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A Nonlinear Programming Optimization Model to Maximize Net Revenue in Cheese Manufacture

Northeast Dairy Foods Research Center Department of Food Science, Cornell University, Ithaca, NY 14853

Corresponding author:
David M. Barbano; e-mail:
dmb37{at}cornell.edu.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

 
A nonlinear programming optimization model was developed to maximize net revenue in cheese manufacture and is described in this paper. The model identifies the optimal mix of milk resources together with the types of cheeses and co-products that maximize net revenue. It works in Excel while it takes the data specified by the user from a user-friendly interface created in Access. The user can specify any number of resources, cheese types, and co-products. To demonstrate the capabilities of the model, we determined the impact of variation in milk price and composition in the period 1998 to 2000 on the optimal mix of resources and optimal type of co-product for Cheddar and low-moisture, part-skim Mozzarella. It was also desired to determine the impact of variation in protein content of nonfat dry milk (NDM) on net revenue, and examine the effect of reconstitution of NDM with water versus milk on net revenue. The optimal mix of resources and the net revenue markedly varied as milk resource prices and composition varied. The net revenue for Mozzarella was much higher than for Cheddar when the price of cream was high. Cheese plants that did not optimize the use of resources in response to variations in prices and composition missed a significant profit opportunity. Whey powder was more profitable than 34% whey protein concentrate and lactose in most months. The use of high-protein NDM led to an appreciable increase in net revenue. When the value of the nonfat portion of raw milk was high, reconstitution of NDM with water rather than milk markedly raised net revenue.

Key Words: cheese • optimization • mathematical programming

Abbreviation key: ACONST = the proportion of lactose, NPN and minerals of separated whey that should be retained during ultrafiltration, CAPH = Calcium phosphate factor, CR = Casein retention factor, CWT = hundred weight, FDB = Fat on a dry basis, FNDM = Percent fat in NDM, FR = Fat retention factor, FREM = Percent fat in removed cream, FSW = Percent fat in separated whey, FWC = Percent fat in whey cream, FWHOLE = Percent fat in raw milk, LPREC = Percent of lactose in the UF permeate that is recovered in lactose powder, M = Percent moisture in the cheese, MAXYD = Maximum cheese yield, PRNDM = Percent total protein in NDM, PROTWPC = Desired percent total protein in WPC, PRREM = Percent total protein in removed cream, PRWHOLE = Percent total protein in raw milk, SALT = Percent salt in the cheese, SEF = solids exclusion factor, SR = solids retention factor, SWREC = Percent recovery of separated whey, TPREC = Percent recovery of WPC, WFR = Percent recovery of fat from the whey, WPC = Whey protein concentrate, WWPC = Percent moisture in WPC

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

 
Standardization of milk composition in cheesemaking was traditionally used to counteract seasonal milk composition variation, ensuring the appropriate fat on a dry basis in cheese. It has also been used to ensure consistent cheese quality. However, standardization in cheese manufacture can also be used to increase profits. The need for optimization in blending of milk resources is greater today because more types of resources are available to cheese producers. The use of NDM in cheese manufacture has markedly increased in the last two decades (Pratt and Erba, 1999; American Dairy Products Institute, 1999). The increased need for optimal standardization is also due to the steady growth of cheese production in the US. Cheese production has approximately doubled in the last 20 yr (International Dairy Foods Association, 1999). Furthermore, the number of cheese plants has decreased while the processing capacity of plants has increased by a higher proportion (International Dairy Foods Association, 1999). The increased plant size in combination with the greater accessibility of all plants to milk resources from more distant areas has increased competition among cheese plants, thus necessitating profit maximization.

Mathematical programming has been widely used to optimize blending of ingredients in food applications (Bender et al., 1982). However, little work has been done in optimization of cheese manufacture. Kerrigan and Norback (1986) developed a linear programming model to maximize net revenue in Cheddar cheese production. Their model used the available milk resources as decision variables and had only four constraints. It also assumed that the only co-products were whey cream and separated whey, with no value assigned to the latter. The constraints ensured that the sum of the resources equaled the batch size, the desired casein-to-fat ratio was attained, and no more cream was removed from the raw milk than was available. The user could also set limits on the amounts of resources used. The authors emphasized that the cheese yield formula used (Van Slyke and Price, 1952) would be valid only for a limited range of fat and CN content in the standardized milk.

Another linear programming model (Craig et al., 1989) was developed to maximize net revenue in Process cheese manufacture. Apart from the constraints that were similar to those above, the model also included some constraints to ensure the acceptability of the product, such as ratios of ingredients and proportions of the different-age cheeses in the blend. Again the only co-product with value was whey cream.

The first model to include co-products produced from separated whey was developed by Samakidis (1994), who was also the first to use the nonlinear Barbano yield formula (Barbano, 1996) to estimate cheese yield. The model maintained its linear structure by the application of an iterative approach to the nonlinear cheese yield formula. Because the model structure was linear, assumptions were made for the composition of whey to estimate the yields of co-products.

