ΠΜΣ: Μαθηματικά των Υπολογιστών και των Αποφάσεων. Μάθημα: Επιχειρησιακή Έρευνα Ακαδημαϊκό Έτος: 2011-12 Διδάσκων: Ν. Τσάντας Homework 1 1. Ασκήσεις: δείτε τις σελίδες 2-8 του παρόντος. 2. Πρόκειται για ατομικές ασκήσεις οι οποίες συνεισφέρουν το 25% του τελικού σας βαθμού. 3. Το παραδοτέο είναι ολόσωμο κείμενο Word όλων των ασκήσεων μαζί σε ελληνική γλώσσα μαζί με τα αρχεία δεδομένων των μοντέλων που κατασκευάσατε στο winqsb (.lpp), στο Lindo (.ltx), στο Excel ή στο POM-QM. 4. Όλα τα παραπάνω αρχεία θα πρέπει να συμπτυχθούν σε ένα zip αρχείο και να αποσταλούν ηλεκτρονικά (e-mail: tsantas@upatras.gr) στον διδάσκοντα μέχρι την 24.00 της 13ης Μαΐου 2012. Στο μήνυμα πρέπει να αναφέρετε απαραίτητα το ονοματεπώνυμό σας. Εργασίες που παραλαμβάνονται εκπρόθεσμα επισύρουν βαθμολογικές κυρώσεις (0,5 βαθμό για κάθε ημερολογιακή ημέρα καθυστέρησης). Εργασίες που υποβάλλονται με καθυστέρηση μεγαλύτερη από 3 ημέρες δεν γίνονται δεκτές. Το αρχείο zip θα το ονομάσετε με λατινικούς χαρακτήρες "eponymo_onoma_hw1.zip". 5. Και το αρχείο Word θα το ονομάσετε με λατινικούς χαρακτήρες "eponymo_onoma_hw1.doc". Τα αρχεία δεδομένων θα τα ονομάσετε με τον αύξοντα αριθμό της άσκησης που αντιστοιχούν (π.χ. 01.lpp, 02.ltx, κλπ). 6. Στο κείμενο της εργασίας, για κάθε άσκηση, πρέπει να υπάρχει μια ενότητα με την εννοιολογική προσέγγιση, δηλαδή την περιγραφή του προβλήματος και των υποθέσεών του. Στη συνέχεια μια ενότητα με την ανάπτυξη του κατάλληλου γραμμικού μοντέλου και τέλος, μια ενότητα με τις απαντήσεις στα υπόλοιπα ερωτήματα της άσκησης (π.χ. ανάλυση ευαισθησίας). 7. Πίνακες αποτελεσμάτων του winqsb ή του Lindo, κ.λπ. πρέπει να είναι μέρος του κειμένου ως εικόνες. 8. Στην πρώτη σελίδα της εργασίας πρέπει να αναφέρετε το όνομά σας. Οι σελίδες είναι Α4, όλα τα margins 2.5, το spacing 1.5 και η γραμματοσειρά Arial 12. 1
1. FABRICS AND FALL FASHIONS From the tenth floor of her office building, Katherine Rally watches the swarms of New Yorkers fight their way through the streets infested with yellow cabs and the sidewalks littered with hot dog stands. Οn this sweltering July day, she pays particular attention to the fashions worn by the various women and wonders what they will choose to wear in the fall. Her thoughts are not simply random musings; they are critical to her work since she owns and manages TrendLines, an elite women's clothing company. Today is an especial1y important day because she must meet with Ted Lawson, the production manager, to decide upon next month's production plan for the fall line. Specifically, she must determine the quantity of each clothing item she should produce given the plant's production capacity, limited resources, and demand forecasts. Accurate planning for next month's production is critical to fall sales since the items produced next month will appear in stores during September, and women generally buy the majority of the fall fashions when they first appear in September. She turns back to her sprawling glass desk and looks at the numerous papers covering it. Her eyes roam across the clothing patterns designed almost six months ago, the lists of materials requirements for each pattern, and the lists of demand forecasts for each pattern determined by customer surveys at fashion shows. She remembers the hectic and sometimes nightmarish days of designing the fall line and presenting it at fashion shows in New York, Milan, and Paris. Ultimately, she paid her team of six designers a total of $860,000 for their work on her fall line. With the cost of hiring runway models, hair stylists, and makeup artists, sewing and fitting clothes, building the set, choreographing and rehearsing the show, and renting the conference hal1, each of the three fashion shows cost her an additional $2,700,000. She studies the clothing patterns and material requirements. Her fall line consists of both professional and casual fashions. She determined the prices for each clothing item by taking into account the quality and cost of material, the cost of labor and machining, the demand for the item, and the prestige of the TrendLines brand name. The fall professional fashions include: The fall casual fashions include: 2
