The XNPV Function
3.8 Blending Problems
Finally,
January 1 liabilities December 1 liabilities December loan payment
amount due on January 1 inventory shipment
10,000 1,000 2,000
$11,000 Constraint 4 may now be written as
2 Multiplying both sides of this inequality by 11,000 yields
20,000 20x1 15x2 22,000 Putting this in a form appropriate for computer input, we obtain
20x1 15x2 2,000 (37)
Combining (33)–(37) with the sign restrictions x1 0 and x2 0 yields the follow-ing LP:
max z 20x1 15x2
s.t. 20x1 15x2 100 (Tape recorder constraint) s.t. 20x1 15x2 100 (Radio constraint) s.t. 50x1 35x2 6,000 (Cash position constraint) s.t. 20x1 15x2 2,000 (Current ratio constraint)
x1, x2 0 (Sign restrictions)
When solved graphically (or by computer), the following optimal solution is obtained:
z 2,500, x1 50, x2 100. Thus, Semicond can maximize the contribution of De-cember’s production to profits by manufacturing 50 tape recorders and 100 radios. This will contribute 20(50) 15(100) $2,500 to profits.
P R O B L E M S
Group A
20,000 20x1 15x2
11,000
1 Graphically solve the Semicond problem. 2 Suppose that the January 1 inventory shipment had been valued at $7,000. Show that Semicond’s LP is now infeasible.
2 Blending various chemicals to produce other chemicals
3 Blending various types of metal alloys to produce various types of steels
4 Blending various livestock feeds in an attempt to produce a minimum-cost feed mix-ture for cattle
5 Mixing various ores to obtain ore of a specified quality
6 Mixing various ingredients (meat, filler, water, and so on) to produce a product like bologna
7 Mixing various types of papers to produce recycled paper of varying quality
The following example illustrates the key ideas that are used in formulating LP models of blending problems.
Sunco Oil manufactures three types of gasoline (gas 1, gas 2, and gas 3). Each type is produced by blending three types of crude oil (crude 1, crude 2, and crude 3). The sales price per barrel of gasoline and the purchase price per barrel of crude oil are given in Table 12. Sunco can purchase up to 5,000 barrels of each type of crude oil daily.
The three types of gasoline differ in their octane rating and sulfur content. The crude oil blended to form gas 1 must have an average octane rating of at least 10 and contain at most 1% sulfur. The crude oil blended to form gas 2 must have an average octane rating of at least 8 and contain at most 2% sulfur. The crude oil blended to form gas 3 must have an octane rat-ing of at least 6 and contain at most 1% sulfur. The octane ratrat-ing and the sulfur content of the three types of oil are given in Table 13. It costs $4 to transform one barrel of oil into one bar-rel of gasoline, and Sunco’s refinery can produce up to 14,000 barbar-rels of gasoline daily.
Sunco’s customers require the following amounts of each gasoline: gas 1—3,000 bar-rels per day; gas 2—2,000 barbar-rels per day; gas 3—1,000 barbar-rels per day. The company considers it an obligation to meet these demands. Sunco also has the option of advertis-ing to stimulate demand for its products. Each dollar spent daily in advertisadvertis-ing a partic-ular type of gas increases the daily demand for that type of gas by 10 barrels. For exam-ple, if Sunco decides to spend $20 daily in advertising gas 2, then the daily demand for gas 2 will increase by 20(10) 200 barrels. Formulate an LP that will enable Sunco to maximize daily profits (profits revenues costs).
Solution Sunco must make two types of decisions: first, how much money should be spent in ad-vertising each type of gas, and second, how to blend each type of gasoline from the three types of crude oil available. For example, Sunco must decide how many barrels of crude 1 should be used to produce gas 1. We define the decision variables
ai dollars spent daily on advertising gas i (i 1, 2, 3)
xij barrels of crude oil i used daily to produce gas j (i 1, 2, 3; j 1, 2, 3) For example, x21is the number of barrels of crude 2 used each day to produce gas 1.
