Multi-objective Optimization Mathematical Model

Multi-objective Optimization Mathematical specimenCHAPTER 3 PRODUCTION COST get to INJURY LEVEL MODELLING 3.1 IntroductionThis chapter describes a multi-objective optimization mathematical warning with finale variables and constraints on them. element 3.2 presents the precedent formulation with aim to minimise the total issue personify and consummation scathe level particularly in a manufacturing sedulousness over a planning prospect. Section 3.3 presents ZC1the cuticle study d au naturel(p)n from literature to validate the proposed model. Section 3.4 presents the method to calculate the work injury terms with consideration of work injury level factor. Section 3.5 gives the summary for this chapter. mystify formulationThe traditional doing planning model is a mathematical optimization model. In such a model, the objective post is the total court, and the decision variable refers to production standard, entry quantity, and outsourcing quantity. The constraint fo rge in the traditional production planning model includes the demand in a planning horizon. In the work of (Xu, 2015), the traditional model includes the work injury cost. The expansion of the model hence mentions the description of the objective function and constraints. The model aims to fulfil the two objective areObjective 1 (ob1) Minimize production cost (CP).Objective 2 (ob2) Minimize work injury level (WIL).Model AssumptionsA mathematical model herein is developed on the following assumptions areThe values of either parameters are certain over the following(a) catamenia t in planning horizon.Actual mash levels, operative hours and warehouse capacity in each check so-and-sonot exceed their respective maximum levels.The sum of role players and tasks are the same over the planning horizon.A single type of product is manufactured over the planning horizon.Trivial solutions will be ignored.Model NotationsThe following notations are used after reviewing the literature and considering practical scenarios (Wang Liang, 2004 Masud Hwang, 1980 Wang Fang, 2001 Chakrabortty Hasin, 2013).t the time period (t=1, 2, 3, , n).CMR the regular unit tangible cost of the product ($/unit).CMO the extra time unit material cost of the product ($/unit).Pt the add of products fabricated (production quantity) during the regular working hours in the period t (unit).Ot the number of products fabricated (extra time production quantity) during the extra time in period t (unit).CLR the regular unit get cost in period t ($/unit).CLO the overtime unit labor cost in period t ($/unit).Ht the regular working man-hour required in period t (man-hour).Et the overtime working man-hour required in period t (man-hour).CI the unit inventory cost ($/unit).It the units of product to be left over as an inventory during period t (unit).CWI total work injury cost over the planning horizon.C2aa3 It-1 the units of leftover products in the previous period of t.dt the product demand in per iod t.D the total demand over the planning horizon.dn the number of working days in period t.W the number of employees.E* allowable overtime hours in period t.Objective function (ob1)To get hold of the ob1, the integrated production planning was used in order to minimize the production cost. The total production cost consists of the material cost, labor cost, inventory cost and work injury cost. Let C represent various costs. The total cost is hence denoted by(3.1)ZC4whereCproduction the total production cost.Cmaterial the material cost.Clabor the labor cost.Cinventory the inventory cost.CWI work injury cost.whereMaterial toll Material cost is the sum of regular material cost and overtime material cost that includes the raw material cost and overhead cost. Raw material directly contributes to the finished product, and the overhead cost includes the utility cost such as electricity, gas and economic rent etc.Labor Cost Labor cost is the sum of all wages paid to employees for the p roduction of products in both regular time and overtime hours. arsenal cost Inventory cost is the holding cost of products in stock.Work injury cost the work injury cost caused by the insistent assembly production over an entire production periodThe first objective function (ob1) of the model is to minimize the cost of production (eq. 3.2).(3.2)(3.3)Moreover, par 3.1 can be written asZC5aa6Where, the first part of equation 3.3, represents the regular material cost (CMR) incurred on the regular production quantity (Pt) and overtime material cost (CMO) on overtime production quantity (Ot) over the planning period. The second part represents the labour cost (workers salary) and it is the combination of the regular unit labor cost (CLR) during regular working hours (Ht) and the overtime unit labor cost (CLO) in overtime working hours (Et). The third part is the unit inventory cost for left over products as an inventory over the period (It) and the final part denotes the accumulate d work injury cost (CWI) during regular working man-hour (Ht) and the overtime working man-hour (Et). Furthermore, the Cwiis calculated on a yearly basis with 21.74 working days in a month and 8-hour shift as per the study by Lin. (2008). It can be seen in equation ().(2)Objective function (ob2)The second objective function (ob2) of the modelis to minimize the work injury levels over the planning horizon as shown belowFurthermoreWhere, equation () represents the accumulated work injury level (WIL) during regular working man-hour (Ht) and the overtime working man-hour (Et) in the time period t. As discussed in literature that increase in regular and overtime production quantity will increase the work injury level because of retentive exposure of worker to the repetitive task. Therefore, higher the production quantity, the longer the working hours and the higher the work injury level.Overall objective functionDecision variablesThe decision variables in the above model are explained belowProduction quantity (Pt) during the regular working time in period t. Overtime production quantity (Ot) in period t.Number of products in inventory (It) in period t.Dependant variablesRegular working man-hour (Ht) required in period t.Overtime working man-hour (Et) required in period t.3.2.3 diffidencesDemand constraint(3.4)(3.4)Where, the sum of regular production quantity (Pt), overtime production quantity (Ot) and inventory levels essentially great than or equal to the market demand (dt) in a period t as shown in equation 3.4. Moreover, the sum of all periods demand (dt) should be greater than or equal to total demand (D) over planning horizon as shown in equation 3.4.Labor hour limit constraint.