She knows that for the next month, she has ordered 45,000 yards of wool, 28,000 yards of acetate, 9,000 yards of cashmere, 18,000 yards of silk, 30,000 yards of rayon, 20,000 yards of velvet, and 30,000 yards of cotton for production. The prices of the materials are listed below. Any material that is not used in production can be sent back to the textile wholesaler for a full refund, although scrap material cannot be sent back to the wholesaler. She knows that the production of both the silk blouse and cotton sweater leaves leftover scraps of material. Specifically, for the production of one silk blouse or one cotton sweater, 2 yards of silk and cotton, respectively, are needed. From these 2 yards, 1.5 yards are used for the silk blouse or the cotton sweater and 0.5 yard is left as scrap material. She does not want to waste the material, so she plans to use the rectangular scrap of silk or cotton to produce a silk camisole or cotton miniskirt, respectively. Therefore, whenever a silk blouse is produced, a silk camisole is also produced. Likewise, whenever a cotton sweater is produced, a cotton miniskirt is also produced. Note that it is possible to produce a silk camisole without producing a silk blouse and a cotton miniskirt without producing a cotton sweater. The demand forecasts indicate that some items have limited demand. Specifically, because the velvet pants and velvet shirts are fashion fads, TrendLines has forecasted that it can sell only 5,500 pairs of velvet pants and 6,000 velvet shirts. TrendLines does not want to produce more than the forecasted demand because once the pants and shirts go out of style, the company cannot sell them. TrendLines can produce less than the forecasted demand, however, since the company is not required to meet the demand. The cashmere sweater also has limited demand because it is quite expensive, and TrendLines knows it can sell at most 4,000 cashmere sweaters. The silk blouses and camisoles have limited demand because many women think silk is too hard to care for, and TrendLines projects that it can sell at most 12,000 silk blouses and 15,000 silk camisoles. The demand forecasts also indicate that the wool slacks, tailored skirts, and wool blazers have a great demand because they are basic items needed in every professional wardrobe. Specifically, the demand for wool slacks is 7,000 pairs of slacks, and the demand for wool blazers is 5,000 blazers. Katherine wants to meet at least 60 percent of the demand for these two items in order to maintain her loyal customer base and not lose business in the future. Although the demand for tailored skirts could not be estimated, Katherine feels she should make at least 2,800 of them. a) Ted is trying to convince Katherine not to produce any velvet shins since the demand for this fashion fad is quite low. He argues that this fashion fad alone accounts for $500,000 of the fixed design and other costs. The net contribution (price of clothing item - materials cost - labor cost) from selling the fashion fad should cover these fixed costs. Each velvet shirt generates a net contribution of $22. He argues that given the net contribution, even satisfying the maximum demand will not yield a profit. What do you think of Ted's argument? b) Formulate and solve a linear programming problem to maximize profit given the production, resource, and demand constraints. Before she makes her final decision, Katherine plans to explore the following questions independently 3