The model presented in this paper is a nonlinear programming model, which is an extension of Samakidis’ model. This is the first time the use of a nonlinear optimization model has been reported for determination of the optimal mix of milk resources to be used for cheesemaking. Its objective is to identify the optimal mix of milk resources and types of cheese products and co-products that will maximize net revenue in cheese manufacture. The selection of a nonlinear model has the advantage that the yield equations are valid under any composition range of the standardized milk, as opposed to the models mentioned above. In addition, the model allows the production of a variety of co-products from the processing of separated whey, and can identify the optimal type of co-product. In order to describe the conversion of standardized milk to cheese and co-products, the model uses yield formulas. There are two different cheese yield formulas that can be selected by the user: the Van Slyke and the Barbano yield formulas. The Barbano formula takes into account the TS content of separated whey to estimate the amount of water-soluble milk solids retained in the aqueous phase of the cheese and thus it can be used for any composition of the standardized milk. This formula produces a more accurate estimate of the nonfat, noncasein milk solids available for whey product manufacture over a wide range of standardized milk composition.

The overall objective of the model is to maximize net revenue for cheese plants and help cheese producers identify opportunities to improve profitability. In this study, there were four problems addressed, each using monthly data from the 3-yr period from January 1998 to December 2000. The first problem was to determine the impact of variation in milk price and composition on the optimal mix of milk resources and net revenue. The second problem was to determine the impact of variation in milk price and composition on the optimal type of co-product produced. The third problem was to determine the impact of extreme differences in protein content of NDM on net revenue. Finally, we wished to compare, with respect to net revenue, the reconstitution of NDM with milk to the reconstitution of NDM with water. For each problem, the nonlinear model was used to identify the optimal solution in each month.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

 
Software/Programming Environment
Excel.
The optimization model is formulated in Excel (Microsoft, Seattle, WA). A code was written in Visual Basic (Microsoft), which converts the data specified by the user to a mathematical programming model. More specifically, the code specifies the objective function, builds the constraints, and calls the solver in Excel. In addition, after the solver has identified the optimal solution, the code converts this solution to organized reports.

Solver.
The nonlinear optimization model is solved using the Premium Solver (Frontline Systems, Inc., Incline Village, NV), which is an enhanced version of the default solver of Excel (Frontline Systems, Inc., 1999). It uses the generalized reduced gradient method to solve nonlinear problems, such as the one presented in our model. The options, which are specified by the user, were set as follows: Iterations = 1000, Precision = 0.00001, Convergence = 0.001, Estimates = Tangent, Derivatives = Forward, and Search = Newton.

User interface.
Although the optimization model is formulated in Excel, the user interacts with the model through the user-friendly Access (Microsoft) interface. This has been done mainly to prevent the user from accidently interfering with the functional parts of the model. When the user has specified all inputs in Access, there is a code in Access which transfers the input data from this interface to Excel. The software system can be obtained as part of a cheese plant management training workshop presented at Cornell University.

Options Provided by the Model
Resources.
The user can specify any number of resources, and the price and composition of each resource. For simplicity, in the example provided below, only three resources are used. The user can specify any batch size of standardized milk. When NDM is included as a resource, the user is allowed to choose if NDM will be reconstituted in milk or in water. If NDM is reconstituted in water, the user must specify the percentage TS after reconstitution. The user must also choose if cream removal is permitted from each whole milk included in the resources.

Cheeses.
The user is given the flexibility to specify any number of cheese types. In a multiple-cheese problem, the model will identify the optimal distribution of resources among the cheese types, the optimum amount of each cheese type, and the optimal co-product(s) for each cheese type. For each cheese type, the user must specify the target moisture (M), salt (SALT), fat on a dry basis (FDB) (minimum and maximum), the retention coefficients in the selected yield formula, and the maximum cheese yield allowed (MAXYD). The maximum cheese yield allowed should reflect the maximum capacity (in pounds per hundred pounds (cwt) of standardized milk) of the cheesemaking equipment.

Co-products.
Whey is always assumed to be separated into whey cream and separated whey. The fat content of both must be specified by the user. Different types of co-products can be produced from separated whey. The user allows the model to select from the following combinations of co-products: liquid separated whey, whey powder, WPC, WPC and lactose powder, whey powder and WPC, whey powder and WPC and lactose powder. To maximize net revenue, the model will select the type and amount of co-products made. One or more of the co-products selected by the user may be produced in the optimal solution. In all cases, some waste is generated, which has a cost ($/lb solids) of disposal that is specified by the user. For each co-product, the user must specify the price and the moisture content. For WPC, the protein content must also be specified. The user must also specify four recovery factors, i.e., the percent recovery of fat in whey (WFR), the percent recovery of separated whey (SWREC), the percent recovery of WPC (TPREC), and the percent of lactose in the UF permeate that is recovered in lactose powder (LPREC). Hurst et al. (1990) suggested an 80% recovery for WPC, while Guu and Zall (1992) reported a 62% recovery of lactose in lactose powder. These are the values used in this study.

Price adjustments.
The user is allowed to explicitly specify adjustments in the prices of milk resources, cheeses, and co-products to account for manufacturing or other costs not reflected in the prices. For instance, the user can account for the cost of reconstitution of NDM and the manufacturing costs for producing cheeses and co-products. These costs are very process and factory specific. We did not include any costs of this type in our demonstration of the model in this paper.

