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Evaluating Production OperationsUnit VIII Production operations require an orderly sequence of operations and the employment of producer goods of various typesProducer goods in the form of production and construction equipment, coupled with organized activity, are the means used to alter physical factors to create consumer goods and services of value.8.1 Characteristics of production operations:Any production operation, whether it represents a single machine or a complete manufacturing facility, can be modeled in terms of its cost and revenue generating characteristics.Use of these models to assist in making operating decisions is critical to the ultimate profitability of the firm.The simplicity of the mathematical form of these models belies their importance to those confronted with the complex operating decisions ordinarily encountered.For any production activity there is an input output relationship, whether it be a gasoline engine, a grocery check-out function, an electric generating plant or a manufacturing facility.The out put is usually expressed in terms of the number of units produced by the system, such as horsepower; number of persons serviced; kilowatts; or the number of items produced.The inputs are generally measured as the costs of resources consumed in the productive activity.8.1.1 Capacity considerations :For every production activity there is a potential maximum level of output that defines the capacity of that activityThis measure of maximum output reflects the upper limit production achievable, assuming that there is no limit on the availability of inputsA gasoline engines capacity is measured by the maximum horsepower achievable, while a manufacturing facility is measured by the maximum number of items per unit time that could be produced if the optimum combination of inputs were possibleThe level of output of a production activity is often stated as a percentage of the maximum capacity of the activity.Thus an engine producing an output of 100 hp when it has a capacity of 200 hp is said to be operating at 50% capacity.The capacity factor is defined as the ratio of the average output to the maximum capacity over some time interval.Figure 8.1 shows the essential elements that specify a capacity factor ,where Capacity factor=(Average actual output)/(maximum capacity)FIGURE 8.1 Output as related to capacityMaximum capacityOut putAverage actual output8.1.2 Physical efficiency : The physical efficiency of a productive entity measures the output as a percentage of the inputs .Plotting the physical efficiency of an engine might lead to a curve as shown in Figure 8.2The shape of the curve indicates that as the level of output increases beyond 80% of capacity more units of fuel ( input ) are required to realize a unit increase of horsepower (output)This effect is a basic concept referred to as the law of diminishing returns.Although this law does not apply to every productive activity, it does describe the behavior of a wide range of these activities.Efficiency (output/input)Output100%90% 80%70%60%50%40% 30%20%10%025%50%75%100%Figure 8.2 Physical efficiency of production activity8.1.3 Cost characteristics:For most production activities there are two types of costs.These are fixed and variable costsAs the level of activity within production unit changes fixed costs will remain relatively constantVariable costs are costs that fluctuate according to changes in the number of units produced.The ability to identify which costs are variable and which ones are fixed is crucial to the analytical frame work to be presented.For any production activity the relationship between the inputs and the outputs must be understoodFigure 8.3 illustrates the fixed and variable