except where otherwise indicated. c) The texti1e wholesaler informs Katherine that the velvet cannot be sent back because the demand forecasts show that the demand for velvet will decrease in the future. Katherine can therefore get no refund for the velvet. How does this fact change the production plan? d) What is an intuitive economic explanation for the difference between the solutions found in parts (b) and (c)? e) The sewing staff encounters difficulties sewing the arms and lining into the wool blazers since the blazer pattern has an awkward shape and the heavy wool material is difficult to cut and sew. The increased labor time to sew a wool blazer increases the labor and machine cost for each blazer by $80. Given this new cost, how many of each clothing item shοιι1d TrendLines produce to maximize profit? f) The textile wholesaler informs Katherine that since another textile customer canceled his order, she can obtain an extra 10,000 yards of acetate. How many of each clothing item should TrendLines now produce το maximize profit? g) TrendLines assumes that it can sell every item that was not sold during September and October in a big sale in November at 60 percent of the original price. Therefore, it can sell all items ίn unlimited quantity during the November sale. (The previously mentioned upper limits οn demand concern only the sales during September and October.) What should the new production plan be to maximize profit? 4
2. AUTO ASSEMBLY Automobile Alliance, a large automobile manufacturing company, organizes the vehicles it manufactures into three families: a family of trucks, a family of small cars, and a family of midsized and luxury cars. One plant outside Detroit, ΜΙ, assembles two models from the family of midsized and luxury cars. The first model, the Family Thrillseeker, is a four-door sedan with vinyl seats, plastic interior, standard features, and excel1ent gas mileage. It is marketed as a smart buy for middle-class families with tight budgets, and each Family Thrillseeker sold generates a modest profit of $3,600 for the company. The second model, the Classy Cruiser, is a two-door luxury sedan with leather seats, wooden interior, custom features, and navigational capabilities. It is marketed as a privilege of affluence for upper-middle-class families, and each Classy Cruiser sold generates a healthy profit of $5,400 for the company. Rachel Rosencrantz, the manager of the assembly plant, is currently deciding the production schedule for the next month. Specifically, she must decide how many Family Thrillseekers and how many Classy Cruisers to assemble in the plant to maximize profit for the company. She knows that the plant possesses a capacity of 48,000 labor-hours during the month. She also knows that it takes 6 labor-hours to assemble one Family Thrillseeker and 10.5 labor-hours to assemble one Classy Cruiser. Because the plant is simply an assembly plant, the parts required to assemble the two models are not produced at the plant. They are instead shipped from other plants around the Michigan area to the assembly plant. For example, tires, steering wheels, windows, seats, and doors all arrive from various supplier plants. For the next month, Rachel knows that she will be able to obtain only 20,000 doors (10,000 left-hand doors and 10,000 right-hand doors) from the door supplier. Α recent labor strike forced the shutdown of that particular supplier plant for several days, and that plant will not be able to meet its production schedule for the next month. Both the Family Thrillseeker and the Classy Cruiser use the same door part. Ιn addition, a recent company forecast of the monthly demands for different automobi1e models suggests that the demand for the Classy Cruiser is limited το 3,500 cars. There is no limit on the demand for the Family Thrillseeker within the capacity limits of the assembly plant. a) Formulate and solve a linear programming problem to determine the number of Family Thrillseekers and the number of Classy Cruisers that should be assembled. Before she makes her final production decisions, Rachel plans to explore the following questions independently except where otherwise indicated. b) The marketing department knows that it can pursue a targeted $500,000 advertising campaign that will raise the demand for the of Classy Cruiser next month by 20 percent. Should the campaign be undertaken'? c) Rachel knows that she can increase next month's plant capacity by using overtime labor. She can increase the plant's labor-hour capacity by 25 percent. With the new assembly plant capacity, how many Family Thrillseekers and how many Classy Cruisers should be assembled? d) Rachel