Optimization vs. evaluation.
As indicated above, the model identifies the optimal mix of resources, products, and co-products which maximizes net revenue. However, the model may be used not only for optimization, but for evaluation of current management practices as well. A cheese producer may tightly restrict the amounts and types of resources available so that they reflect the current formulation. In this way, the yield of cheese and co-products provided by the model can be compared with the actual yields observed in the plant to evaluate the efficiency of current manufacturing practices.

Description of the Model
Objective function.
The model identifies the combination of milk resources, cheeses, and co-products that maximize the net revenue. In mathematic terms, the objective of the model is to identify the set of variables (vector X) that maximizes the value of the objective function (Z), which is the net revenue. The objective function has the form Z = c x X, where c is the vector of the objective function coefficients (it includes the prices of products and the costs of resources).

Decision variables.
The decision variables (Table 1) of the model can be divided into 3 groups: The standardized milk composition variables (x2 and x3, x14 to x16, x23 to x25), the variables that represent the amount of each resource used and each product produced (x4 to x13), and the variables that are used to determine the composition of whey (x17 to x22, x26 to x30), which are important in determining the amounts of cheese and co-products produced. All the variables of the model represent weights, except for x20, x22, x28 and x29, which are ratios of weights. The batch size (x1) is not a variable in single-cheese problems, where it is user specified. However, in multiple-cheese problems the batch size of standardized milk of each cheese constitutes a variable.

Constraints.
The variables of the model are subject to a set of constraints that are described by A x X b, where A is the matrix of technical coefficients and b is the vector of the right hand sides of the constraints. The constraints, described individually below, can be divided into 4 groups: Those that represent the mass balance for each component (fat, protein, etc.) between the resources used and the standardized milk (constraints 1 to 11), those that describe the conversion of standardized milk to cheese and co-products (12 to 19), those that are related to the composition of the whey (20 to 31), and the constraints that represent limitations on the amounts of milk resources used and products made (32 to 42). The constraints related to the composition of whey are important in determining the cheese yield (when the Barbano yield formula is used) and the amounts of co-products. The constraints of the model follow:

1. Batch size of standardized milk. The batch size is equal to the sum of the amounts of resources used minus the amount of removed cream:

2. Maximum amount of cream removed. There is one constraint of this type for each different raw milk included in the resources. This constraint ensures that no more cream is removed from the raw milk than is possible. The maximum amount of cream removed depends on the fat contents of raw milk (FWHOLE) and removed cream (FREM):

3. Fat balance. The amount of fat in the standardized milk is equal to the sum of the amounts of fat contributed by each resource: For example, if the fat content of NDM is FNDM, then

4. Casein balance. The amount of CN in the standardized milk is equal to the sum of the amounts of CN contributed by each resource. This constraint has the same form as the one for fat balance.

5–10. Balance for total protein, true protein, NPN, anhydrous lactose, minerals and TS. These six constraints ensure a mass balance is achieved for each of the above components between the resources used and their amount in the standardized milk. For instance, given the total protein contents of raw milk (PRWHOLE), removed cream (PRREM), and NDM (PRNDM), the constraint for total protein can be written as:

11. Maximum cheese yield per hundred pounds of standardized milk. This constraint imposes an upper limit on cheese yield to reflect limitations of the maximum cheese handling capacity of the equipment.

12. Cheese yield. This constraint determines the cheese yield. The user must select one of the two theoretical cheese yield formulas that are available. The first yield formula is the Van Slyke formula (Van Slyke and Price, 1952), in which cheese yield depends on fat and CN content of the standardized milk, the retention coefficients for fat (FR), casein (CR), and other milk solids plus salt (SR), and the moisture content of the cheese:

This equation can be rearranged as:

The Van Slyke formula was developed for full fat Cheddar cheese (FR=0.93, CR=0.96 and SR=1.09) and hence it should not be used for other cheeses without appropriate changes in the retention factors. It may also be used for Mozzarella with different fat retention and solids retention factors (i.e., FR=0.85, CR=0.96 and SR=1.13).

The second yield formula is the Barbano formula (Barbano, 1996), which takes into account the composition of the whey produced in order to determine the amount of whey solids retained in the aqueous phase of the cheese.

where the solids exclusion factor (SEF) is the proportion of the moisture of the cheese that is available to dissolve the whey solids. The proportion is less than 100% because a portion of the cheese moisture is bound to protein, hence it is not available for the dissolution of lactose, minerals, and NPN. The calcium phosphate factor (CAPH) is used to account for the calcium phosphate that is bound to CN micelles, thus it is retained in the cheese. Because cheese yield depends on the composition of whey, this constraint is nonlinear when the Barbano yield formula is used.

13. Whey cream yield. The amount of fat that is not recovered in the cheese, but is recovered from the whey is either fat in the whey cream [which is calculated from the percent fat in whey cream (FWC) multiplied by the weight of whey cream] or fat in the separated whey. Therefore, the yield of whey cream depends on the percent recovery of fat from whey and the fat content of separated whey (FSW).

14. Yield of separated whey. This constraint determines the yield of separated whey, which is equal to the batch size less the non-salt portion of cheese, the amount of whey cream, and the fat of standardized milk that is not retained in the cheese nor in whey products and thus ends up in the waste (fatnotrec).

