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121 Food manufacture 1 A food is manufactured by refining raw oils and blending them together The raw oils are of two categories Vegetable oils VEG 1 VEG 2 Nonvegetable oils OIL 1 OIL 2 OIL 3 Model Building in Mathematical Programming Fifth Edition H Paul Williams 2013 John Wiley Sons Ltd Published 2013 by John Wiley Sons Ltd VEG 1 VEG 2 OIL 1 OIL 2 OIL 3 January 110 120 130 110 115 February 130 130 110 90 115 March 110 140 130 100 95 April 120 110 120 120 125 May 100 120 150 110 105 June 90 100 140 80 135 The final product sells at 150 per ton Vegetable oils and nonvegetable oils require different production lines for refining In any month it is not possible to refine more than 200 tons of vegetable oils and more than 250 tons of nonvegetable oils There is no loss of weight in the refining process and the cost of refining may be ignored It is possible to store up to 1000 tons of each raw oil for use later The cost of storage for vegetable and nonvegetable oil is 5 per ton per month The final product cannot be stored nor can refined oils be stored There is a technological restriction of hardness on the final product In the units in which hardness is measured this must lie between 3 and 6 It is assumed that hardness blends linearly and that the hardnesses of the raw oils are VEG 1 88 VEG 2 61 OIL 1 20 OIL 2 42 OIL 3 50 What buying and manufacturing policy should the company pursue in order to maximise profit At present there are 500 tons of each type of raw oil in storage It is required that these stocks will also exist at the end of June This problem and the subsequent problem is based on a larger model built for the margarine producer Van den Bergs and Jurgens and discussed in Williams and Redwood 1974 131 Food manufacture 1 Blending problems are frequently solved using linear programming Linear programming has been used to find minimum cost blends of fertilizer metal alloys clays and many food products to name only a few Applications are described in Fisher and Schruben 1953 and Williams and Redwood 1974 for example The problem presented here has two aspects Firstly it is a series of simple blending problems Secondly there is a purchasing and storing problem To understand how this problem may be formulated it is convenient to consider first the blending problem for only one month This is the singleperiod problem that has already been presented as the second example in Section 12 1311 The singleperiod problem If no storage of raw oils were allowed the problem of what to buy and how to blend in January could be formulated as follows Maximize PROFIT 110x1 120x2 130x3 110x4 115x5 150y subject to VVEG x1 x2 200 NVEG x3 x4 x5 250 UHRD 88x1 61x2 2x3 42x4 5x5 6y 0 LHRD 88x1 61x2 2x3 42x4 5x5 3y 0 CONT x1 x2 x3 x4 x5 y 0 The variables x1 x2 x3 x4 x5 represent the quantities of the raw oils that should be bought respectively that is VEG 1 VEG 2 OIL 1 OIL 2 and OIL 3 y represents the quantity of PROD that should be made The objective is to maximize profit which represents the income derived from selling PROD minus the cost of the raw oils The first two constraints represent the limited production capacities for refining vegetable and nonvegetable oils The next two constraints force the hardness of PROD to lie between its upper limit of 6 and its lower limit of 3 It is important to model these restrictions correctly A frequent mistake is to model them as 88x1 61x2 2x3 42x4 5x5 6 and 88x1 61x1 2x3 42x4 5x5 3 Such constraints are clearly dimensionally wrong The expressions on the left have the dimension of hardness multiplied by quantity whereas the figures on the right have the dimensions of hardness Instead of the variables xi in the above two inequalities expressions xiy are needed to represent proportions of the ingredients rather than the absolute quantities xi When such replacements are made the resultant inequalities can easily be reexpressed in a linear form as the constraints UHRD and LHRD Finally it is necessary to make sure that the weight of the final product PROD is equal to the weight of the ingredients This is done by the last constraint CONT which imposes this continuity of weight The singleperiod problems for the other months would be similar to that for January apart from the objective coefficients representing the raw oil costs 1312 