Review Problem #1 • INTERNATIONAL SHIPPING. The Takahashi Transport Company (TTC) leases excess space on commercial vessels to the United States at a reduced rate of $10 per square foot. The only condition is that goods must be packaged in standard 30 -inch-high crates. • TTC ships items in two standard 30 -inch-high crates, – an eight-square-foot crate (two feet by four feet) and – a four-square-foot (two feet by two feet) specially insulated crate. • It charges customers – $160 to ship an eight-square-foot crate and – $100 to ship the insulated four-square-foot crate. • Allowing for the cost of $10 per square foot, TTC makes – $80 per standard eight-foot crate and (160 – 10*8) – $60 on the four-foot crate (100 – 10*4) 1
• TTC stores the crates until space becomes available on a cargo ship, at which time TTC receives payment from its customers. TTC has been able to lease – 1200 square feet of cargo space on the Formosa Frigate cargo ship, which leaves for the United States in two days. • As of this date, TTC has – 140 eight-square-foot crates and – 100 insulated four-square-foot crates awaiting shipment to the United States. • It has – 48 hours to finish loading the crates, and • it estimates the average loading time to be – 12 minutes (. 2 hour) per eight-square-foot crate and – 24 minutes (. 4 hour) per four-square-foot crate (owing to the special handling of the insulated crates). 2
• Formulate and solve a linear program for TTC to optimize its profit on the upcoming sailing of the Formosa Frigate. 8 -SQFT 4 -SQFT Profit 80 # of 8 FT 1 <= 140 1 <= 100 0. 2 0. 4 <= 48 8 4 <= 1200 # of 4 FT Load Time Space 60 3
• What are the optimal values of the slack on each constraint in the optimal solution? Express this result in words. Constraints Ce ll Name Cell Value Formula Status $D $4 # of 8 FT 120 $D$4<=$F$4 Not Binding 20 $D $5 # of 4 FT 60 $D$5<=$F$5 Not Binding 40 $D $6 Load Time 48 $D$6<=$F$6 Binding 0 1200 $D$7<=$F$7 Binding 0 $D $7 Space Slack 4
Review Problem # 1 - continued a. Determine the range of optimality for each revenue coefficient. Adjustable Cells Final Reduced Objective Allowable Name Value Cost Coefficient Increase Decrease 8 -SQFT 120 0 80 40 50 4 -SQFT 60 0 60 100 20 The ranges of optimality are: 80+40, 80 -50 60+100, 60 -20 5
Review Problem # 1 - continued b. Determine the range of feasibility for the number of square feet available, the loading time, the number of standard containers available, and the number of insulated containers available. Constraints Final Shadow Constraint Allowable Name Value Price R. H. Side Increase Decrease # of 8 FT 120 0 140 1 E+30 20 # of 4 FT 60 0 100 1 E+30 40 Load Time 48 66. 66666667 48 12 12 1200 8. 33333 1200 120 480 Space The ranges of feasibility are: 140+infinity, 140 -20 100+infinity, 100 -40 48+12, 48 -12 1200+120, 1200 -480 6
c. Determine the shadow price for each resource for which a range of feasibility was calculated in part (b). Do you think that any of these should be treated as a sunk cost? Given your answer, explain the meaning of each shadow price. • As long as extra space could be leased for 10 + 8. 33 = $18. 33 it should be leased. Explanation: The leasing cost per sq-ft was included in determining the profit per sq-ft (for an 8 -sq ft crate it is 160/8 -10). Thus, TCC should be willing to pay up to $8. 33 above $10 per sq ft, but not more). • Loading time is a sunk cost. Extra loading time will add $66. 67 to the profit per hour added 7
e. Suppose that at the last second, the Formosa Frigate decided to raise its ss charge per square foot from $10 to $12. Would the loptimal solution c a in change? ed s s iscu as d w ul Two coefficients in the objectiveefunction are changing. % r 00 Percent change: For t 8 -square foot crate=((80 -64)/(80 -30))=. 32 e 1 if h y foot crate=((60 -52)/(60 -40))=. 40 l For 4 -square r on e Cov Total=. 72 < 1. 00 The optimal solution will not change. 8
Review Problem #2 BAKERY. Mary Custard's is a pie shop that specializes in custard and fruit pies. It makes delicious pies and sells them at reasonable prices so that it can sell all the pies it makes in a day. Every dozen custard pies nets Mary Custard's $ 15 and requires 12 pounds of flour, 50 eggs, and 5 pounds of sugar (and no fruit mixture). Every dozen fruit pies nets a $25 profit and uses 10 pounds of flour, 40 eggs, 10 pounds of sugar, and 15 pounds of fruit mixture. On a given day, the bakers at Mary Custard's had 150 pounds of flour, 500 eggs, 90 pounds of sugar, and 120 pounds of fruit mixture. 9