15. Maximum FDB in the cheese. This constraint ensures that the FDB of the cheese does not exceed the upper limit set by the user. When the Van Slyke cheese yield formula is used:

Therefore, FDB FDB_max leads to

When the Barbano yield formula is used:

Therefore, FDB FDB_max leads to

16. Minimum FDB in the cheese. This constraint is similar to the previous one, ensuring that the FDB of the cheese does not fall below the lower limit set by the user.

When the Van Slyke yield formula is used, the following formula applies:

When the Barbano yield formula is used, this formula applies:

17. WPC yield. This constraint determines the yield of WPC based on the true protein content of the separated whey, the percent recovery of WPC, and the moisture content of WPC (WWPC). The yield formula uses the proportion of lactose, NPN, and minerals of separated whey that should be retained during ultrafiltration (ACONST) in order to achieve the desired total protein content of WPC (PROTWPC). If WPC and lactose powder are the only co-products, the constraint holds as an equality, while in all other cases the constraint holds as an inequality.

If this equation is solved for ACONST, the following relationship is derived:

18. Lactose yield. The amount of lactose powder produced depends on the amount of lactose in separated whey that is not retained in the WPC and on the percentage of lactose in the UF permeate that is recovered in lactose powder. If WPC and lactose are the only co-products, the constraint holds as an equality.

19. Amount of waste solids. The waste solids consist of the amount of NPN and minerals in separated whey that are not retained in the WPC, the fat not recovered, the lactose of UF permeate that is not retained in lactose powder, the solids of separated whey that are not recovered, and the WPC that is not recovered, as is shown in this formula:

20. Separated whey protein. The amount of protein in the recovered separated whey influences the WPC yield. The total protein in the standardized milk is distributed among the cheese, whey cream, and separated whey.

21. Protein in the nonfat portion of whey. The protein in the nonfat portion of whey is equal to the protein in standardized milk less the protein retained in the cheese.

22. Proportion of protein in the nonfat portion of whey. The proportion of protein in the nonfat portion of whey (x20), which is defined in this constraint, is used to determine the amount of protein in whey cream. This is used to determine the protein of separated whey (constraint 20), which is essential for calculation of the WPC yield.

23. Separated whey TS. The solids of standardized milk are distributed among the cheese, whey cream, and separated whey. Also a portion of the standardized milk solids is the fat that is not recovered in the cheese or whey and hence ends up in the waste (fatnotrec).

24. Total solids in the nonfat portion of whey. The solids in the nonfat portion of whey are equal to the standardized milk solids less the cheese solids less the fat of whey.

25. Proportion of TS in the nonfat portion of whey. The proportion of TS in the nonfat portion of whey (x22), which is defined in this constraint, is used to determine the amount of TS in whey cream.

26. Nonfat portion of whey. The nonfat portion of whey is the portion of whey cream and separated whey that does not contain fat.

27. Ratio of lactose to water. This constraint identifies the ratio of lactose to water in the standardized milk. This ratio serves to determine the amount of lactose that is retained in the cheese and in the whey cream (in constraint 29).

28. Ratio of NPN to water. As in the previous constraint, the ratio of NPN to water in the standardized milk is identified, which serves to determine the amount of NPN retained in the cheese and in the whey cream (in constraint 30).

29. Separated whey lactose. The amount of lactose in standardized milk is distributed among the cheese, whey cream and separated whey.

30. Separated whey NPN. The amount of NPN in the standardized milk is distributed among the cheese, whey cream and separated whey.

31. Balance for solids of separated whey. This constraint defines the solids of recovered separated whey that remain after the production of whey powder (x30) and are fully converted to WPC, lactose powder and waste solids. The TS of separated whey are written as the standardized milk solids less the solids of cheese and whey cream less the fatnotrec.

This equation can be rearranged as:

When the only co-products are WPC and lactose powder, the constraint can take a simpler form:

32–37. Lower and upper limits for the resources. These constraints allow the user to set limits in pounds per batch on the use of resources.

38–39. Lower and upper limit for cheese. These constraints allow the user to define the weight range of the cheese type produced. They represent minimum and maximum limitations on the marketing potential of cheese, opposed to constraint 11, which reflects limitations due to equipment handling capacity of the cheese curd.

40–42. Upper limits for co-products. These constraints allow the user to set limits on the amount of whey powder, WPC and lactose produced.

Linear vs. nonlinear model.
In a linear programming model, all the elements of the matrix A and the vectors b and c in the constraints and objective functions are known constants, while in this nonlinear programming model some of the elements depend on the value of at least one other variable. The advantage of a linear programming model is the existence of a standard solving procedure, the Simplex method (Hillier and Lieberman, 1995), which is guaranteed to identify the global optimum of any linear problem if one exists. On the other hand, there is no method that is guaranteed to identify the global optimum of all nonlinear models. Most algorithms for nonlinear models require a starting vector X and the local optimums they find markedly depend on this starting vector. To overcome this difficulty, the model presented here first approximates the solution of each problem with a linear model and then uses the solution of the linear model as a starting point for the nonlinear model. In this way, the initial vector X is very likely to be sufficiently close to the global optimum of the nonlinear model so as to ensure that the solution provided to the user is a global optimum.