The multiperiod problem The decisions of how much to buy each month with a view to storing for use later can be incorporated into a linear programming model To do this a multiperiod model is built It is necessary each month to distinguish the quantities of each raw oil bought used and stored These quantities must be represented by different variables We suppose the quantities of VEG 1 bought used and stored in each successive month are represented by variables with the following names BVEG 11 BVEG 12 and so on UVEG 11 UVEG 12 and so on SVEG 11 SVEG 12 and so on It is necessary to link these variables together by the relation quantity stored in month t 1 quantity bought in month t quantity used in month t quantity stored in month t Initially month 0 and finally month 6 the quantities in store are constants 500 The relation above involving VEG 1 gives rise to the following constraints BVEG 11 UVEG 11 SVEG 11 500 SVEG 11 BVEG 12 UVEG 12 SVEG 12 0 SVEG 12 BVEG 13 UVEG 13 SVEG 13 0 SVEG 13 BVEG 14 UVEG 14 SVEG 14 0 SVEG 14 BVEG 15 UVEG 15 SVEG 15 0 SVEG 15 BVEG 16 UVEG 16 500 Similar constraints must be specified for the other four raw oils It may be more convenient to introduce variables SVEG 10 and so on and SVEG 16 and so on into the model and fix them at the value 500 In the objective function the buying variables will be given the appropriate raw oil costs in each month The storage variables will be given the cost of 5 or profit of 5 Separate variables PROD 1 PROD 2 and so on must be defined to represent the quantity of PROD to be made in each month These variables will each have a profit of 150 The resulting model will have the following dimensions as well as the single objective function 6 x 5 30 buying variables 6 x 5 30 using variables 5 x 5 25 storing variables 6 product variables Total 91 variables 6 x 5 30 blending constraints as in the singleperiod model 6 x 5 30 storage linking constraints Total 60 constraints 6 x 5 30 blending constraints as in the singleperiod model 6 x 5 30 storage linking constraints Total 60 constraints It is also important to realize the use to which a model such as this might be put for mediumterm planning By solving the model in January buying and blending plans could be determined for January together with provisional plans for the succeeding months In February the model would probably be resolved with revised figures to give firm plans for February together with provisional plans for succeeding months up to and including July By this means the best use is made of the information for succeeding months to derive an operating policy for the current month 141 Food manufacture 1 The optimal policy is given in Table 141 The profit income from sales cost of raw oils derived from this policy is 107 843 This figure includes storage costs for the last month There are alternative optimal solutions 121 Food manufacture 1 A food is manufactured by refining raw oils and blending them together The raw oils are of two categories Vegetable oils VEG 1 VEG 2 Nonvegetable oils OIL 1 OIL 2 OIL 3 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Each oil may be purchased for immediate delivery January or bought on the futures market for delivery in a subsequent month Prices at present and in the futures market are given below in ton VEG 1 VEG 2 OIL 1 OIL 2 OIL 3 January 110 120 130 110 115 February 130 130 110 90 115 March 110 140 130 100 95 April 120 110 120 120 125 May 100 120 150 110 105 June 90 100 140 80 135 The final product sells at 150 per ton Vegetable oils and nonvegetable oils require different production lines for refining In any month it is not possible to refine more than 200 tons of vegetable oils and more than 250 tons of nonvegetable oils There is no loss of weight in the refining process and the cost of refining may be ignored It is possible to store up to 1000 tons of each raw oil for use later The cost of storage for vegetable and nonvegetable oil is 5 per ton per month The final product cannot be stored nor can refined oils be stored There is a technological restriction of hardness on the final product In the units in which hardness is measured this must lie between 3 and 6 It is assumed that hardness blends linearly and that the hardnesses of the raw oils are VEG 1 88 VEG 2 61 OIL 1 20 OIL 2 42 OIL 3 50 What buying and manufacturing policy should the company