Problem 2 - solution • Formulate a linear program that will give the optimal production schedule of pies for the day. X 1 = the number of dozen custard pies baked X 2 = the number of dozen fruit pies baked MAX 15 X 1+25 X 2 S. T. 12 X 1 + 10 X 2 <= 150 (Flour) 50 X 1 + 40 X 2 <= 500 (Eggs) 5 X 1 + 10 X 2 <= 90 (Sugar) 15 X 2 <= 120 (Fruit mixture) XI, X 2 >= 0 10
Problem 2 - solution • Solve for the optimal production schedule Adjustable Cells Final Reduced Objective Allowable Value Cost Coefficient Increase Decrease x 1 4. 67 0 15 0. 5 7 x 2 6. 67 0 25 14 0. 4 Bake 56 (4 2/3 dozen) custard pies and 80 (6 2/3 dozen) frui Profit = $236. 67 11
c. If Mary Custard's could double its profit on custard pies, should more custard pies be produced? Explain. $30 is within the range of optimality - no change. d. If Mary Custard's raised the price (and hence the profit) on all pies by $0. 25 ($3. 00 per dozen), would the optimal production Cover only if the 100% rule was discussed in class schedule for the day change? Would the profit change? Percent changes: Custard pies 3/16. 25 = 18. 46% Fruit pies 3/5 = 60. 00%. Total = 78. 46% Adjustable Cells Final Reduced Objective Allowable Value Cost Coefficient Increase Decrease x 1 4. 67 0 12 16. 25 2. 5 x 2 6. 67 0 10 5 13 No change in the quantities produced; change in the profit. 12
e. Suppose Mary Custard's found that 10% of its fruit mixture had been stored in containers that were not air-tight. For quality and health reasons, it decided that it would be unwise to use any of this portion of the fruit mixture. How would this affect the optimal production schedule? Explain. 12 pounds of fruit mixture are not available (10% of 120); yet there is a slack of 20 pounds and thus the optimal solution will not be affected. Final Shadow Allowable Constraint Allowable Name Value Price R. H. Side Increase Decrease Flour 122. 67 0 150 1 E+30 27. 333 Eggs 500 0. 083 500 117. 143 80 Sugar 90 2. 17 90 8 40 Fruit 100 0 120 1 E+30 20 13
f. Mary Custard's currently pays $2. 50 for a five-pound bag of sugar from its bakery supply vendor. (The $0. 50 per pound price of sugar is included in the unit profits given earlier. ) Its vendor has already made its deliveries for the day. If Mary Custard's wishes to purchase additional sugar, it must buy it from Donatelli's Market, a small, independent grocery store that sells sugar in one-pound boxes for $2. 25 a box. Should Mary Custard's purchase any boxes of sugar from Donatelli's Market? Explain The shadow price for sugar is $2. 17. Since cost of sugar is included, extra sugar is worth $2. 17 + $0. 50 = $2. 67. Since this is greater than $2. 25, Mary Custard's should purchase it. Final Shadow Constraint Allowable Name Value Price R. H. Side Increase Decrease Flour 122. 67 0 150 1 E+30 27. 333 Eggs 500 0. 083 500 117. 143 80 Sugar 90 2. 17 90 8 40 Fruit 100 0 120 1 E+30 20 14
Review Problem # 3 • This problem focuses on modeling a blending problem, multiple changes and the application of the 100% rule. • The Party Nut Company has on hand 550 pounds of peanuts, 150 pounds of cashews, 90 pounds of Brazil nuts. It packages and sells four varieties of mixed nuts in standard 8 -ounce (half pound) cans. The mix requirements and the unit profit per can are shown in the table below. What mix of products (how many cans of each product) should be produced and sold? 15
Data Mix Contents Profit per can 1 Peanuts only $0. 26 2 No more than 50% peanuts At least 15% cashew $0. 40 3 Cashew only $0. 51 4 At least 30% cashew At least 20% brazil nuts $0. 52 16
Solution • Definitions: • M 1, M 2, M 3, M 4 = the number of cans produced of mix 1, 2, 3, 4 respectively. • Pi, Ci, Bi = the amount (in pounds) of peanuts, cashew, brazil use in mix ‘i’. 17
The model – Max. 26 M 1+. 40 M 2+. 51 M 3+. 52 M 4 ST. P 1 =. 5 M 1 P 2 £ (50%)(. 5 M 2) C 2 ³ (15%)(. 5 M 2) C 3 =. 5 M 3 C 4 ³ (30%)(. 5 M 4) B 4 ³ (20%)(. 5 M 4) P 1 + P 2 £ 550 C 2 + C 3 + C 4 £ 150 B 4 £ 90 Required quantities Available resources 18