costs that represent the input and the output to be realizedFigure 8.3 fixed and variable costs as inputsTotal costVariable costsFixed costsInput (costs)Maximum capacityOutputThe total cost of providing a particular level of output is seen to be the sum of the fixed costs and the variable costs incurred for that output level.The total cost ( fixed cost and variable cost ) over the entire range of possible outputs forms the basis for the analysis presented.Incremental cost represents the change in total cost that occurs for a change in the level of outputAverage unit cost is the total input cost divided by the level of output.8.2 Break-even analysis:There are two aspects to a production activityOne consists of assembling facilities, labor and material for production.The other consists of marketing the goods or services that have been produced.The success of the activity depends upon a positive net difference between receipts for goods and services sold and the input necessary to produce and market them.Breakeven analysis can be used to evaluate the situation.8.2.1 Linear breakeven analysis:If receipts and costs are assumed to be linear functions of the quantity of product made and sold, an analysis of their relationship to profit is greatly simplifiedThe patterns of income and costs will appear as in Figure 8.4 Fixed production costs are represented by the line HLThe sum of variable production costs and marketing costs is represented by the line HKIncome from sales is represented by the line OJ.This linear representation is an approximation of actual operation conditions and is only a model.MJKLHONQN*PN in units per yearIncome or cost in dollarsFigure 8.4 General graphic representation for income , cost ,units of output and profitFor an illustration of mathematical method, let N=number of units or products made and sold per yearR= the amount received per unit of product in dollars; R is equal to the slope of OJ;I= RN, the annual income from sales in dollars I=RN is the equation of line OJF= fixed cost in dollars per year, represented by OH and HLV=variable cost per unit of product ; V is equal to the slope of HK TC= the sum of fixed and variable cost of N units of product, F+VN;TC=F+VN is the equation of line HKP= annual profit in dollars per year ;P=I-TC ; negative values of P represent loss;N*=break-even point ; at this point P=0Q=capacity of plant in units per year.The breakeven point occurs where total income is equal to total costIn Figure 8.4,the break even point occurs where lines OJ and HK intersect.At this point I=TC and RN=F+VNSolving for N* ,the break even point where N*=F/(R-V)..(8.2) yields the level of output designated N* in Figure 8.4The quantity (R-V) is commonly referred as the marginal contribution.In this model the price, R , is constant, as is the variable cost , V.Since each item produced incurs the variable cost amount V, and each sale generates a revenue amount, R, the difference represents a residual amount per unit that can be used to cover the fixed costs.Once the fixed costs are covered, the marginal contribution represents the amount of profit resulting from the sale of each additional unit of productionIf F/(V-R) is substituted for N in I=RN or TC=F+VN, the ordinate at the break even point may be foundThis value, in terms of dollars of income or cost will beI= R{F/(R-V)} and TC=F+ V.F/(R-V)Since P is the annual profit, it is often desirable to have a relationship that expresses P as a function of the number of units made and sold.This relationship may be derived as follows: P=I-TC =RN-(F+VN) = (R-V) N-FThe profit-volume function in Figure 8.5 is a means for displaying this relationship.Note that the slope of the profit line is equal to the marginal contribution, R-V $Annual profitFAnnual loss0