pursue in order to maximise profit At present there are 500 tons of each type of raw oil in storage It is required that these stocks will also exist at the end of June This problem and the subsequent problem is based on a larger model built for the margarine producer Van den Bergs and Jurgens and discussed in Williams and Redwood 1974 131 Food manufacture 1 Blending problems are frequently solved using linear programming Linear programming has been used to find minimum cost blends of fertilizer metal alloys clays and many food products to name only a few Applications are described in Fisher and Schruben 1953 and Williams and Redwood 1974 for example The problem presented here has two aspects Firstly it is a series of simple blending problems Secondly there is a purchasing and storing problem To understand how this problem may be formulated it is convenient to consider first the blending problem for only one month This is the singleperiod problem that has already been presented as the second example in Section 12 1311 The singleperiod problem If no storage of raw oils were allowed the problem of what to buy and how to blend in January could be formulated as follows Maximize PROFIT 110x1 120x2 130x3 110x4 115x5 150y subject to VVEG x1 x2 200 NVEG x3 x4 x5 250 UHRD 88x1 61x2 2x3 42x4 5x5 6y 0 LHRD 88x1 61x2 2x3 42x4 5x5 3y 0 CONT x1 x2 x3 x4 x5 y 0 The variables x1 x2 x3 x4 x5 represent the quantities of the raw oils that should be bought respectively that is VEG 1 VEG 2 OIL 1 OIL 2 and OIL 3 y represents the quantity of PROD that should be made The objective is to maximize profit which represents the income derived from selling PROD minus the cost of the raw oils The first two constraints represent the limited production capacities for refining vegetable and nonvegetable oils The next two constraints force the hardness of PROD to lie between its upper limit of 6 and its lower limit of 3 It is important to model these restrictions correctly A frequent mistake is to model them as 88x1 61x2 2x3 42x4 5x5 6 and 88x1 61x1 2x3 42x4 5x5 3 Such constraints are clearly dimensionally wrong The expressions on the left have the dimension of hardness multiplied by quantity whereas the figures on the right have the dimensions of hardness Instead of the variables xi in the above two inequalities expressions xiy are needed to represent proportions of the ingredients rather than the absolute quantities xi When such replacements are made the resultant inequalities can easily be reexpressed in a linear form as the constraints UHRD and LHRD Finally it is necessary to make sure that the weight of the final product PROD is equal to the weight of the ingredients This is done by the last constraint CONT which imposes this continuity of weight The singleperiod problems for the other months would be similar to that for January apart from the objective coefficients representing the raw oil costs 1312 The multiperiod problem The decisions of how much to buy each month with a view to storing for use later can be incorporated into a linear programming model To do this a multiperiod model is built It is necessary each month to distinguish the quantities of each raw oil bought used and stored These quantities must be represented by different variables We suppose the quantities of VEG 1 bought used and stored in each successive month are represented by variables with the following names BVEG 11 BVEG 12 and so on UVEG 11 UVEG 12 and so on SVEG 11 SVEG 12 and so on It is necessary to link these variables together by the relation quantity stored in month t 1 quantity bought in month t quantity used in month t quantity stored in month t 298 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Initially month 0 and finally month 6 the quantities in store are constants 500 The relation above involving VEG 1 gives rise to the following constraints BVEG11 UVEG11 SVEG11 500 SVEG11 BVEG12 UVEG12 SVEG12 0 SVEG12 BVEG13 UVEG13 SVEG13 0 SVEG13 BVEG14 UVEG14 SVEG14 0 SVEG14 BVEG15 UVEG15 SVEG15 0 SVEG15 BVEG16 UVEG16 500 Similar constraints must be specified for the other four raw oils It may be more convenient to introduce variables SVEG 10 and so on and SVEG 16 and so on into the model and fix them at the value 500 In the objective function the buying variables will be given the appropriate raw oil costs in each month The storage variables