool break.

4) The life span of a tool is considered based on empirical models with the experimentation of similar conditions.

5) Tool vendors are located nearby resulting in a fixed lead time. Hence, all tool demands are satisfied in real-time without any need for back orders.

6) Labor cost is included within machine usage cost and considered to be a constant.

7) Inventory is considered as the average between the initial and final inventory.

8) The demand of part product is random, however, it presents patterns of seasonality, cyclic tendencies that can be followed through forecasting methods such as triple exponential smoothening as premised in (Holt-Winter’s Method). Some examples of these patterns are festive yearly seasons, economicfactors, recession and inflation.

9) Machine time used is inclusive of both operational, setup and installation time.

10) Price per unit product is constant

11) Ordering costs are constant

12) The holding cost is a constant and includes both the warehouse and preservation costs.

13) Initial purchases are below demand quantity.

Objective FunctionThe focus of this research is to optimize production by obtaining the minimum total cost of operations supported by the inventory system as a complete manufacturing process. This can be represented by an objective function which consists of the cost optimizations elements, these are; the cost of operations, cost of holding, cost of shortage, annual ordering and raw material costs.

T o t a l c o s t = o p e r a t i o n a l c o s t + s h o r t a g e c o s t + h o l d i n g c o s t + r a w m a t e r i a l + o r d e r i n g c o s t

TC ( $ ) = C o p ( $ ) + C s h ( $ ) + C h ( $ ) + C B ( $ ) + C o ($)

*TC = total cost of production in over a production time period i.

The next sub-sections present the details of the elements of the objective function:

Machining time is vital in determining the factor of tool life. The cutting time during production is defined by the material volume to be cut (Length × Width × Height) and the depths of cut (radial and axial) as presented in

To estimate the life span of a tool using Taylor’s empirical model as presented in Equation (2), it is necessary to determine the tool cost. The following equations are a derivation of the original Taylor’s model:

t 0 = k v α f β d r w ε

In application for milling operations, the tool life expression ( t 0 ) was extended by adding the radial width of cut with α, β and ε serving as parametric constants. Equations based on this type of operation could be devised for either roughing or finishing operations i.e. t 0 r or t 0 f . However, this research is focused on the roughing operation as presented in Equation (3) below:

t 0 r = k v r α f r β d r γ w r λ

The parametric constants defined in this model need to be obtained through experimentation due to the fact that the machining conditions such as the tool material-make, workpiece, cutting fluid, rigidity of the assembly and the system vibration amongst other factors, vary significantly from one machine to the other. These can be determined using a statistical design of experiment (DOE) [

The cost of tools utilization per hour is defined here as the ratio of the current market price to the tool life as presented in Equation (5). The tool life is usually expressed in terms of the operations the tool is subjected to over time:

c h r = c p r t 0 r

In this research, the primary focus is the cutting operation. Apart from identifying and analyzing the cutting operations, a special consideration is also given to the respective quantities of cutting tools ( Q p ) deployed during the overall production task. A combination of all these results in the models below:

operationalcost = C o p = c ( r , f ) ⋅ Q p

where,

c r = ( k 0 ⋅ t o p r + c h r ⋅ t o p r )

c f = ( k 0 ⋅ t o p f + c h f ⋅ t o p f )

Q p = Q r ( roughing ) + Q f ( finishing )

where Q f = 0 in roughing only operations.

Thus:

C o p ( $ ) = ( k 0 ⋅ t o p + c h r ⋅ t o p ) ⋅ Q p

The holding cost represents all the costs associated with the storage of the inventory until its depletion. These costs usually include tied-up capital, space, insurance, protection, taxes attributed to storage amongst others. The holding cost can be assessed either continuously or on a period-by-period basis. A common consideration for the holding cost is that an initial inventory level exists at the beginning of every period. Based on this, the inventory cost at the beginning of a period I can be obtained from:

C h ( $ ) = c h { I i + Q s }

where I i is considered as the average inventory level required to avoid shortage of cutting tools. This is computed based on the summation of the initial and final inventory levels divided by two.

I i = ( I initial + I final ) 2

Furthermore, I i ¯ as presented below, represents the total amount of tools required to meet the demand for a given product.