The optimization model we developed is nonlinear for the following reasons:

1. When the Barbano cheese yield formula is used, the cheese yield depends on the TS content of the separated whey. Because the latter is unknown and is equal to the ratio of two unknowns in the model (i.e., solids of separated whey:amount of separated whey), this constraint is nonlinear.
   
2. The amounts of WPC and lactose powder produced depend on the composition of separated whey. The higher the true protein and lactose content of separated whey, the higher the WPC and lactose powder yield. Since the concentration of each component in the separated whey is the ratio of two unknowns (amount of the component : amount of separated whey), the constraints that determine the concentration of each component are nonlinear. Moreover, when the user selects to produce both WPC and whey powder, the amount of WPC produced is proportional to the ratio of the solids left after the whey powder production (variable x30) to the TS of separated whey (x12 + x30). This is another source of "nonlinearity".
   
3. When the user decides to produce more than one cheese, the total standardized milk batch size is known but the batch size for each cheese is unknown, since this is the solution that the user is looking for. Therefore, in this case, the constraints that include the batch size as a coefficient (constraints 3 to 4, 12 to 13, and 20 to 21) are nonlinear.
   

The advantage of the nonlinear model structure is the accurate determination of the amount of cheese and co-products and of their composition. A linear model would not be able to monitor the distribution of each component (fat, protein, etc.) among the various products. Moreover, the nonlinear model offers the user the option of producing more than one co-product.

If one wished to use only a linear model, one would have to make assumptions about the composition of the separated whey (e.g., 7% TS). However, these assumptions might significantly deviate from the real values in cases of high fortification levels (high TS in the standardized milk). Moreover, the composition of whey depends on the type of the cheese, because the latter affects the type of resources used. For instance, Cheddar has higher fat content than Mozzarella and hence the standardized milk (and consequently the whey) for Cheddar would have more fat than that for Mozzarella.

Reports produced.
The solution of the optimization model is the optimal vector X, which is the set of the optimal values of the decision variables. However, this set of values does not help the user evaluate the results and make decisions. Therefore, a code has been developed that converts the solution identified by the solver to well organized reports. There are five different reports produced:

1. Flow report: The amount of each product and the weight of each milk solids component included in each product are shown in the flow report for the optimal solution. It can be used by a cheesemaker to compare these amounts to the actual yields.
   
2. Composition report: The concentration of each component in the standardized milk and in each product is shown in the composition report for the optimal solution. The optimal FDB for the cheese is also shown. This report gives important information about the composition of all products and waste.
   
3. Cost-Revenue report: The contribution of each milk resource to cost, the contribution of each product to revenue, and the net revenue are shown in the cost-revenue report for the optimal solution.
   
4. Mass balance report: The amount of each milk component included in each resource used and in each product produced is shown in the mass balance report. The user can check the mass balance of each component among the resources and the products to ensure that the model has identified a solution that maintains mass balance. The report also shows how each component is distributed among the different products. For instance, it will indicate the weight of milk fat that ends up in the waste, which a cheesemaker can compare to his actual losses.
   
5. Sensitivity analysis: Apart from the optimal solution, it is also important to know under what conditions the solution remains optimal. The sensitivity analysis will report the magnitude of changes in prices that does not alter the optimal milk resources and type of co-product. For a milk resource that is not in the optimal mix, the allowable decrease in its price will indicate the magnitude of the decrease in its price required to render its use profitable. Similarly, for a resource included in the optimal mix, the allowable increase will indicate the magnitude of increase in its price that may exclude it from the optimal mix. Sensitivity analysis, as applied here, is a characteristic of linear programming. Because our model is nonlinear, the optimal solution is first linearized before sensitivity analysis is applied. This means that the sensitivity analysis in our nonlinear model has limited accuracy. The accuracy is higher when the price ranges reported in the sensitivity analysis are narrow.
   

Prices of Milk Resources
The prices of milk resources (Figure 1, raw milk; Figure 2, fresh cream, condensed skim, and NDM) used (USDA, 1998) refer to the Northeast area of the United States for the 3-yr period 1998–2000. The raw milk price was calculated for each month from the Federal Milk Market class III price, the component prices (fat, true protein and other solids), and the average milk composition for that month. Since premiums are usually paid by cheese manufacturers (Jacobson and Wasserman, 1992), an average $0.50/cwt premium was assumed for each month (New York State Dept. of Agriculture and Markets, 1999). The premium was included in the prices of raw milk and of the nonfat portion of raw milk. The prices of NDM (low/medium heat) and condensed skim were taken from the same source (USDA, 1998).

View larger version (17K): [in this window] [in a new window]   Figure 1. Class III price (dollars/100 lb) of raw milk () and of the nonfat portion of raw milk (•) for Northeast United States calculated at the average milk composition per month for the period January 1998 to December 2000 with a $0.50 per hundred weight average competitive premium added for all months.

View larger version (19K): [in this window] [in a new window]   Figure 2. Class II price (dollars/lb fat) of cream (), class III price (dollars/lb solids) of condensed skim (), and price (dollars/lb) of low/medium heat NDM (•) for Northeast United States for the period January 1998 to December 2000.