will be given the cost of 5 or profit of 5 Separate variables PROD 1 PROD 2 and so on must be defined to represent the quantity of PROD to be made in each month These variables will each have a profit of 150 The resulting model will have the following dimensions as well as the single objective function 6 5 30 buying variables 6 5 30 using variables 5 5 25 storing variables 6 product variables Total 91 variables 6 5 30 blending constraints as in the singleperiod model 6 5 30 storage linking constraints Total 60 constraints 6 5 30 blending constraints as in the singleperiod model 6 5 30 storage linking constraints Total 60 constraints It is also important to realize the use to which a model such as this might be put for mediumterm planning By solving the model in January buying and blending plans could be determined for January together with provisional plans for the succeeding months In February the model would probably be resolved with revised figures to give firm plans for February together with provisional plans for succeeding months up to and including July By this means the best use is made of the information for succeeding months to derive an operating policy for the current month 141 Food manufacture 1 The optimal policy is given in Table 141 The profit income from sales cost of raw oils derived from this policy is 107 843 This figure includes storage costs for the last month There are alternative optimal solutions Table 141 Buy Use Store January Nothing 222 tons VEG 1 4778 tons VEG 1 1778 tons VEG 2 3222 tons VEG 2 250 tons OIL 3 500 tons OIL 1 500 tons OIL 2 250 tons OIL 3 February 250 tons OIL 2 200 tons VEG 2 4778 tons VEG 1 250 tons OIL 3 1222 tons VEG 2 500 tons OIL 1 750 tons OIL 2 March Nothing 1593 tons VEG 1 3185 tons VEG 1 407 tons VEG 2 815 tons VEG 2 250 tons OIL 2 500 tons OIL 1 500 tons OIL 2 April Nothing 1593 tons VEG 1 1593 tons VEG 1 407 tons VEG 2 407 tons VEG 2 250 tons OIL 2 500 tons OIL 1 250 tons OIL 2 May 500 tons OIL 3 1593 tons VEG 1 500 tons OIL 1 407 tons VEG 2 500 tons OIL 3 250 tons OIL 2 June 6593 tons VEG 1 1593 tons VEG 1 500 tons 5407 tons VEG 2 407 tons VEG 2 each oil 750 tons OIL 2 250 tons OIL 2 stipulated
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121 Food manufacture 1 A food is manufactured by refining raw oils and blending them together The raw oils are of two categories Vegetable oils VEG 1 VEG 2 Nonvegetable oils OIL 1 OIL 2 OIL 3 Model Building in Mathematical Programming Fifth Edition H Paul Williams 2013 John Wiley Sons Ltd Published 2013 by John Wiley Sons Ltd VEG 1 VEG 2 OIL 1 OIL 2 OIL 3 January 110 120 130 110 115 February 130 130 110 90 115 March 110 140 130 100 95 April 120 110 120 120 125 May 100 120 150 110 105 June 90 100 140 80 135 The final product sells at 150 per ton Vegetable oils and nonvegetable oils require different production lines for refining In any month it is not possible to refine more than 200 tons of vegetable oils and more than 250 tons of nonvegetable oils There is no loss of weight in the refining process and the cost of refining may be ignored It is possible to store up to 1000 tons of each raw oil for use later The cost of storage for vegetable and nonvegetable oil is 5 per ton per month The final product cannot be stored nor can refined oils be stored There is a technological restriction of hardness on the final product In the units in which hardness is measured this must lie between 3 and 6 It is assumed that hardness blends linearly and that the hardnesses of the raw oils are VEG 1 88 VEG 2 61 OIL 1 20 OIL 2 42 OIL 3 50 What buying and manufacturing policy should the company pursue in order to maximise profit At present there are 500 tons of each type of raw oil in storage It is required that these stocks will also exist at the end of June This problem and the subsequent problem is based on a larger model built for the margarine producer Van den Bergs and Jurgens and discussed in Williams and Redwood 1974 131 Food manufacture 1 Blending problems are frequently solved using linear programming Linear programming has been used to find minimum cost blends of fertilizer metal alloys clays and many food products to name only a few Applications are described in Fisher and Schruben 1953 and Williams and Redwood 1974 for example The problem presented here has two aspects Firstly it is a series of simple