I i ¯ = Q s + I i

In addition, considering the likely physical limitations of a warehouse in terms of its holding capacity, the capacity of storage can be represented as:

I i ¯ ≤ A i

Considering that the average inventory, we can add it directly to the objective function as follows:

C h ( $ ) = c h ⋅ I i ¯ (6)

The shortage cost ( C s h ) represents the costs incurred during the time production has a depleted inventory. This is the cost incurred between this time and the period of a re-order. It includes costs of not using the machine and equipment involved in production.

C s h = c s h ⋅ F t (7)

F t = D y i Q s

where y i service as an indicator for when shortage is present in the equation. It can be activated or switched off during computation and holds two statuses which are 0 or 1. Whereas F t stands for the amount of missing tools in system.

The material cost of purchasing the workpiece used in production is included for a more inclusive and comprehensive review of the total costs involved in production. It is a factor of the quantity of material and the cost per unit material purchase.

C B ( $ ) = Q B ⋅ c B

The ordering cost is described as a function of the unit cost of ordering tools, cycle time between orders and a variableterm. The variable term z i is used to consider the alternate conditions which are assumed constant in this present model. Such conditions for variability in unit ordering, shipping, delays and alternate reasons which could influence this cost could be expanded through this variable in future research modifications of this model. However, in this research work, the ordering cost is considered as constant of value 1.

C O ( $ ) = C f D Q z i

The objective function formed to minimize the production cost that involves the operation cost, shortage cost, holding cost, material cost and ordering cost is shown in the Equation (11). The cost of labor is not included in this function. The focal point of this function lies in the operational costs which is derived from the machining parameters and tool conditions.

MinTC ( Q p T c ) i = C o p + C s h + C h + C B + C O

Minimised Total Cost ( Q p , T c ) i = ( k 0 t o p + c h r t o p ) Q p + c s h ∑ i = 0 n D y i Q s + ∑ i = 0 n c h i I i ¯ + Q B ⋅ c B + C f D Q o z i

Tool restrictions

Some decision variables found in the model are the radial width of cut for the roughing operation ( w r ) and the axial depth of cut ( d r ). This radial width ( w r ) is connected to the number of passes and the width of the volume to machine (W):

w r ⋅ N w = W

Also, the axial depth of cut ( d r ) which is connected to the number of axial passes ( N p ) and the height (H) of the volume to machine is a decision variable:

d r ⋅ N p = H

Limits are given to the width of radial cuts ( w r ) by the diameter of the tool ( D t ):

w r ≤ D t

The depths of axial cuts ( d r ) are also limited by the length of the tool flutes ( h f max ):

d r ≤ h f max

Machine-tool constraints

The power needed for the operation relies on the material removal rate (M_{RR}) and the specific power rating of the workpiece material as seen in Equation (12). It can thus be inferred that at maximum machine power, the maximum metal removal rate (M_{RR}) can be determined. This rate depends on the width of cut ( w r ), the feed ( f r ) to the depth ( d r ), as shown in the equation below:

w r ∗ f r ∗ d r ≤ M R R

The material removal rate ( M R R ) can be denoted as:

M R R ≤ energy coefficent ( η ) ⋅ maximum machine cutting power Unit power for the workpiece material (12)

In this case,

• The cutting speed rate is constrained by the machine’s maximum spindle speed.

1000 ⋅ v r π ⋅ D t ≤ N max ;

• The achievable power is constrained by the maximum safe operational force. 60 ⋅ H P r v r ≤ F max ;

• The cutting feed rate is constrained by the machine’s maximum feed rate. f r ≤ f m max

where f m max is the maximum feed of the machine, F max are the maximum forces obtained from the machine and N max is the maximum spindle speed of the machine.

Inventory and shortage restrictions

With restrictions of inventory as follows:

( I i − 1 + Q p ) − I i = Q t i

I i ≤ A i

Restrictions of shortage:

Q t − Q p = F t

Q p − I i ¯ ( y i ) ≤ 0

Finally, non-negativity is assumed for all the decision variables and binary variables:

F t , I i ¯ , A i ≥ 0 and y i = 0 , 1

This project focuses on optimizing the process thought for quantity of tool in an inventory. Therefore the study remains within basic machine limitations or overlooks tool immersion and surface finish as constraints.