(3.5)where, equation (3.5) represents the regular working man-hour (Ht) in period t should be less than or equal to 8 hours per day, monthly working days (dn) as tumefy as number of employees (W). Overtime working man-hour (Et) should not exceed the allowable hours (E*) by law.Pro duction rate constraint. Assume that the unit time is one hour, and the congener between the produced units and labor can be expressed as(3.6)C7whereRh the production rate during regular working time.Re the production rate during overtime.Non-negative constraints. The number of produced product, the number of demand and the unit labor cost are non-negative, respectively that is(3.7)Model implementationTo validate the model efficiency, the specific case study approximately the aggregate production planning of single product is selected. This case study is drawn from the literature and the author s own experience in industry (Chakrabortty Hasin, 2013).Case study descriptionTo validate the proposed model, the real life data of Comfit Composite Knit Limited (CCKL) is interpreted. The company manufactures knit ware product. The production planning is more specifically about the production of hooded jacket over a couple of months planning horizon. Table 3.1 3.2 give the monthly produc t demand, and cerebrate cost data are as follows.Table 3.1 Product demand over planning horizonPeriod (t)MayJuneDemand (dt) (units)14003000Table 3.2 Cost data of case studyRegular time unit material cost (CMR)14 ($/unit)Overtime unit material cost (CMO)28 ($/unit)Inventory Cost (CI)3.5 ($/unit)Regular time unit labor cost (CLR)8 ($/unit)Overtime unit labor cost (CLO)12 ($/unit).Table 3.3 Model Constraint DataInitial Inventory level- I0500End inventory in period- I2400Labor hour (Ht 0+ Et) 225 man-hoursProduction rate (Rh)0.033 man-hour/unitIn given case study, the company makes knit ware product (Hooded Jacket). In manufacturing of product, the job requires a worker posture in a standing perplex to process the product on a machine. The worker need to place the product part in a machine to stitch it , for this reason worker has to lean forward to focus on the product parts. The neck may bend to get a better view of stitching if required. To perform this task, the stop number arms are need to be elevated to the height of the work table. To place the product part in a right way the proboscis rotation is required (Fig. 3.1).3.3 Work injury cost (Cwi) calculationWork injury cost C8(Cwi) is calculated by using the model proposed by Lin (2008). This model is shown here (Eq. 3.8)(3.8)whereCWI the cost of work injuriesn the coefficient of multiplier associated with each variable X1 to X7.X1 the type of origin Manufacturing M61 1 Mills and Semi-medium0 otherwiseX2 the type of business M81 1 Metal Foundries and Mills 0 otherwiseX3 the type of business M91 1 Agricultural Equipment 0 otherwiseX4 the type of business M92 1 if it is railroad car Shops, Manufacturing0 otherwiseX5 workers age.X6 gender 1 if it is male 0 otherwiseX7 the level of work injury. the error term.The work injury levels of different body parts are presented in Table 3.13 (Lin, 2008).Table 3.13 Work injury level rangeParts of BodyLevel of work injuryUpper artillery1-6Forearm1-3Wrist1-4Neck1-6Tru nk1-6Leg1-7The statistics software SPSS is used (Lin, 2008) to determine the coefficient of every variable in equation 3.8. In the first footfall, all data regarding each variable were redefined. In the second step, work injury cost (dependent variable) was adjusted by power transformation. Hence, the work injury cost model is expressed by the following equations (Lin, 2008).(3.9)(3.10)(3.11)(3.12)(3.13)(3.14)After the second step, Equation 3.9 to 3.14 were again adjusted to calculate the work injury cost. The manufacturing type of business is considered, therefore X1=X2=X3=0 and X4=1. It has been noticed that the age and gender coefficient were small and can be neglected. Furthermore, the equation states that work injury levels were the major part in work injury cost (Xu, 2015). The revised work injury cost model equations are as follows(3.15)(3.16)(3.17)(3.18)(3.19)(3.20)From the above discussion, it was noticed that to calculate the work injury cost the first step is to measure the work injury level of a given posture. Moreover, in order to measure the work injury level (WIL), DELMIAV5 production software (Lin,2008) is used. In the first step, Human Builder tool is used for posture visualization. In the second step, posture simulation is done by using Posture editor program tool. In the last step, to measure work injury level for the particular posture RULA (Hedge, 2001) is applied.3.5 SUMMARYIn this chapter, multi objective optimization model was tailored to achieve desired objectives. First objective was to minimize the total production cost over the planning horizon with consideration of work injury cost factor. snatch objective was to minimize the work injury levels over the planning horizon. In Section 3.2 multi objective optimization was made along with decision variables and constraints on them. Assumptions and notations were taken from Chakrabortty Hasin. (2013), Wang et al. (2005) and Xu. (2015). In the next Section 3.3 the case study was presen ted to validate the model. In Section 3.4 work injury cost calculation method was presented with its all variables and work injury level range. Thus, both objectives 1 2 mentioned in chapter 1 have been achieved by proposed model. More detail regarding the results will be discussed in next chapter.