blending problems Secondly there is a purchasing and storing problem To understand how this problem may be formulated it is convenient to consider first the blending problem for only one month This is the singleperiod problem that has already been presented as the second example in Section 12 1311 The singleperiod problem If no storage of raw oils were allowed the problem of what to buy and how to blend in January could be formulated as follows Maximize PROFIT 110x1 120x2 130x3 110x4 115x5 150y subject to VVEG x1 x2 200 NVEG x3 x4 x5 250 UHRD 88x1 61x2 2x3 42x4 5x5 6y 0 LHRD 88x1 61x2 2x3 42x4 5x5 3y 0 CONT x1 x2 x3 x4 x5 y 0 The variables x1 x2 x3 x4 x5 represent the quantities of the raw oils that should be bought respectively that is VEG 1 VEG 2 OIL 1 OIL 2 and OIL 3 y represents the quantity of PROD that should be made The objective is to maximize profit which represents the income derived from selling PROD minus the cost of the raw oils The first two constraints represent the limited production capacities for refining vegetable and nonvegetable oils The next two constraints force the hardness of PROD to lie between its upper limit of 6 and its lower limit of 3 It is important to model these restrictions correctly A frequent mistake is to model them as 88x1 61x2 2x3 42x4 5x5 6 and 88x1 61x1 2x3 42x4 5x5 3 Such constraints are clearly dimensionally wrong The expressions on the left have the dimension of hardness multiplied by quantity whereas the figures on the right have the dimensions of hardness Instead of the variables xi in the above two inequalities expressions xiy are needed to represent proportions of the ingredients rather than the absolute quantities xi When such replacements are made the resultant inequalities can easily be reexpressed in a linear form as the constraints UHRD and LHRD Finally it is necessary to make sure that the weight of the final product PROD is equal to the weight of the ingredients This is done by the last constraint CONT which imposes this continuity of weight The singleperiod problems for the other months would be similar to that for January apart from the objective coefficients representing the raw oil costs 1312 The multiperiod problem The decisions of how much to buy each month with a view to storing for use later can be incorporated into a linear programming model To do this a multiperiod model is built It is necessary each month to distinguish the quantities of each raw oil bought used and stored These quantities must be represented by different variables We suppose the quantities of VEG 1 bought used and stored in each successive month are represented by variables with the following names BVEG 11 BVEG 12 and so on UVEG 11 UVEG 12 and so on SVEG 11 SVEG 12 and so on It is necessary to link these variables together by the relation quantity stored in month t 1 quantity bought in month t quantity used in month t quantity stored in month t Initially month 0 and finally month 6 the quantities in store are constants 500 The relation above involving VEG 1 gives rise to the following constraints BVEG 11 UVEG 11 SVEG 11 500 SVEG 11 BVEG 12 UVEG 12 SVEG 12 0 SVEG 12 BVEG 13 UVEG 13 SVEG 13 0 SVEG 13 BVEG 14 UVEG 14 SVEG 14 0 SVEG 14 BVEG 15 UVEG 15 SVEG 15 0 SVEG 15 BVEG 16 UVEG 16 500 Similar constraints must be specified for the other four raw oils It may be more convenient to introduce variables SVEG 10 and so on and SVEG 16 and so on into the model and fix them at the value 500 In the objective function the buying variables will be given the appropriate raw oil costs in each month The storage variables will be given the cost of 5 or profit of 5 Separate variables PROD 1 PROD 2 and so on must be defined to represent the quantity of PROD to be made in each month These variables will each have a profit of 150 The resulting model will have the following dimensions as well as the single objective function 6 x 5 30 buying variables 6 x 5 30 using variables 5 x 5 25 storing variables 6 product variables Total 91 variables 6 x 5 30 blending constraints as in the singleperiod model 6 x 5 30 storage linking constraints Total 60 constraints 6 x 5 30 blending constraints as in the singleperiod model 6 x 5 30 storage linking constraints Total 60 constraints It is also important to realize the use to which a model such as this might be put for mediumterm planning By