PN* NQ Units per yearSlope (R-V)FIGURE 8.5 Profit-volume graphFor example , to find the annual profit when R= $11, F=$4000, V=$5 and N= 800, calculate P=(R-V) N-F =($11-$5)800-$4000 =$6(800)-$4000=$800All important elements contributing to the profitability of a production activity are represented on the profit-volume function.This simple graphical framework assists in the understanding of the relationship among price, fixed costs, variable costs and profit.For example, a decrease in fixed cost by the amount (F-A) with no other changes, as shown in Figure 8.6 defines a profit line AB parallel to the original line FC.This reduction in fixed costs shifts the break even point from N* to N1.A change in either the price per unit, R, or the variable cost per unit or both affects only the marginal contribution amount, (R-V).Any change in this amount with no change in the fixed cost could shift the line FC to the line FD as illustrated in Figure 8.6A change in the amount (R-V) alters the slope of FC forcing the original profit line to rotate up or down while pivoting around the point F.In Figure 8.6, reducing the marginal contribution (R-V ) decreases the slope of the profit line with the accompanying shift of the break-even point from N* to N2BCD O AFN1N*N2Annual profitAnnual lossFigure 8.6 Changing parameters for the profit-volume graphThe profit volume function allows for the analysis of simultaneous changes in price, fixed cost and variable cost, so that the resulting change in profit and shift in the break-even point can be easily assessed. Within this frame work the advantage to a firm for investing resources in marketing programs or cost reduction programs can be evaluated.Also, the firm can study profit trade offs connected with the change in the number of items sold due to the raising and lowering of prices.Even though the analysis presented was based on the assumption that fixed cost, variable cost, and income are linear functions of the quantity made and sold, it is useful in evaluating the effect on profit of proposals for new operations not yet implemented and for which no data exist.$12 108642010002000300040005000Output in units per year ,NCost and income per unitAverage cost per unit=(F/N)+VVariable cost per unit ,V Fixed cost per unitF/N= Figure 8.8 Fixed cost ,variable cost ,and income per unitIncome per unit , RIn Figure 8.8 production cost per unit, variable cost per unit and income per unit are plotted as N varies from 0 to 6,000.It is noted that the fixed cost per unit may be infinite.Thus, in determining average unit costs, fixed cost has little meaning unless the number of units to which it applies is known.Income for most enterprises is directly proportional to the number of units sold. However, the income per unit may easily be exceeded by the sum of the fixed and the variable cost per unit for low production volumes.This is shown by comparing the average-cost per unit curve and the income per unit curve in Fig 8.88.3 Operation and production decisions :Because there are unforeseen variations in the demand for a product and /or its cost of production, the firm must continually readjust its mode of operation.Although these variations may affect only one production unit or the entire production facility, the analysis methods are generally applicable to any production activity, whether it be large or small.There are usually a variety of actions that can be undertaken by management that will adjust prices and costs, the goal being the proper management of the elements to improve profits.This goal may include the minimization of losses.This section presents a frame work for representing the interactions among price, fixed cost, variable cost, incremental cost and profit.By using this frame work, analyses can be made that will provide great assistance in deciding the appropriate management actions.8.3.1 Break-even for profit maximization: For every productivity activity, it is vital that the cost characteristics of that activity be well understood. It is observed from the example in Figure 8.8 that the average unit cost decreases with each additional unit of output.The rate of decrease of this amount is important in determining cost competitiveness at various levels of output. In this example, a fixed price of $ 3.00 per unit breaks even at an average unit cost of $3.00 or at N*=$3000/($3.00-$2.10)=3,333 units while a fixed price of $ 4.00 per unit breaks even at N*=$3000/( $4.00-$2.10)=1,579 units.The average unit-cost curve defines the break-even point for a fixed price equal to any point on the curve.Also of interest is the incremental cost over the range of output.For this example the incremental cost is a constant $2.10 per unit.As long as the incremental income received for each additional unit exceeds the incremental cost, additional production should be undertaken.8.3.2 Making pricing decisions : There are numerous situations where it may be appropriate to modify prices for various levels of output.These situations can arise when there are either temporary reductions or increases in the demand for output.When there are reductions in demands, the goals should be to increase demand so that the production operation can remain relatively stable.In other cases, a production activity operating at 100% capacity may be required to increase output through overtime and extra shiftsThe prices set should help compensate for these variations in demand and yet assure a profitable outcome.8.3.3 Stopping production : In certain situations the demand for, or the price of the output of, a production activity may be so low that substantial losses are being recorded.At some point, the temporary closing down of production becomes a viable alternative to be considered.8.3.4 Operating Multiple production units:Some production operations consist of multiple production facilities or units that all produce the same product.An electric generating system may have a combination of nuclear, fossil and hydro generating units. Other examples include manufacturing facilities that contain multiple and distinct production lines and pumping operations having engines or motors that can be used in combination.Since all the output of each unit is identical when viewed by the system, a common problem is how to assign most economically the various levels of output to each of the separate units within the system.The task is to determine for each level of the total system output the various operating levels of the individual unitsIncremental costs are the basis on which these type of operating decisions are made.Assuming that the price for the output is sufficient so that it does not affect the assignment of the output levels, the objective is to minimize the total cost of production.In many situations incremental costs can facilitate the determination of how this minimization is to be achieved.There are three possible situations that require different decision rules for the assignment of multiple-unit output. When incremental costs are constant for different output levels, the principle is to:Assign the unit with the lowest incremental cost and load it to full capacity before bringing the next productive unit into operation. Continue this procedure until the system output is achieved.When incremental costs are always increasing for different output levels, the principle is to:Assign the initial unit of output to the production unit having the smallest incremental cost. Then assign each additional unit of output to the production unit having the lowest incremental cost, given its present level, of output.