The model was implemented in “LINGO” by LINDO® systems. “LINGO” is a commercially available optimization modelling software used for building and solving mathematical optimization models. Its package build-up provides a language tool needed to build and design models as well as include needed solving tools within a compact integrated environment. It could be applied to linear, nonlinear, quadratic, integer and stochastic optimization problems. A seven core computer with 8 gigabytes of memory was used for the LINDO analysis to minimize processing time. Note that for each result the LINGO solver provided, it declared the solution as being only a local optimum, i.e. it could not fully guarantee a global optimum.

The illustration in below depicts the policy of the model for the annual optimization of production in a manufacturing company whose primary task and operations is focused at metal works. This new tool policy includes the forecast of sales and the annual tool planning for production. The main objectives of this illustration is to show how the model can meet a high level of service in sales and a low inventory level of production. As the proposed model is founded on the principle of the Economic order quantity model (EOQ) [

From the figure, the tool inventory system shows two different ordering quantity cycles which changes based on the variation of tools inventory depletion. By evaluating an accurate estimate of the production demand and correlating it with the productive tool life for machining tools, it is possible to determine the optimal order quantity ( Q t ) and optimal-order cycle ( T c ). A steep quantity slope due to reduced tool life or frequent tool damage would reduce the order cycle length. This can be prevented by accurately estimating the reorder point from the lead time in obtaining new tools [

to react to depleting stock. Beyond this threshold the risks of reaching shortage is high.

It is worth noting that a machining tool has an expected productive life around its mean value; therefore, there exists more risk of failure cost for prolonged use of machining tool over this value. However, a reduction in running time of these tools will potentially increase the replacement quantity and thus re-ordering cost. Therefore, production inventory estimation is contingent on requirement needs based on trends and controlling the utilization time for the tools could be used to regulate the inventory system without influencing the integrity and quality of the product.

The example shown below was adapted from the work of Wang, Kim, Katayama and Hsueh [

Machine data | |||
---|---|---|---|

Nmax (rpm) | 1200 | max feed speed | 8000 |

HPs(w/mm^{3}/min) | 59.2 | max cutting forces (kN) | 2000 |

Hpmax (kw) | 22,000 | max spindle speed (rpm) | 12,000 |

Energy efficiency η (%) | 0.85 |

Cutting data | Cost data | ||
---|---|---|---|

feed per tooth (mm/tooth) | 0.1 - 0.6 | Quantity of workpiece purchased (kg) | 500 |

cutting speed (m/min) | 50 - 300 | price of the tool ($) | 49.5 |

Height (mm) | 30 | cost per minute operation ($/min) | 4 |

Length (mm) | 240 | raw material cost ($) per unit | 0.5 |

Width (mm) | 60 | Inventory cost | 0.5 |

axial depth of cut (mm) | 4 | Per ordering cost | $25 |

α | 0.2 | Holding cost per unit year | $9 |

β | 4.0 | Time of operation (min) | 6 |

γ | 1.0 | Initial inventory | 100 |

Taylor constant K | 2.086E+12 |

Solving the developed model using the LINGO® solver in Equation (11) gave the following cost breakdown as presented in

This elevated holding cost is due to the short cycle period of 5.5 days for reordering. A set of approximately 65 orders per year was obtained. The average forecast quantity of 7 units per order cycle was obtained from Equation (5). This resulted in an optimal order quantity Q 0 of about 38 units to minimize total costs of production. This can also be seen from

quantity was obtained at the intersection of the ordering, sum of operations and safety holding costs. The results also indicated that a safety stock level of 28 units indicating the least amount in the inventory is needed to prevent a shortage situation. Furthermore, a value for re-ordering of stock to match lead time was deduced to be 52 units.

Cutting data | Cost data | ||
---|---|---|---|

Cutting feed rate (mm/min) | 4.01 | Mean Demand/year | 5000 |

feed per tooth (mm/tooth) | 0.05 | Quantity of workpiece purchased (kg) | 200 |

cutting speed (m/min) | 41 | price of the tool ($) | 90 |

Height (mm) | 25 | cost per minute operation ($/min) | 4 |

Length (mm) | 100 | raw material cost ($) per unit | 0.75 |

Width (mm) | 60 | Inventory cost | 0.5 |

axial depth of cut (mm) | 0.063 | Per ordering cost | $30 |

α | 5 | Holding cost per unit year | $6 |

β | 1.75 | Time of operation (min) | 6 |

γ | 3.5 | ||

λ | 0 | ||

Taylor constant K | 5.0E+11 |

in Equation (11). From the figure, the main contributors to the total production costs were identified as the ordering and machine operations cost. However, this example presents a lower inventory holding costs. This is due to the lower cost of purchasing HSS tools and its influence in the combined optimization function. Due to the successful minimization of the model, the holding costs incurred for safety stock as well as the shortage costs remain insignificant in the overall summation.