knows that overtime labor does not come without an extra cost. What is the maximum amount she should be willing to pay for all overtime labor beyond the cost of this labor at regular time rates? Express your answer as a lump sum. e) Rachel explores the option of using both the targeted advertising campaign and the overtime labor-hours. The advertising campaign raises the demand for the Classy Cruiser by 20 percent, and the overtime labor increases the plant's labor-hour capacity by 25 percent. How many Family Thrillseekers and how many Classy Cruisers should be assembled using the advertising campaign and overtime labor-hours if the profit from each Classy Cruiser sold continues to be 50 percent more than for each Family Thrillseeke sold? 5
f) Knowing that the advertising campaign costs $500,000 and the maximum usage of overtime labor-hours costs $1,600,000 beyond regular time rates, is the solution found in part (e) a wise decision compared to the solution found in part (a)? g) Automobile Alliance has determined that dealerships are actual1y heavily discounting the price of the Family Thrillseekers το move them off the lot. Because of a profit-sharing agreement with its dealers, the company is therefore not making a profit of $3,600 on the Family Thrillseeker but is instead making a profit of $2,800. Determine the number of Family Thril1seekers and the number of Classy Cruisers that should be assembled given this new discounted price. h) The company has discovered quality problems with the Family Thril1seeker by randomly testing Thril1seekers at the end of the assembly line. Inspectors have discovered that in over 60 percent of the cases, two of the four doors on a Thril1seeker do not seal properly. Because the percentage of defective Thri11seekers determined by the random testing is so high, the floor supervisor has decided to perform quality control tests on every Thril1seeker at the end of the line Because of the added tests, the time it takes to assemble one Family Thril1seeker has increased from 6 to 7.5 hours. Determine the number of units of each model that shou1d be assembled given the new assembly time for the Family Thril1seeker. i) The board of director's of Automobile Alliance wishes to capture a larger share of the luxury sedan market and therefore would like to meet the ful1 demand for Classy Cruisers. They ask Rachel to determine by how much the profit of her assembly plant would decrease as compared to the profit found in part (a). They then ask her to meet the full demand for Classy Cruisers if the decrease in profit is not more than $2,000,000. j) Rachel now makes her final decision by combining all the new considerations described in parts (j), (g), and (h). What are her final decisions on whether to undertake the advertising campaign, whether to use overtime labor, the number of Family Thril1seekers to assemble, and the number of Classy Cruisers to assemble? 6
3. AGRICULTURAL ALLOCATION MODEL Α university has offered a rent subsidy alterative to the families living in the graduate student housing complex. Near the complex are 20 acres that can be farmed. The university wil1 allow the residents' organization to farm all or part of this land. Any derived profits can be apportioned to reducing the rent for the graduate families. Families may keep whatever produce they wish for their own use. The university has agreed to buy produce from the students, up to certain limits, for use within the dining services. The students have agreed to plant lettuce, potatoes, tomatoes, and soybeans. Table below summarizes for each group over the growing season the projected yield per acre, the student demand, the maximum university demand, and estimated profit per unit. Note that there is no student or university demand for soybeans. Rather, this crop would be sold directly to a commodity broker for profit. Formulate and solve the LP model that would enable the students to determine the number of acres they should allocate to each crop so as to maximize total profit. Assume that student demands are to be satisfied exactly. Profit is earned