solving the model in January buying and blending plans could be determined for January together with provisional plans for the succeeding months In February the model would probably be resolved with revised figures to give firm plans for February together with provisional plans for succeeding months up to and including July By this means the best use is made of the information for succeeding months to derive an operating policy for the current month 141 Food manufacture 1 The optimal policy is given in Table 141 The profit income from sales cost of raw oils derived from this policy is 107 843 This figure includes storage costs for the last month There are alternative optimal solutions 121 Food manufacture 1 A food is manufactured by refining raw oils and blending them together The raw oils are of two categories Vegetable oils VEG 1 VEG 2 Nonvegetable oils OIL 1 OIL 2 OIL 3 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Each oil may be purchased for immediate delivery January or bought on the futures market for delivery in a subsequent month Prices at present and in the futures market are given below in ton VEG 1 VEG 2 OIL 1 OIL 2 OIL 3 January 110 120 130 110 115 February 130 130 110 90 115 March 110 140 130 100 95 April 120 110 120 120 125 May 100 120 150 110 105 June 90 100 140 80 135 The final product sells at 150 per ton Vegetable oils and nonvegetable oils require different production lines for refining In any month it is not possible to refine more than 200 tons of vegetable oils and more than 250 tons of nonvegetable oils There is no loss of weight in the refining process and the cost of refining may be ignored It is possible to store up to 1000 tons of each raw oil for use later The cost of storage for vegetable and nonvegetable oil is 5 per ton per month The final product cannot be stored nor can refined oils be stored There is a technological restriction of hardness on the final product In the units in which hardness is measured this must lie between 3 and 6 It is assumed that hardness blends linearly and that the hardnesses of the raw oils are VEG 1 88 VEG 2 61 OIL 1 20 OIL 2 42 OIL 3 50 What buying and manufacturing policy should the company pursue in order to maximise profit At present there are 500 tons of each type of raw oil in storage It is required that these stocks will also exist at the end of June This problem and the subsequent problem is based on a larger model built for the margarine producer Van den Bergs and Jurgens and discussed in Williams and Redwood 1974 131 Food manufacture 1 Blending problems are frequently solved using linear programming Linear programming has been used to find minimum cost blends of fertilizer metal alloys clays and many food products to name only a few Applications are described in Fisher and Schruben 1953 and Williams and Redwood 1974 for example The problem presented here has two aspects Firstly it is a series of simple blending problems Secondly there is a purchasing and storing problem To understand how this problem may be formulated it is convenient to consider first the blending problem for only one month This is the singleperiod problem that has already been presented as the second example in Section 12 1311 The singleperiod problem If no storage of raw oils were allowed the problem of what to buy and how to blend in January could be formulated as follows Maximize PROFIT 110x1 120x2 130x3 110x4 115x5 150y subject to VVEG x1 x2 200 NVEG x3 x4 x5 250 UHRD 88x1 61x2 2x3 42x4 5x5 6y 0 LHRD 88x1 61x2 2x3 42x4 5x5 3y 0 CONT x1 x2 x3 x4 x5 y 0 The variables x1 x2 x3 x4 x5 represent the quantities of the raw oils that should be bought respectively that is VEG 1 VEG 2 OIL 1 OIL 2 and OIL 3 y represents the quantity of PROD that should be made The objective is to maximize profit which represents the income derived from selling PROD minus the cost of the raw oils The first two constraints represent the limited production capacities for refining vegetable and nonvegetable oils The next two constraints force the hardness of PROD to lie between its upper limit of 6 and its lower limit of 3 It is important to model these restrictions correctly A frequent mistake is to model them as 88x1 61x2 2x3 42x4 5x5 6 and 88x1 61x1 2x3 42x4 5x5 3 Such constraints are clearly dimensionally wrong The expressions on the left have the dimension of hardness