The prices of Cheddar and co-products were midwest prices (Figure 3), as there was no Northeast price quoted. The price of low-moisture, part-skim Mozzarella was assumed to equal that of Cheddar. Because the price of lactose powder was very close to that of whey powder, it was not included in Figure 3. The handling cost of waste was assumed to be $0.05 per pound.

View larger version (19K): [in this window] [in a new window]   Figure 3. Prices (dollars/lb) of block, full fat Cheddar (), whey powder (•), and WPC 34% () for Midwest United States for the period January 1998 to December 2000.

Design of the Analysis
All the problems addressed by the model use data from the 3-yr period 1998–2000. For all problems, four milk resources were available: raw milk, fresh cream, condensed skim milk (34% TS) and NDM (96.5% TS). Cream removal from raw milk was allowed. For all problems, the monthly data for prices and milk composition were used. The parameters used for all problems are shown in Table 3.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

 
It should be noted that the retrospective analyses presented in problems 1 through 4 assume that the outcome of the optimization decisions in one month would not influence the prices of ingredients in subsequent months. If optimum behavior was practiced on a wide enough scale in the cheese industry, it could alter trends in ingredient prices from their historic paths. The amount of alteration in ingredient prices would be hard to predict and is beyond the scope of this discussion. The impact would depend on the price elasticities of supply and demand for each ingredient.

Problem 1. Variation of Milk Prices and Composition
Fortification strategies used.
The optimal resource mix (i.e., maximum net revenue) varied as prices and milk composition varied during the period 1998 – 2000. This was observed for both Cheddar and Mozzarella (Tables 4 and 5). For Cheddar, in 13 out of the 36 mo, the optimal strategy was to remove cream from the raw milk. On the other hand, in 12 mo double standardization with additional fresh cream and condensed skim produced the highest net revenue, while in the remaining 11 mo double standardization with NDM plus cream was the optimal strategy. For Mozzarella the optimal strategy was double standardization (i.e., less cream removal and solids nonfat addition) in all months except for November 2000, when the optimum was to only remove cream. In 21 mo condensed skim was used, while in 15 mo NDM was used.

Key factors causing milk resource strategy to change.
By observing how the optimal mix of milk resources changed (Tables 4 and 5) with prices (Figures 1, 2, and 3) and composition variation (Table 2), one could conclude that cream removal is profitable for Cheddar only when the price of the cream portion of raw milk is high and the price of Cheddar is sufficiently low. Both of these circumstances favor selling the fat as cream rather than incorporating it in the cheese. This conclusion is less clear in the case of part-skim Mozzarella, where cream removal is applied in every month in order to achieve the desired FDB without exceeding the maximum cheese yield allowed.

The price of the nonfat portion of raw milk seems to be critical also for the choice between NDM (96.5% TS) and condensed skim (34% TS). When the price of the nonfat portion of raw milk is high, it favors the use of condensed skim because the latter can supplement a greater amount of nonfat portion of raw milk than NDM reconstituted in milk. This conclusion is indicated by the results, though it does not always hold true because the choice between these two resources is also driven by their relative prices.

In all 11 mo in which NDM was in the optimal mix for Cheddar, it was also in the optimal mix for Mozzarella. The same was observed for condensed skim. This implies that the choice between NDM and condensed skim may be unaffected by the type of cheese when the price behavior of cheeses is similar.

The significant variation in the optimal mix over time can certainly be attributed to the fluctuations in prices. Because the composition pattern is assumed to be the same every year, the source of variation can be ascertained by comparing the optimal mixes for the same month of different years. It is readily observed that the optimal mix for Cheddar was never the same in any two months in all 3 yr. For some of the other months, the optimal mix greatly changed in different years. For instance, in December 1998, when the price of cheese was high ($1.90 per lb), the optimal strategy was to double-standardize with raw milk, NDM, and fresh cream. Despite the high price of the nonfat portion of raw milk ($12.53 per cwt), NDM was preferred to condensed skim due to its lower price ($1.16 per lb vs. $1.34 per lb solids). The sensitivity analysis report showed that the price of condensed skim would have to fall from $1.34 to $1.28 per lb solids to render it more attractive than NDM. In December 1999, the optimal strategy was again double standardization, but condensed skim replaced NDM due to its lower price ($0.93 per lb solids vs. $1.02 per lb). Sensitivity analysis showed that an increase in the price of condensed skim from $0.93 to $0.99 per lb solids would be necessary to shift the choice back to NDM. In December 2000, when the price of cheese ($1.12 per lb) was markedly low and the price of cream was high ($1.99 per lb fat), the optimal decision was to remove cream from the raw milk. The prices of condensed skim and NDM would have to fall dramatically (to $0.66 per lb solids and $0.69 per lb, respectively) to alter the optimal mix. The pronounced changes in strategy in the same months of different years were observed for Mozzarella as well, though the optimal strategy was the same in July and August in all 3 yr for this cheese.