on crops sold to the university (over and above student demands) and on the soybean crop. Include the following conditions: a) Lettuce, potatoes, tomatoes, and soybeans require 5,000, 2,500, 6,500 and 3,500 gallons of water per acre, respectively, over and above natural rainfall during the growing season. Available water is limited to 40,000 gallons. b) Because of required crop rotation, the number of acres planted in soybeans can be no more than 50 percent of the combined total acres planted in the other three crops. c) Composted manure is to be used for fertilizer. The requirements are 2,500 lb. per acre for lettuce, 1,000 lb. per acre for potatoes, 3,000 lb. per acre for tomatoes, and 900 lb. per acre for soybeans; however only 40,000 lb. of fertilizer are available overall. d) Much as the graduate students and their families enjoy working the farm, academic demands (and other pleasures) limit total available labor to 1,000 hours over the growing season. Lettuce requires 75 hours per acre, potatoes 35 hours per acre, tomatoes 60 hours per acre and soybeans 90 hours per acre. You are additionally required to contact extended sensitivity analysis on the total available water, available manure, demand and on at least one of the objective function coefficients. 7
4. SOLID WASTE MANAGEMENT In years' past, the problems associated with the disposal of solid wastes (residues) from production processes were considered to be incidental to the problems associated with the production of the primary products. Recently, both the social and economic consequences of solid wastes have forced the explicit consideration of solid waste management. The following example clearly shows that decisions regarding the most efficient manner to produce a set of primary products completely interact with decisions pertaining to the efficient utilization of solid wastes. Α firm is involved in blending four final products as illustrated in Figure 3-2. The firm produces two primary products, each of which is manufactured through a chemical process that blends three raw materials. Table 3-8 shows how these materials must be blended in order to produce each of the two products. For example, it shows that at least 20 percent of the weight of product Α must be accounted for by material 1, at least 40 percent by material 2, and no more than 10 percent by material 3. Table 3-8 also shows the market prices for each product; assume that the firm. within its normal level of plant operations can sell as many units of each product as it wants without affecting these prices. 8
In order to guarantee the ready supply of materials, the firm has agreed to purchase certain minimal amounts of each material during each planning period. Moreover, the physical capacities of the firm's manufacturing facilities limit the amount of each material it can handle during the planning period. The upper and lower bounds on the amount of each material that can be processed each period, along with the unit cost of each material, are shown in Table 3-9. The nature of the manufacturing process is such that only a fraction of each material going into each product contributes directly to the primary products. The fraction not utilized in the primary products, known as the coefficient of waste, can be either (1) recycled (the material's chemical properties have changed as a result of the initial manufacturing operation) into a second related manufacturing process and turned into secondary products C and D or. (2) disposed of in part or in total at a specific cost to the firm. Table 3-10 shows the waste coefficients. For example, 10 percent of the amount of material 1 initially processed into product Α is left over as a residue, and 20 percent of that processed into product Β results in a residue. Assume that secondary product C can be blended by mixing any amounts of the residues from materials 1, 2, and 3 derived from product Α with the original (unprocessed) material 1, as long as the latter is exactly 20 percent of the mix by weight. Similarly, secondary product D can be blended by mixing any amounts of the