multiplied by quantity whereas the figures on the right have the dimensions of hardness Instead of the variables xi in the above two inequalities expressions xiy are needed to represent proportions of the ingredients rather than the absolute quantities xi When such replacements are made the resultant inequalities can easily be reexpressed in a linear form as the constraints UHRD and LHRD Finally it is necessary to make sure that the weight of the final product PROD is equal to the weight of the ingredients This is done by the last constraint CONT which imposes this continuity of weight The singleperiod problems for the other months would be similar to that for January apart from the objective coefficients representing the raw oil costs 1312 The multiperiod problem The decisions of how much to buy each month with a view to storing for use later can be incorporated into a linear programming model To do this a multiperiod model is built It is necessary each month to distinguish the quantities of each raw oil bought used and stored These quantities must be represented by different variables We suppose the quantities of VEG 1 bought used and stored in each successive month are represented by variables with the following names BVEG 11 BVEG 12 and so on UVEG 11 UVEG 12 and so on SVEG 11 SVEG 12 and so on It is necessary to link these variables together by the relation quantity stored in month t 1 quantity bought in month t quantity used in month t quantity stored in month t 298 MODEL BUILDING IN MATHEMATICAL PROGRAMMING Initially month 0 and finally month 6 the quantities in store are constants 500 The relation above involving VEG 1 gives rise to the following constraints BVEG11 UVEG11 SVEG11 500 SVEG11 BVEG12 UVEG12 SVEG12 0 SVEG12 BVEG13 UVEG13 SVEG13 0 SVEG13 BVEG14 UVEG14 SVEG14 0 SVEG14 BVEG15 UVEG15 SVEG15 0 SVEG15 BVEG16 UVEG16 500 Similar constraints must be specified for the other four raw oils It may be more convenient to introduce variables SVEG 10 and so on and SVEG 16 and so on into the model and fix them at the value 500 In the objective function the buying variables will be given the appropriate raw oil costs in each month The storage variables will be given the cost of 5 or profit of 5 Separate variables PROD 1 PROD 2 and so on must be defined to represent the quantity of PROD to be made in each month These variables will each have a profit of 150 The resulting model will have the following dimensions as well as the single objective function 6 5 30 buying variables 6 5 30 using variables 5 5 25 storing variables 6 product variables Total 91 variables 6 5 30 blending constraints as in the singleperiod model 6 5 30 storage linking constraints Total 60 constraints 6 5 30 blending constraints as in the singleperiod model 6 5 30 storage linking constraints Total 60 constraints It is also important to realize the use to which a model such as this might be put for mediumterm planning By solving the model in January buying and blending plans could be determined for January together with provisional plans for the succeeding months In February the model would probably be resolved with revised figures to give firm plans for February together with provisional plans for succeeding months up to and including July By this means the best use is made of the information for succeeding months to derive an operating policy for the current month 141 Food manufacture 1 The optimal policy is given in Table 141 The profit income from sales cost of raw oils derived from this policy is 107 843 This figure includes storage costs for the last month There are alternative optimal solutions Table 141 Buy Use Store January Nothing 222 tons VEG 1 4778 tons VEG 1 1778 tons VEG 2 3222 tons VEG 2 250 tons OIL 3 500 tons OIL 1 500 tons OIL 2 250 tons OIL 3 February 250 tons OIL 2 200 tons VEG 2 4778 tons VEG 1 250 tons OIL 3 1222 tons VEG 2 500 tons OIL 1 750 tons OIL 2 March Nothing 1593 tons VEG 1 3185 tons VEG 1 407 tons VEG 2 815 tons VEG 2 250 tons OIL 2 500 tons OIL 1 500 tons OIL 2 April Nothing 1593 tons VEG 1 1593 tons VEG 1 407 tons VEG 2 407 tons VEG 2 250 tons OIL 2 500 tons OIL 1 250 tons OIL 2 May 500 tons OIL 3 1593 tons VEG 1 500 tons OIL 1 407 tons VEG 2 500 tons OIL 3 250 tons OIL 2 June 6593 tons VEG 1 1593 tons VEG 1 500 tons 5407 tons VEG 2 407 tons VEG 2 each oil 750 tons OIL 2 250 tons OIL 2 stipulated