For the numerical example, a set of 22 orders per year over an approximate 16 days cycle was obtained. The average forecast quantity of 13 units per order cycle was obtained from Equation (5). This resulted in an optimal order quantity of about 54 units to minimize total costs of production. This can also be seen from

An in-depth sensitivity analysis of numerical sample case 2 is performed to establish influencing factors relevant to changes in operational costs, determine the influence of demand on inventory optimization and ascertain the key cost drivers of the model using actual industrial stochastic demand data.

From an overview of the factors influencing the total cost of consequence, operational machining costs are selected to assess the effect of cutting factors. The

most important variables for minimizing this cost area are the cutting speed and the feed rate. These significantly affect the tool life criterion and could create a rising cost avalanche during production. By fixing other factors involved to optimal levels, a plot of the relationship between speed and feed to total cost is shown in the

From

Similar observations are seen from the feed in

The estimated demand ideally influences total production cost since inclusive components costs do increase (

proximity (

From

A sensitivity analysis was carried out to ascertain the key cost drivers of the model using actual industrial stochastic demand data. The application of Holt-Winters (HW) forecasting was used to establish the trend for future demands. The current Holt-Winters method of exponential smoothing displays trend and seasonality and is characterized by three smoothing equations: the smoothing for level, equation for the trend and equation for seasonality [

The basic equations for the Holt-Winters additive method are:

Equation for level (Overall smoothing):

L t = α ( Y t S t − s ) + ( 1 − α ) ( L t − 1 + b t − 1 ) (12)

Equation for trend (Trend Smoothing):

b t = β ( L t − L t − 1 ) + ( 1 − β ) b t − 1 (13)

Equation for seasonality (seasonal index):

S t = γ Y t L t + ( 1 − γ ) S t − s (14)

Forecast for m period equals:

F t + m = L t + b t m + S t − s m (15)

where Y t are the observed value, L t represent the smoothing of variable in time t, b t the trend estimation and S t is the estimation of seasonality. The smoothing constants α, β, γ are in the interval [0, 1], m is the number of forecast periods and s stands for the duration of seasonality (e.g. in months or quarters in a year).

Manufacturing industrial data figures utilized to establish the adequacy of the model can be found in

Sensitivityindex = D max − D min / D max (16)

where D min and D max are the minimum and maximum output costs resulting from changing dependent variables to the model.

The machine data and user inputs selected for sensitivity analysis and the applied maximum and minimum values are presented in

The results of the sensitivity analysis for the variables in the model is shown in

Cost variables | Minimum | Maximum | Sensitivity (%) | ||
---|---|---|---|---|---|

Quantity | Cost value | Quantity | Cost value | ||

Machine Uptime (hr) | 1 | $17,646.62 | 8 | $89,777.68 | 80.34% |

Demand ($/year) | 1500 | $3547.49 | 8000 | $7644.05 | 53.59% |

Ordering Costs ($/unit) | 10 | 3764.443 | 50 | 7737.640 | 51.35% |

Machining tool life (min) | 5 | 9646.872 | 160 | 5019.712 | 47.97% |

Machine unit cost (Ko) ($) | 3 | $5784.02 | 20 | $9962.62 | 41.94% |

Cutting Feed (mm/tooth) | 0.04 | 5708.877 | 0.1 | 8360.683 | 31.72% |

Cutting Speed (mm/min) | 30 | 5092.900 | 45 | 6741.792 | 24.46% |

Machining tool cost per ($/unit) | 60 | $5423.33 | 120 | $6494.90 | 16.50% |

Lead time (days) | 3.5 | $6117.26 | 14 | $6560.08 | 6.75% |

Shortage Costs ($/unit) | 5 | $6112.16 | 40 | $6119.85 | 0.13% |

that Machining uptime has a significant influence on the total cost. An increase in cost of approximately ±10% per additional 1000 units demand was also identified as significant. Machine tool life and cutting feed rate have been identified as the most influential cutting variables to total costs. The cutting speed rate had marginal effect on both costs and tool life. Other costs variable such as shortage costs per unit and tool costs had low sensitivity values as their effects were mitigated from the minimization process. Optimization trend of ordered quantity over changing demand displayed a correlation with stochastic changes premised on a smoothing factor. This is shown in

From the results in

The implications of the research to industrial practice, shows an adequate method to estimate tools needed in a production system based on machining and clearly define the correlations between costs to process machining conditions. An enterprise could thus estimate create a production process which optimizes machining conditions while at the same time finding the optimal settings to tool allocation for storage.