residues from materials 1, 2, and 3 derived from product Β with the original material 2, as long as the latter is exactly 30 percent by weight of the mix. No waste occurs in the manufacturing process associated with the production of these secondary products. The net revenue per pound (after all associated secondary production expenses) for products C and D are, respectively, $0.60 and $0.10. Residue waste materials that are derived from the production of products Α and Β and that are not used to manufacture products C and D must be disposed of. Because the properties of the residue from different material-product combinations differ, the cost of disposing of each residue material not employed in a secondary product differs depending on the primary products from which it was derived. Table 3-11 shows these disposal costs. 9
Formulate the appropriate ΙΡ model to decide about a) how many units of materials 1, 2, 3 to purchase, b) how much of each material to allocate to primary products Α and Β, c) how much of each material to allocate to secondary products C and D, d) how much of each the residue materials obtained from producing products Α and Β should be disposed of immediately rather than recycled into products C and D. Contact sensitivity analysis on minimum and maximum material limits. 10
5. EVA'S DIET Η Εύα κάθε μεσημέρι γευματίζει σ ένα συγκεκριμένο εστιατόριο (γρήγορης εξυπηρέτησης). Λόγω βάρους κι επιπέδων χοληστερόλης πρέπει να ελέγχει τη διατροφή της (όπως και ο καθένας μας) και την τρέχουσα περίοδο είναι σε αυστηρή δίαιτα. Υπάρχουν δύο εναλλακτικές προτάσεις γεύματος (όχι αποκλειόμενες μεταξύ τους), ας τις ονομάσουμε γεύμα τύπου Α (π.χ. πιάτο με λευκό κρέας) και γεύμα τύπου Β (π.χ. πιάτο με ζυμαρικά), και πρέπει να διαλέξει τον κατάλληλο συνδυασμό από αυτές τις προτάσεις. Διαβάζοντας, πληροφορείται ότι μια μερίδα Α ζυγίζει 37 γραμμάρια, δίνει 120 θερμίδες και 5 γραμμάρια λιπαρά. Μία μερίδα Β ζυγίζει 65 γραμμάρια, δίνει 160 θερμίδες και 10 γραμμάρια λιπαρά. Επίσης, κάθε γεύμα έχει έναν δείκτη γευστικότητας από 0-100 για κάθε γραμμάριο φαγητού. Η Εύα, βαθμολογεί με 85 μονάδες γευστικότητας κάθε γραμμάριο γεύματος τύπου Α και με 95 μονάδες κάθε γραμμάριο γεύματος τύπου Β. Η συνολική γευστικότητα που απολαμβάνει προκύπτει από το άθροισμα των επιμέρους. Η Εύα, σε ημερήσια βάση, έχει περιθώριο να καταναλώσει το πολύ 450 θερμίδες και μέχρι 25 γραμμάρια λίπους ενώ πρέπει να πάρει τουλάχιστον 120 γραμμάρια φαγητού. a) Υποθέτοντας ότι μπορεί να καταναλώνει και κλασματικές μερίδες γεύματος, να διαμορφώσετε το κατάλληλο μοντέλο γραμμικού προγραμματισμού για να απαντήσετε στο ερώτημα «πόσες μερίδες κάθε τύπου πρέπει να τρώει η Εύα σε ημερήσια βάση έτσι ώστε να ικανοποιεί τους διατροφικούς περιορισμούς και να μεγιστοποιεί τις συνολικές μονάδες του δείκτη γευστικότητας». b) Χρησιμοποιείστε τη γραφική μέθοδο για να σκιαγραφήσετε την εφικτή περιοχή και να βρείτε τη βέλτιστη λύση και την άριστη τιμή του μοντέλου που διαμορφώσατε. Να εξηγήσετε με πληρότητα και σαφήνεια πώς προκύπτει ο χώρος των εφικτών λύσεων και πώς υπολογίζεται η άριστη τιμή του μοντέλου που διαμορφώσατε. Να διατυπώσετε με σαφήνεια τα αποτελέσματα της επίλυσης και να δώσετε τις ημερήσιες τιμές των γραμμαρίων συνολικής τροφής που θα καταναλώσει, των συνολικών θερμίδων καθώς και των γραμμαρίων συνολικού λίπους που προσλαμβάνει η Εύα στην άριστη λύση. c) Να χρησιμοποιήσετε κάποιο λογισμικό για να επιλύσετε αλγεβρικά το μοντέλο σας και να επιβεβαιώσετε την άριστη λύση και όλα τα άλλα αποτελέσματα του προηγούμενου ερωτήματος. Να συμπεριλάβετε στο αρχείο Word της εργασίας που θα παραδώσετε, εικόνες της λύσης και της αναφοράς ευαισθησίας. d) Αν το γεύμα τύπου Α είχε δείκτη γευστικότητας ίσο με 83 μονάδες ανά γραμμάριο, αλλάζει η βέλτιστη λύση; Αν ναι, βρείτε με τη χρήση του λογισμικού, πόσο είναι η νέα αυτή βέλτιστη λύση (και η νέα βέλτιστη τιμή). e) Αν το ανώτερο όριο ημερήσιων θερμίδων μειωθεί στις 440 θερμίδες, πώς μεταβάλλονται οι μονάδες γευστικότητας που απολαμβάνει η Εύα; Η απάντηση πρέπει να δοθεί χωρίς να επιλύσετε ξανά το μοντέλο σας, αλλά στηριζόμενοι στις αναφορές που έχετε ήδη. 11