Multi-objective Optimization Mathematical Model

Multi- nonsubjective Optimization Mathematical influenceCHAPTER 3 PRODUCTION COST WORK INJURY LEVEL mannequin 3.1 IntroductionThis chapter describes a multi- object lens optimization mathematical sit down with ratiocination variables and controls on them. de tell 3.2 presents the manakin formulation with aim to minimize the total roil live and locomote soil take particularly in a manufacturing industry every(prenominal) oer a think vista. Section 3.3 presents ZC1the case argonna drawn from literature to validate the proposed stumper. Section 3.4 presents the method to calculate the go bad defect exist with consideration of give speck train factor. Section 3.5 gives the summary for this chapter. poseur formulationThe traditional harvest-festivalion cooking impersonate is a mathematical optimization mold. In such a model, the physical object move is the total approach, and the decision variable refers to proceedsion quantity, roll quantity, and outs ourcing quantity. The constraint function in the traditional return home process model includes the drive in a preparedness horizon. In the work of (Xu, 2015), the traditional model includes the work lesion monetary value. The expansion of the model hence mentions the description of the objective function and constraints. The model aims to achieve the two objective are intent 1 (ob1) minify business toll (CP).Objective 2 (ob2) Minimize work injury level (WIL).Model AssumptionsA mathematical model herein is developed on the following assumptions areThe values of only parameters are certain over the next compass point t in planning horizon.Actual cranch levels, working(a) hours and warehouse capacity in each degree lavatorynot draw their respective maximum levels.The rate of proles and tasks are the same over the planning horizon.A single type of mathematical product is manufactured over the planning horizon.Trivial solutions will be ignored.Model NotationsThe fol lowing 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 term period (t=1, 2, 3, , n).CMR the uninterrupted building block material live of the product ($/ building block).CMO the extra time unit material cost of the product ($/unit).Pt the fare of products fabricated (production quantity) during the unbroken working hours in the period t (unit).Ot the look of products fabricated (extra time production quantity) during the extra time in period t (unit).CLR the fixture unit comprehend cost in period t ($/unit).CLO the extra time unit labor cost in period t ($/unit).Ht the perpetual working man-hour call for in period t (man-hour).Et the extra time working man-hour required in period t (man-hour).CI the unit livestock cost ($/unit).It the units of product to be left over as an stock-take during period t (unit).CWI total work injury cost over the pla nning horizon.C2aa3 It-1 the units of leftover products in the previous period of t.dt the product implore in period t.D the total guide 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 achieve the ob1, the integrated production planning was used in score to minimize the production cost. The total production cost consists of the material cost, labor cost, inventory cost and work injury cost. Let C contain 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 cost Material cost is the sum of regular material cost and overtime material cost that includes the raw material cost and overhead cost. crude material directly contributes to the finished product, and the overhead cost includes the utility cost such as electricity, gas and rent etc.Labor damage Labor cost is the sum of all wages paid to employees for the production of products in both regular time and overtime hours.Inventory cost Inventory cost is the holding cost of products in stock.Work injury cost the work injury cost caused by the repetitive forum 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, equality 3.1 weed 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 inven tory cost for left over products as an inventory over the period (It) and the final part denotes the accumulated 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 landing field 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 downstairsFurthermoreWhere, 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 long exposure of worker to the repetitive task. Therefore, higher the production quantity, the long-lasting the working hours and the h igher 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 ConstraintsDemand constraint(3.4)(3.4)Where, the sum of regular production quantity (Pt), overtime production quantity (Ot) and inventory levels essentially great than or twin 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 little than or equal to 8 hours per day, monthly working day s (dn) as well as number of employees (W). Overtime working man-hour (Et) should not exceed the allowable hours (E*) by law.Production rate constraint. Assume that the unit time is one hour, and the relation between the produced units and labor can be verbalised 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 theatre about the aggregate production planning of single product is selected. This case study is drawn from the literature and the causation s own experience in industry (Chakrabortty Hasin, 2013).Case study descriptionTo validate the proposed model, the real life data of Comfit involved Knit Limited (CCKL) is taken. The club manufactures knit ware product. The production planning is more specifically about the production of hooded jacket over a couple of months planning horizon. set back 3.1 3.2 give the monthly product 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- I0 five hundredEnd 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 attitude in a standing position to process the product on a machine. The worker need to place the product separate in a machine to stitch it , for this reason worker has to lam forward to focus on the product parts. The neck may influence to get a better view of stitching if required. To perform this task, the speed arms are need to be elevated to the height of the work table. To place the product part in a right appearance the body rotation is required (Fig. 3.1).3.3 Work injury cost (Cwi) deliberationWork 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 factor associated with each variable X1 to X7.X1 the type of business Manufacturing M61 1 Mills and Semi-medium0 other thanX2 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 Machine Shops, Manufacturing0 otherwiseX5 workers age.X6 sexuality 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 range split of BodyLevel of work injuryUpper Arm1-6forearm1-3Wrist1-4Neck1-6Trunk1-6Leg1-7The statistics software SPSS is used (Lin, 2008) to make up ones mind the coefficient of every variable in equation 3.8. In the first step, 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, equality 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 foll ows(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 personate. Moreover, in order to measure the work injury level (WIL), DELMIAV5 production software (Lin,2008) is used. In the first step, benignant Builder tool is used for posture visualization. In the second step, posture simulation is done by using Posture Editor 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. Second 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. Assumpt ions 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 presented to validate the model. In Section 3.4 work injury cost deliberateness 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 compass point regarding the results will be discussed in next chapter.