When incremental costs are not constant or not constantly increasing with output level, the principle is to compute the total incremental cost for a given level of output from the total production system.8.4 ECONOMIC OPERATION OF EQUIPMENT :Most production facilities may be operated at rates below, equal to, or above their normal capacity.Facilities are usually inefficient and costly to operate when utilized at a rate of output below their normal capacity.For example, if a production department is allotted more space than is needed for its efficient operation, a number of losses may be expected.The fixed costs such as maintenance, taxes, insurance, and interest will be higher than necessaryHeat, light and janitor service will be wasted in the unused space, and the cost of supervision and material handling may be higher than necessary.The load that is considered proper to impose on equipment is usually indicated by its manufacturerSuch indicated normal loads are often not the optimum for economy.Operation of equipment above its normal capacity will result in increased production at the expense of increased power consumption and shortened life.In many cases, the equipment in question should be over-loaded so that over-all economy may be realized.8.4.2 The economy of loading above normal capacity:The life of a piece of equipment is usually inversely proportional to the load imposed upon it.However, the output is directly proportional to the load imposed.When this is the case, there exists an optimum load that will determine the level of operation for maximum economy.

In some cases, particularly with short lived assets, the economic life may be determined by experimentIf the effects of varied loadings upon the life, maintenance and operating costs of a piece of equipment are known or can be estimated with reasonable accuracy, it is easy to determine the load that will result in the greatest economy. 8.4.3 Economic load distribution between machines:In many cases two or more machines are available for the same kind of production.Thus two or more boilers may be available to produce steam or two or more turbines may be available to produce powerIf the total load is less than the combined capacity of the available machines, the economy of operation will usually depend upon the portion of the total load that is carried by each machine.8.5 MODELS FOR PROCUREMENT OPERATIONS:When the decision has been made to procure a certain item , it becomes necessary to determine the procurement quantity that will result in a minimum cost.The demand for the item may be met by procuring a years supply at the beginning of each year or by procuring a days supply at the beginning of each day.Neither of these extremes may be the most economical in terms of the sum of costs associated with procurement and holding the item in inventory.Models for inventory operations are used to determine the most economical procurement quantity.8.5.1 The Economic Purchase Quantity:If the demand for an item is met by purchasing once per year, the cost incident to purchasing will occur once, but the large quantity received will result in a relatively high inventory holding cost for the year.

Conversely, if orders are placed several times per year, the cost incident to purchasing will be incurred several times per year, but since small quantities will be received, the cost of holding the them in inventory will be relatively small.If the decision is to be based on economy of the total operation, the purchase quantity that will result in a minimum annual cost must be determined.Let TC= total yearly cost of providing the item;D=yearly demand for the item;N=number of purchases per year;t=time between purchasesQ=purchase quantityCi=item cost per unit ( purchase price);Cp=purchase cost per purchase order;Ch= holding cost per unit per year made up of such items as interest, insurance, taxes, storage space, and handling.If it is assumed that the demand for the item is constant throughout the year, the purchase lead time is zero and no shortages are allowed, the resulting inventory flow may be represented graphically as in Figure 8.16The total yearly cost will be the sum of the item cost for the year, the purchase cost for the year and the holding cost for the year. That isTC= IC +PC+HC .(8.6)The item cost for the year will be the item cost per unit times the yearly demand in units , or QQQtttFIGURE 8.16 Graphical representation of inventory flow for purchasingIC=Ci(D)..(8.7)The purchase cost for the year will be the cost per purchase times the number of purchases per year or PC=Cp(N). But since N is the yearly demand divided by the purchase quantity, PC=Cp(D)/Q ..(8.8)Since the interval t begins with Q units in stock and ends with none, the average inventory during the cycle will be Q/2.Therefore, the holding cost for the year will be the holding cost per unit times the average number of units in stock for the year, or HC=Ch(Q)/2.(8.9)The total yearly cost of providing the required item is the sum of the item cost, purchase cost, and holding cost, orTC=Ci(D)+Cp(D)/Q+Ch(Q)/2 .(8.10)The purchase quantity resulting in a minimum yearly cost may be found by differentiating with respect to Q, setting the result equal to zero, and solving for Q as follows:dTC/dQ=-Cp(D)/Q2+Ch/2=0Q2=2Cp(D)/ChQ= 2Cp(D)/Ch .. ( 8.11)

8.5.2 The Economic Production Quantity:When the decision has been made to produce a certain item, it becomes necessary to determine the optimum production quantityThis may be done in a manner similar to determining economic purchase quantitiesThe difference in analysis occurs because a purchased lot is received at one time whereas a production lot accumulates as it is made. LetTC=total yearly cost of providing the itemD= yearly demand for the itemN=number of production runs per year;t=time between production runs