Key factors in deciding to raise or lower cheese FDB.
The optimal strategy in most months was to reach the upper limit of FDB, which was 53% for Cheddar and 42% for Mozzarella. Only in 6 mo was attaining the low end of FDB for Cheddar (50%) optimal. In 4 (July 1998 to October 1998) of those 6 mo, the choice of the low end of FDB could be attributed to the excessively high price of cream, which did not favor keeping more than a minimum of fat in cheese. In those 4 mo, the low end of FDB was optimal for Mozzarella as well. In the other two months, where the low FDB was optimal for Cheddar, the choice of the minimum FDB could be attributed to the very low price of cheese and the high value of fat in cream. Both of these prices favored selling fat in the cream rather than incorporating more fat in the cheese. However, in contrast to Cheddar, in those two months the high end of FDB was optimal for Mozzarella. This difference can be attributed to the fact that each pound of fat corresponds to a higher yield for Mozzarella, compared with Cheddar, and hence a higher price of cream is required to render the low FDB optimal in the case of Mozzarella.

Maximizing net revenues.
Apart from the pronounced variation in the optimal mix, there was also marked fluctuation in the net revenue throughout the whole time period. More specifically, the net revenue varied from $0.158 to $5.449 per 100 lb standardized milk for Cheddar and from $1.958 to $7.661 for Mozzarella. This marked variation can be attributed to the pronounced variation in the prices of raw milk, cream, and cheese. There was a consistent trend for higher net revenues in the second half of all years for both Cheddar and Mozzarella (Tables 4 and 5). This seasonal trend can be attributed to the seasonal pattern of raw milk and cheese prices, which tend to increase in the second half of each year, with the upward trend of cheese price dominating.

The net revenues of the two cheeses were consistently changing in the same direction throughout the 3-yr period. However, the difference in net revenue between Mozzarella and Cheddar markedly varied during the 3-yr period, following closely the changes in cream price (Figure 4). The higher the cream price the greater the observed difference in net revenue between Mozzarella and Cheddar. A higher cream price enhanced net revenues for Mozzarella cheese makers much more than for Cheddar makers. The difference in net revenue between Mozzarella and Cheddar cheese manufacture (y, in dollars per 100 lb standardized milk) exhibited a linear relationship with the fresh cream price (x, in dollars per lb fat):

View larger version (16K): [in this window] [in a new window]   Figure 4. The difference (•) in net revenue (dollars/100 lb standardized milk) between Mozzarella and Cheddar (i.e., Mozzarella minus Cheddar) and the class II price (dollars/lb fat) of cream () for each month in the period January 1998 to December 2000.

We also compared the net revenue when a cheese plant was using the optimal mix at all times versus when it was following the same strategy over time (Table 6). When NDM was excluded from the resources, the net revenue fell by $0.05 for Cheddar and $0.08 for Mozzarella per 100 lb batch. These decreases in net revenue translate to $1000 and $1600 per day for Cheddar and Mozzarella, respectively, for a cheese factory that processes 2,000,000 lb standardized milk per day. The decrease in revenue was much more pronounced when condensed skim was excluded from the resources as well. In this case, which allowed only single standardization, the decrease in revenue was $0.23 for Cheddar and $0.58 for Mozzarella per 100 lb batch, which translates to $4600 and $11,600 per day, respectively. Therefore, a cheese plant that was not optimizing the selection and use of milk resources in response to variation in prices and milk composition lost significant potential profit.

The amount of waste solids is higher in the case of WPC plus lactose than whey powder because of the water-soluble solids (NPN and minerals) that are not retained during ultrafiltration, material lost during filtration (WPC not recovered), and lactose that is not recovered during crystallization, all based on the assumed recoveries. Therefore, the assumed cost of disposal of waste solids ($0.05/lb) might be expected to influence the choice of co-product, with a higher cost favoring whey powder. However, sensitivity analysis showed that the decrease in the cost of handling waste required to alter the choice of co-product was relatively high, $0.09 / lb on average in the 3-yr period. Consequently, the assumed cost of waste solids did not significantly influence the choice of co-product.

Many cheese plants wish to know the price difference between WPC and whey powder necessary to render 34% WPC more profitable than whey powder. To answer this question, the average yields of the three co-products and the average amount of waste solids throughout the 3-yr period were used (Table 7). Given these average yields for the case of Cheddar, WPC is more profitable than whey powder if the prices of co-products (i.e., P_WPC = price of 34% WPC, P_LAC = price of lactose powder, C_WAS = cost of waste processing, and P_WP = price of whey powder) satisfy the relationship:

If the price of lactose is set at its average level, $0.186, and the cost of disposal of waste solids is $0.05, then the above relationship can be written as:

This inequality has to hold true in order for 34% WPC to be more profitable than whey powder. If the inequality does not hold true, then whey powder is more profitable. Indeed, this condition is verified in 32 of the 36 mo, while in the remaining 4 mo it is refuted by only a minimal amount. A very similar condition is derived in the case of Mozzarella as well (P_WPC > 3.883 x P_WP – 0.202) and is verified in 35 of the 36 mo.

Nevertheless, it should be emphasized that the above condition does not take into account the additional processing cost of WPC production as described by Hurst et al. (1990), i.e., the operation cost of ultrafiltration. If one wants to be more precise, one should add this cost divided by the average yield of WPC (1.724) to the right-hand side of the above inequality. The condition suggested here indicates that cheese plants should not determine the optimal co-product based on the price difference between WPC and whey powder but rather they should assess whether the above relationship holds true. The price difference would be a good indicator only if the two co-products had similar yields.