WINQSB solution and sensitivity analysis • • M=infinity Assume the profit drops by 10 cents per can for mix 1, increases by 10 cents per can for mix 2, and by 20 cents per can for mix 3. Would the optimal production plan change? (. 16 -. 26)/(0 -. 26)+ (. 50 -. 40)/(M-. 40)+(. 71 -. 51)/(2. 67 -. 51) =. 4772 < 1. The optimal solution will not change. 19
WINQSB solution and sensitivity analysis • • • Assume the available amount of all the materials used decreases by 10% each. Would the optimal solution change. Would the total profit change? By how much? Observe the last three constraints, and check the 100% rule: – 55/(0 -550)+ (-15)/(0 -150)+(-9)/(0 -90) =. 3 < 1. The shadow prices will remain the same, but the solution will change, because the changes are made in binding constraints (no slack). The new profit = old profit + S[shadow price(change in the constraint’s RHS)] = 1086 + (52)(-55)+5. 33(-15)+(0)(-9) = 977. 45 20
Modeling and Sensitivity Analysis – Example 1 APPAREL INDUSTRY. Exclaim! Jeans is setting up a production schedule for the coming week. Exclaim! Can make four jean products: men’s and women’s jackets and pants. Although it can make different sizes of each, the variation in material usage and labor between sizes is negligible. Each jacket and pair of pants goes through cutting and stitching operations before being boxed. The following table gives the profit, denim, cutting time, stitching time, and boxing time required per 100 items, as well as the total resource availabilities during the week. 21
APPAREL INDUSTRY Item Profit ($) Denim (yd. ) Cutting (hr. ) Stitching (hr. ) Boxing (hr. ) Men’s Jacket Women’s Jacket Men’s Pants Women’s Pants $2, 000 150 3 4 . 75 2, 800 125 4 3 . 75 1200 2 2 . 50 150 2 2 . 50 2500 36 36 8 Available 22
APPAREL INDUSTRY Solution • Develop and solve a linear programming model. For Exclaim! Jeans which will maximize its profit for the week. Decision variables and objective function Max 2000 MJacket+2800 WJacket+1200 MPants+1500 WPants Constraints Material constraint 150 MJacket+125 WJacket+200 MPants+150 WPants £ 2500 Cutting time constraint 3 MJacket+ 4 WJacket+ 2 MPants+ 2 WPants £ 36 Stitching time constraint 4 MJacket+ 3 WJacket+ 2 MPants+ 2 WPants £ 36 Boxing time constraint . 75 MJacket+. 75 WJacket+. 50 MPants+ . 50 WPants £ 7. 5 23
APPAREL INDUSTRY Solution • Develop and solve a linear programming model. For Exclaim! Jeans which will maximize its profit for the week. 24
APPAREL INDUSTRY Solution • How much should the profit for Men’s Jacket increase before it becomes part of the production plan? + 25
APPAREL INDUSTRY Solution • Assume the unit profit of women’s pants increases by 20%. If all pants produced can still be sold should Exclaim change its production plan in order to sell more women’s pants? Explain. No. 20%(1500)= 300 < 366. 67 26
APPAREL INDUSTRY Solution • Management is considering allocating one hour of overtime in one department. Where should this hour be allocated? How much should be paid per hour? Both Cutting and Boxing are candidates at the shadow price contribution shown. Prefer the boxing department because 1 hour of overtime contributes $800 to the profit (more than $550 in the cutting department). +1 +1 +1 27
APPAREL INDUSTRY Solution • Suppose that, in addition to the existing restrictions, management wishes to produce at least 300 of each item. Add these constraints to your linear program and re-solve the problem. What is the result? To what do you attribute this result? Additional constraints 28
APPAREL INDUSTRY Solution • If the minimum required of 300 women’s pants is increased to 350, what will happen to the profit? Cover only if the 100% rule was discussed in class 29 3. 5
APPAREL INDUSTRY Solution • Managements wants to relax one of the constraints on the minimum of 300 units production by 30 units. Which constraint should be selected? What is the new profit? Discussion: Material constraint 2600 150 MJacket+125 WJacket+200 MPants+150 WPants £ 2500 Minimum required constraint MJacket ³ 32 Cover only if the 100% rule was discussed in class 30