This study has presented an inventory control analysis premised on stochastic demand of machining tools. The model developed in this research is an optimum cost model governed by experimented machining conditions. Furthermore, the model is proposed to serve as a tool management policy for manufacturing industries. The fundamental considerations of the developed model includes the cutting tool, machining conditions and applied constraints. All of these are related to realistic conditions adapted to a practical production environment.

Optimization of the model solution was performed using Lingo, a commercially available optimization software. The simulation results obtained indicates that the feed rate is the most significant cutting parameter following its effect on machining costs. This in turn has a significant incremental effect on the overall production cost. Machine uptime and demand also showed significant effect on the total production cost. The process of optimizing the demand quantity, resulted in a cost-efficient optimum demand range characterized with a low safety stock and optimum ordering quantity. Additionally, an increase in the level of demand beyond the optimum range indicated that higher safety stock would be required. The application of Holt-Winters exponential smoothing forecasting technique further validates the outputs from this research.

Some limitations apply to the results of this research. Some of which are that the study focuses mainly on the use of a single cutting operation. In addition, an in-depth breakdown of cost influences from personnel, tool refurbishing and penalty costs are not considered. Fluctuations in ordering, holding, and unexpected inventory conditions are also not assessed.

In conclusion, several practical applications of this model can be obtained. This beginning with a single production operation concept as illustrated in this research to a more robust multi-operation manufacturing assembly. Future considerations of this research involve the development of a multiple-operational production approach with the inclusion of labor and penalty costs.

The authors declare no conflicts of interest regarding the publication of this paper.

Pantoja, F.G., Songmene, V., Kenné, J.-P., Olufayo, O.A. and Ayomoh, M. (2018) Development of a Tool Cost Optimization Model for Stochastic Demand of Machined Products. Applied Mathematics, 9, 1395-1423. https://doi.org/10.4236/am.2018.912091

1) Cost function components and intermediate variables

C o p : Total cost of operation ($)

C s h : Total shortage cost ($)

C B : Total workpiece material cost ($)

C h : Total holding cost ($)

C o : Total ordering cost ($)

TC: Total production cost ($)

2) Decision variables

Q p : Quantity of cutting tools required for production (tool/order)

Q r : Quantity of cutting tools required for roughing operation (tool/order) in period i

Q f : Quantity of cutting tools for finishing operation (tool/order)

T c : Tool ordering cycle length i.e. (length of time between placement of replenishment of tool orders) (time units)

3) Cost function components and intermediate variables

c h r , c h f : Tool utilization cost per hour ($/hr)

c h : Expected unit holding cost ($/tool)

c B : Cost per unit of workpiece ($/unit)

c p r , c p f : Tool Market price ($/tool)

k 0 : Cost of machine use per hour ($/hr)

c s h : Expected total annual shortage cost ($/tool)

L: Length per unit of workpiece to be cut (mm)

W: Width per unit of workpiece to be cut (mm)

H: Height per unit of workpiece to be cut (mm)

w r : Width (radial) of cut (mm)

d r : Depth (axial) of cut (mm)

D t : Diameter of the tool (mm)

t o p r , t o p f : Cutting time per unit of workpiece, roughing and finishing operations, (min/unit)

t 0 r , t 0 f : Tool life, roughing and finishing operations (min)

k: Taylor’s equation constant

v r , v f : Cutting speed, roughing and finishing operations (m/min)

f f f r : Feed per tooth, roughing and finishing operations (mm/tooth)

α , β , γ , λ : Tool life parameters

Q t : Demand for tool required per year (in units)

Q s : Quantity of tool shortage (units)

Q B : Quantity of material ordered (units)

D: Demand rate of products (products/year)

F t : Amount of missing tool in system (units)

I i : Average inventory at the end of the current period (units)

I i − 1 : Previous inventory (units)

A i : Total capacity of inventory (units)

I i ¯ : Possible holding inventory (units)

z i : Factor indicating variable ordering cost (1,0)