Oil Blending
E X A M P L E 1 2
T A B L E 12
Gas and Crude Oil Prices for Blending
Sales Price Purchase Price
Gas per Barrel ($) Crude per Barrel ($)
1 70 1 45
2 60 2 35
3 50 3 25
Knowledge of these variables is sufficient to determine Sunco’s objective function and con-straints, but before we do this, we note that the definition of the decision variables implies that
x11 x12 x13 barrels of crude 1 used daily
x21 x22 x23 barrels of crude 2 used daily (38) x31 x32 x33 barrels of crude 3 used daily
x11 x21 x31 barrels of gas 1 produced daily
x12 x22 x32 barrels of gas 2 produced daily (39) x13 x23 x33 barrels of gas 3 produced daily
To simplify matters, let’s assume that gasoline cannot be stored, so it must be sold on the day it is produced. This implies that for i 1, 2, 3, the amount of gas i produced daily should equal the daily demand for gas i. Suppose that the amount of gas i produced daily exceeded the daily demand. Then we would have incurred unnecessary purchasing and production costs. On the other hand, if the amount of gas i produced daily is less than the daily demand for gas i, then we are failing to meet mandatory demands or incurring un-necessary advertising costs.
We are now ready to determine Sunco’s objective function and constraints. We begin with Sunco’s objective function. From (39),
Daily revenues from gas sales 70(x11 x21 x31) 60(x12 x22 x32)
50(x13 x23 x33) From (38),
Daily cost of purchasing crude oil 45(x11 x12 x13) 35(x21 x22 x23)
25(x31 x32 x33) Also,
Daily advertising costs a1 a2 a3
Daily production costs 4(x11 x12 x13 x21 x22 x23 x31 x32 x33) Then,
Daily profit daily revenue from gas sales
daily cost of purchasing crude oil
daily advertising costs daily production costs
(70 45 4)x11 (60 45 4)x12 (50 45 4)x13
(70 35 4)x21 (60 35 4)x22 (50 35 4)x23
(70 25 4)x31 (60 25 4)x32
(50 25 4)x33 a1 a2 a3 T A B L E 13
Octane Ratings and Sulfur Requirements for Blending
Octane Sulfur
Crude Rating Content (%)
1 12 0.5
2 6 2.0
3 8 3.0
Thus, Sunco’s goal is to maximize
z 21x11 11x12 x13 31x21 21x22 11x23 41x31
31x32 21x33 a1 a2 a3 (40)
Regarding Sunco’s constraints, we see that the following 13 constraints must be satis-fied:
Constraint 1 Gas 1 produced daily should equal its daily demand.
Constraint 2 Gas 2 produced daily should equal its daily demand.
Constraint 3 Gas 3 produced daily should equal its daily demand.
Constraint 4 At most 5,000 barrels of crude 1 can be purchased daily.
Constraint 5 At most 5,000 barrels of crude 2 can be purchased daily.
Constraint 6 At most 5,000 barrels of crude 3 can be purchased daily.
Constraint 7 Because of limited refinery capacity, at most 14,000 barrels of gasoline can be produced daily.
Constraint 8 Crude oil blended to make gas 1 must have an average octane level of at least 10.
Constraint 9 Crude oil blended to make gas 2 must have an average octane level of at least 8.
Constraint 10 Crude oil blended to make gas 3 must have an average octane level of at least 6.
Constraint 11 Crude oil blended to make gas 1 must contain at most 1% sulfur.
Constraint 12 Crude oil blended to make gas 2 must contain at most 2% sulfur.
Constraint 13 Crude oil blended to make gas 3 must contain at most 1% sulfur.