APPAREL INDUSTRY Solution • Managements wants to relax one of the constraints on the minimum of 300 units production by 30 units. Which constraint should be selected? What is the new profit? 31
APPAREL INDUSTRY Solution • Managements wants to relax one of the constraints on the minimum of 300 units production by 30 units. Which constraint should be selected? What is the new profit? Change in objective value = (Shadow price)(Change in the right hand side) = (-800)(-. 3) = +240. The new objective value is: 22, 500+240 = 22, 740. The change in the right hand side is (-. 3) because: (i) Relaxing this constraint means a reduction(!) in the right hand (ii) A change of 30 units translates to. 3 32
Modeling and Sensitivity Analysis– Example 2 • AGRICULTURE: BP Farms is a 300 -acre farm located near Lawrence, Kansas, owned and operated exclusively By Bill Phashley. For the upcoming growing season, Bill will grow wheat, corn, oats, and soybeans. The following table gives relevant data concerning expected crop yields, labor required, expected expenses, and water required. Also included is the price per bushel Bill expects to r 4 eceivre when the crops are harvested. 33
AGRICALTURE Yield Labor Expenses (bush. /acre (hr/acre ($/acre) ) Water (acreft. /acre) Price ($/bushel ) Wheat $50 2 $3. 20 300 5 $75 6 $2. 55 Oats • • • 4 Corn • 210 180 3 $30 1 $1. 45 Soybeans 240 10 $60 4 $3. 10 Bill wishes to produce at least 30, 000 bushels of wheat and 30, 000 bushels of corn, but not more than 25, 000 bushels of oats. Bill has $25, 000 to invest in his crop. Bill plans to work 12 hours a day during the 150 -day season. He does not wish to exceed the base water supply of 1200 acre-feet allocated to him. 34
AGRICALTURE – Solution • How many acres should Bill allocate to each crop? Discussion: • Since the return depends on the acres allocated to each crop, the decision variables are Wheat, Corn, Oats, and Soybeans expressed in acres. • The objective function represents the return, that is sales minus expenses. In general for each crop: Return from selling a crop = {[Price per bushel][Yield in bushels per acre] – [Expenses in $ per acre]}[Acres] For example: Return from selling wheat = [(3. 20$/bu. )(210 bu. /acre) – 50](“Wheat” in acres) 35
AGRICALTURE – Solution • How many acres should Bill allocate to each crop? 36
AGRICALTURE – Solution • If the selling price of oats remains $1. 45 a bushel, (1) to what level must the yield increase before oats should be planted? (2) If the yield remains 180 bu. /acre, to what level would the price of oats have to rise before oats should be planted? We need to use the range of optimality for the coefficient of Oats in the objective function. For Oats to be planted its return per acre should be at least 261+451. 5 = $712. 5. (1) If the price remains $1. 45 a bushel, the yield should increase to 712. 5/1. 45 = 491. 38 bushels per acre. (2) If the yield remain 180 bushels/acre the price should increase to 712. 5/180 = $3. 95 per bushel 37
AGRICALTURE – Solution • If there were no constraint on the minimum production of corn, would corn be planted? No constraint on the minimum production of corn translates to a constraint of the form 300 Corn ³ 0. Notice that this is always true (and in fact already exists as a non-negativity constraint). A change in the right hand side of this constraint has occurred (from 30000 down to zero). Since the current solution calls for a production of 42, 857 bushels, the optimal solution remains unchanged, and corn is still planted. 38
AGRICALTURE – Solution • Bill can lease an adjacent 40 -acre parcel for $10000. Should Bill lease this property for this price? A change occurs in the “Acres Total” constraint. The right hand side is changing from 300 to 340. The maximum value of the right hand side for the shadow price to remain unchanged is 300+18. 57=318. 57. So the shadow price changes if the total acre is 340. Still, we can answer the question. At 318. 57 acres the objective value (total return) increases by (Shadow price)(Constraint change) = (702)(18. 57) = $13, 036. This amount is greater than the cost of $10, 000. Clearly, with a chang of 40 acres the return exceeds the cost. Bill should lease the property. 39
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