It is also interesting to compare the profits of cheese plants that differ in the type of co-products they produce. In the 3-yr period studied, Cheddar cheese plants producing whey powder achieved higher net revenue than their counterparts that produced WPC and lactose powder by $0.09 per 100 lb batch (Table 7), which can translate to $1800 per day for plants processing 2,000,000 lb standardized milk per day. The difference in net revenue was approximately the same for Mozzarella plants, i.e., $0.08 per 100 lb milk.

Problem 3. Impact of NDM Composition on Net Revenue
Extremes of NDM protein content (32.5% to 38.5% true protein) were tested to determine their impact on the optimal mix of resources and net revenue. The high-protein NDM was included in the optimal mix in approximately twice as many months for both cheeses (i.e., 24 vs. 11 for Cheddar and 29 vs. 15 for Mozzarella), compared with the average NDM (Table 8). It also led to an average increase in net revenue for the 3-yr period of $0.07 for Cheddar and $0.10 for Mozzarella per 100 lb standardized milk. As expected, the impact on net revenue was greater when the optimal mix included a higher amount of NDM. The above results indicate the existence of profit opportunity, stemming from the fact that the price of NDM is not based on its protein content. Because the high-protein NDM proved to be profitable in most months, a cheese plant that finds an abundant supply of high-protein NDM might choose to stop using condensed skim, which is perishable and has a fairly volatile price. In this way, the cheese plant can insulate itself against price volatility.

Because NDM reconstituted with water was only used during 5 of the mo, while condensed skim was being used during 12 mo, one could conclude that condensed skim was more profitable than NDM with water for the given time period. Indeed, the total net revenue for the 3 yr when condensed skim was available as a resource was $0.06 for Cheddar and $0.10 for Mozzarella per 100 lb standardized milk higher than the total revenue when NDM with water was available. This can translate to $1200 and $2000, respectively, per day difference in net revenue for a cheese plant that processes 2,000,000 lb standardized milk per day.

However, if water was allowed as an ingredient at any level (i.e., reconstitution to less than 34% TS) in the 5 mo mentioned above, it would lead to a pronounced increase in net revenue, compared with the revenue obtained from the use of either condensed skim or NDM reconstituted with milk (Table 9). This increase was slightly higher for Mozzarella in all 5 mo. On average, the increase in net revenue from the use of water as an ingredient was $1.07 for Mozzarella and $1.02 for Cheddar per 100 lb of standardized milk, which means $21,400 and $20,400 per day, respectively, for the two cheeses. The use of water as an ingredient obviated the use of any amount of raw milk in all 5 mo for both cheeses, while it still led to the high end of FDB and cheese yield. The amount of water used in the optimal mix was approximately 80 lb per 100 lb of resources in all 5 mo for both cheeses.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

 
The use of the model presented in this paper may increase profits for cheese plants by helping them make management decisions that respond optimally to milk price and composition variation and recognize the existence of opportunities for improved profitability in the cheese market. The benefit from the model will be more reliable when cheese plants use input data that reflects their own resource prices and technical parameters. The results of this study demonstrate the potential of using the model in decision-making in cheese manufacture.

The optimal mix of milk resources and the net revenue varied markedly for both Cheddar and low-moisture, part-skim Mozzarella, as prices and composition of milk resources varied throughout the 3-yr period. While the net revenues for the two cheeses were consistently changing in the same direction in the whole time period, the net revenue for Mozzarella was increasing much more steeply than that for Cheddar when the cream price increased. A positive linear relationship was observed for the difference in net revenue between Mozzarella and Cheddar and the cream price. A cheese plant that did not optimize the selection of milk resources in response to milk price and composition variation lost appreciable potential profit.

Considering only resource costs and product prices without including manufacturing costs, whey powder was more profitable than the combination of 34% WPC and lactose powder in most months. The cost of handling waste did not affect the choice of co-product in most months. A relationship between the prices of 34% WPC and whey powder was derived to determine the optimal type of co-product based on their prices. This condition was verified in almost all months for both Cheddar and Mozzarella.

The high-protein NDM was included in the optimal mix of resources in twice as many months as the average NDM, leading to an increase in net revenue for both cheeses. This indicates the existence of a profit opportunity, which emerges from the fact that the price of NDM is not based on its protein content. Finally, the reconstitution of NDM with water proved to considerably increase net revenue in months in which the price of the nonfat portion of raw milk was high. Therefore, partial replacement of the skim portion of raw milk with NDM reconstituted in water should be applied in periods in which the availability of raw milk is low and its price is high.

	ACKNOWLEDGEMENTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

 
The authors thank Sharon Trerise, Craig Alexander, and Ioannis Samakidis for technical support and the Northeast Dairy Foods Research Center for partial financial support.

	FOOTNOTES

 
¹ Use of names, names of ingredients, and identification of specific models of equipment is for scientific clarity and does not constitute any endorsement of product by authors, Cornell University, and the Northeast Dairy Foods Research Center.

Received for publication February 23, 2002. Accepted for publication June 19, 2002.

	REFERENCES

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS ACKNOWLEDGEMENTS REFERENCES

 


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