To express Constraint 1 in terms of decision variables, note that
Daily demand for gas 1 3,000 gas 1 demand generated by advertising
Gas 1 demand generated by advertising
gd ao s l
1 lar
de s m pe
a n
n t
d
10a1†
Thus, daily demand for gas 1 3,000 10a1. Constraint 1 may now be written as x11 x21 x31 3,000 10a1 (41 ) which we rewrite as
x11 x21 x31 10a1 3,000 (41) Constraint 2 is expressed by
x12 x22 x32 10a2 2,000 (42) dollars
spent
†Many students believe that gas 1 demand generated by advertising should be written as 1 1
0a1. Analyzing the units of this term will show that this is not correct. 1
1
0has units of dollars spent per barrel of demand, and a1
has units of dollars spent. Thus, the term 1 1
0a1would have units of (dollars spent)2per barrel of demand. This cannot be correct!
Constraint 3 is expressed by
x13 x23 x33 10a3 1,000 (43) From (38), Constraint 4 reduces to
x11 x12 x13 5,000 (44)
Constraint 5 reduces to
x21 x22 x23 5,000 (45)
Constraint 6 reduces to
x31 x32 x33 5,000 (46)
Note that
Total gas produced gas 1 produced gas 2 produced gas 3 produced
(x11 x21 x31) (x12 x22 x32) (x13 x23 x33) Then Constraint 7 becomes
x11 x21 x31 x12 x22 x32 x13 x23 x33 14,000 (47) To express Constraints 8–10, we must be able to determine the “average” octane level in a mixture of different types of crude oil. We assume that the octane levels of different crudes blend linearly. For example, if we blend two barrels of crude 1, three barrels of crude 2, and one barrel of crude 3, the average octane level in this mixture would be
8
Generalizing, we can express Constraint 8 by
10 (48 )
Unfortunately, (48 ) is not a linear inequality. To transform (48 ) into a linear inequality, all we have to do is multiply both sides by the denominator of the left-hand side. The re-sulting inequality is
12x11 6x21 8x31 10(x11 x21x31) which may be rewritten as
2x11 4x21 2x31 0 (48)
Similarly, Constraint 9 yields
8
Multiplying both sides of this inequality by x12 x22 x32and simplifying yields
4x12 2x22 0 (49)
Because each type of crude oil has an octane level of 6 or higher, whatever we blend to manufacture gas 3 will have an average octane level of at least 6. This means that any val-ues of the variables will satisfy Constraint 10. To verify this, we may express Constraint 10 by
12x13 6x23 8x33 6
x13 x23 x33
12x12 6x22 8x32
x12 x22 x32
12x11 6x21 8x31
x11 x21 x31
Total octane value in gas 1
1
50 12(2) 6(3) 8(1)
2 3 1 Total octane value in mixture
Multiplying both sides of this inequality by x13 x23 x33and simplifying, we obtain
6x13 2x33 0 (50)
Because x13 0 and x33 0 are always satisfied, (50) will automatically be satisfied and thus need not be included in the model. A constraint such as (50) that is implied by other constraints in the model is said to be a redundant constraint and need not be included in the formulation.