Q=production quantity;Ci=item cost per unit (production cost)Cs=set up cost per production run;Ch=holding cost per unit per year made up of such items as interest, insurance, taxes, storage space, and handling;R= production rate.If it is assumed that the demand for the item is constant, the production rate is constant during the production period, the production lead time is zero, and no shortages are allowed, then the resulting inventory flow may be represented graphically as in Figure 8.18 QQQtttFIGURE 8.18 Graphical representation of inventory flow for productionThe total yearly cost will be the sum of the item cost for the year, the set up cost of the year, and the holding cost for the year. That is,TC=IC+SC+HC ..(8.12)The item cost for the year will be the item cost per unit times the yearly demand in units, or IC=Ci(D).The set up cost for the year will be the cost per set-up times the number of set-ups per year, or SC=Cs(N)But since N is the yearly demand divided by the production quantity, SC=Cs(D)/Q.(8.14)When the items are added to inventory at the rate of R units per year and are taken from inventory at a rate D units per year ( where R is greater than D), the net rate of accumulation is (R-D) units per year. The time required to produce D units at the rate R units per year is D/R yearsIf D units are made in a single lot, the maximum accumulation in inventory will be (R-D)(D/R).Since no units will be in storage at the end of the year, the average number of units in inventory will be {(R-D).D/R +0}/2=(R-D)(D/2R).If N lots are produced per year, the average number of units in storage will be (R-D).(D/2RN)But since N=D/Q, the average number of units in storage may be expressed as (R-D).(Q/2R)The holding cost for the year will be the holding cost per unit times the average number of units in storage for the year, orHC=Ch (R-D).(Q/2R). (8.15)The total yearly cost of producing the required item is the sum of the item cost, set-up cost and holding cost, or TC=Ci (D) +Cs(D)/Q +Ch(R-D).(Q/2R)(8.16)The production quantity resulting in a minimum yearly cost may be found by differentiating with respect to Q, setting the result equal to zero , and solving for Q as follows:dTC = -Cs (D) + Ch( R-D)= 0 dQ Q2 2R

Q2=Cs (D).2R = Cs(D).2 Ch(R-D) Ch{1-(D/R)} Q = 2Cs (D) (8.17) Ch {1-(D/R)}8.5.3 The make or buy decision :The question of whether to manufacture or purchase a needed item may be resolved by the application of minimum cost analysis for multiple alternatives.The alternative of producing may be compared with the alternative of purchasing if the minimum -cost procurement quantity for each is computed and used to find the respective total cost values.Choice of the total cost value that is a minimum identifies the best of the two alternatives.KEY POINTS:The progress in the physical sciences and engineering has pushed back limiting factors of a physical nature and made possible the development of complex operational systems to meet human needs.Decision makers are becoming increasingly aware that experience , intuition and judgment must be augmented with analyses based on mathematical models if systems are to be operated more economically.Engineering proposals can not meet human needs if they remain in the form of plans and specifications ; they must be converted to products through construction or production operations.By examining the relationships among the cost of inputs , the price obtained for the output , and the levels of output , it is possible to determine optimum operating conditions.Any productivity ,whether represents a single machine or a complete manufacturing facility , can be modeled in terms of its cost and revenue generating characteristics.For every production activity there is a potential maximum level of output that defines its capacity ; its efficiency then derives from the output as a percentage of the input.Fixed and variable costs are the primary considerations in production operations ,and the ability to distinguish these cost types is essential in economy studies of production systems

Breakeven analysis is a very useful tool in the analysis of production and service operations and may be applied under the assumption of linearity or non- linearity.Economy of scale in production operations occurs when fixed cost is amortized over a larger and larger number of output units.The firm must continually readjust its mode of operation owing to unforeseen variations in demand , changes in the price obtained , and increases or decreases in the cost of input factors.Operations economy often depends upon the loading or rate of use of equipment, making loading and load distribution an important consideration.

Models for procurement operations make it possible to determine the economic purchase quantity and the economic production quantity as well as the minimumcost procurement source for a needed item. THE END