Constraint 11 may be written as
0.01 Then, using the percentages of sulfur in each type of oil, we see that
Total sulfur in gas 1 mixture Sulfur in oil 1 used for gas 1
sulfur in oil 2 used for gas 1
sulfur in oil 3 used for gas 1
0.005x11 0.02x21 0.03x31
Constraint 11 may now be written as
0.01
Again, this is not a linear inequality, but we can multiply both sides of the inequality by x11 x21 x31and simplify, obtaining
0.005x11 0.01x21 0.02x31 0 (51) Similarly, Constraint 12 is equivalent to
0.02
Multiplying both sides of this inequality by x12 x22 x32and simplifying yields
0.015x12 0.01x32 0 (52)
Finally, Constraint 13 is equivalent to
0.01
Multiplying both sides of this inequality by x13 x23 x33and simplifying yields the LP constraint
0.005x13 0.01x23 0.02x33 0 (53) Combining (40)–(53), except the redundant constraint (50), with the sign restrictions xij 0 and ai 0 yields an LP that may be expressed in tabular form (see Table 14). In Table 14, the first row (max) represents the objective function, the second row represents the first constraint, and so on. When solved on a computer, an optimal solution to Sunco’s LP is found to be
z 287,500
x11 2222.22 x12 2111.11 x13 666.67 x21 444.44 x22 4222.22 x23 333.34 x31 333.33 x32 3166.67 x33 0
a1 0 a2 750 a3 0 0.005x13 0.02x23 0.03x33
x13 x23 x33
0.005x12 0.02x22 0.03x32
x12 x22 x32
0.005x11 0.02x21 0.03x31
x11 x21 x31
Total sulfur in gas 1 mixture
Thus, Sunco should produce x11 x21 x31 3,000 barrels of gas 1, using 2222.22 barrels of crude 1, 444.44 barrels of crude 2, and 333.33 barrels of crude 3. The firm should produce x12 x22 x32 9,500 barrels of gas 2, using 2,111.11 barrels of crude 1, 4222.22 barrels of crude 2, and 3,166.67 barrels of crude 3. Sunco should also pro-duce x13 x23 x33 1,000 barrels of gas 3, using 666.67 barrels of crude 1 and 333.34 barrels of crude 2. The firm should also spend $750 on advertising gas 2. Sunco will earn a profit of $287,500.
Observe that although gas 1 appears to be most profitable, we stimulate demand for gas 2, not gas 1. The reason for this is that given the quality (with respect to octane level and sulfur content) of the available crude, it is difficult to produce gas 1. Therefore, Sunco can make more money by producing more of the lower-quality gas 2 than by producing extra quantities of gas 1.
Modeling Issues
1 We have assumed that the quality level of a mixture is a linear function of each input used in the mixture. For example, we have assumed that if gas 3 is made with2
3crude 1 and1
3crude 2, then octane level for gas 3 (23) (octane level for crude 1) (13) (oc-tane level for crude 2). If the oc(oc-tane level of a gas is not a linear function of the fraction of each input used to produce the gas, then we no longer have a linear programming prob-lem; we have a nonlinear programming problem. For example, let gi3 fraction of gas 3 made with oil i. Suppose that the octane level for gas 3 is given by gas 3 octane level g13.5 (oil 1 octane level) g23.4 (oil 2 octane level) g33.3 (oil 3 octane level). Then we do not have an LP problem. The reason for this is that the octane level of gas 3 is not a linear function of g13, g23, and g33. We discuss nonlinear programming in Chapter 11.
2 In reality, a company using a blending model would run the model periodically (each day, say) and set production on the basis of the current inventory of inputs and current demand forecasts. Then the forecast levels and input levels would be updated, and the model would be run again to determine the next day’s production.
T A B L E 14
Objective Function and Constraints for Blending
x11 x12 x13 x21 x22 x23 x31 x32 x33 a1 a2 a3
21 11 1 31 21 11 41 31 21 1 1 1 (max)
1 0 0 1 0 0 1 0 0 10 0 0 3,000
0 1 0 0 1 0 0 1 0 0 10 0 2,000
0 0 1 0 0 1 0 0 1 0 0 10 1,000
1 1 1 0 0 0 0 0 0 0 0 0 5,000
0 0 0 1 1 1 0 0 0 0 0 0 5,000
0 0 0 0 0 0 1 1 1 0 0 0 5,000
1 1 1 1 1 1 1 1 1 0 0 0 14,000
2 0 0 4 0 0 2 0 0 0 0 0 0
0 4 0 0 2 0 0 0 0 0 0 0 0
0.005 0 0 0.01 0 0 0.02 0 0 0 0 0 0
0 0.015 0 0 0 0 0 0.01 0 0 0 0 0
0 0 0.005 0 0 0.01 0 0 0.02 0 0 0 0