PRODUCT MIX PROBLEMS
Problem 1.1
A garment manufacturer has a production line making two styles of shirts. Style I requires 200 grams of cotton thread, 300 grams of o f dacron thread, and 300 grams of linen thread. Style II requires 200 grams of cotton thread, 200 grams of dacron thread and 100 grams of linen thread. he manufacturer makes a net profit of !s. 1".#0 on Style 1, !s. 1#."0 on Style II. $e has in hand an in%entory of 2& kg of cotton thread, 2' kg of dacron thread and 22 kg of linen thread. $is immediate pro(lem is to determine a production schedule, gi%en the current in%entory to make a ma)imum profit. *ormulate the + model. Solution
+et
x1 -
um(er of Style I shirts, x2 - um(er of Style II shirts. Since the o(/ecti%e is to ma)imise the profit, the o(/ecti%e function is gi%en (y a)imie - 1".#0 x1 4 1#."0 x2 Su(/ect to constraints5 200 x1 4 200 x2 6 2& 2&,000 7a)imum 8ty of 9otton thread a%aila(le: 300 x1 4 200 x2 6 2' 2',000 7a)imum 8ty. of ;a ;acron th thread a% a%aila(le: 300 x1 4 100 x2 6 22 22,000 7a)imum 8ty. of +ines thread a%aila(le: 7on=egati%ity constraint> x1, x2 < 0
Problem 1.2
A firm makes two types of furniture5 chairs and ta(les. he contri(ution for each product as calculated (y the accounting department is !s. 20 per chair and !s. 30 per ta(le. ?oth products are processed on three machines 1, 2 and 3. he time required (y each product and total time a%aila(le per week on each machine are as follows5
$ow should the manufacturer schedule his produc tion in order to ma)imise contri(ution@ Solution
+et
x1 -
um(er of chairs to (e produced x1 - um(er of ta(les to (e produced Since the o(/ecti%e is to ma)imise the profit, the o(/ecti%e function is gi%en (y a)imise - 20 x1 4 30 x2 Su(/ect to constraints5 3 x1 4 3 x2 6 3' otal time of machine 1>
# x1 4 2 x2 6 #0 2 x1 4 ' x2 6 '0 x1, x2 < 0
otal time of machine 2> otal time of machine 3> on=negati%ity constraint>
Problem 1.3
he A?9 manufacturing company can make two products 1 and 2. Bach of the products requires time on a cutting machine and a finishing machine. !ele%ant data are5 Product
9utting $ours per unit> *inishing $ours per unit> rofit !s. per unit> a)imum sales unit per week>
P 1
P 2
2 3 '
1 3 & 200
he num(er of cutting hours a%aila(le per week is 3"0 and the num(er of finishing hours a%aila(le per week is C10. $ow much should (e produced of each product in order to achie%e ma)imum profit for the company@ Solution
+et
x1 -
um(er of roduct 1 to (e produced x1 - um(er of roduct 2 to (e produced Since the o(/ecti%e is to ma)imise profit, the o(/ecti%e function is gi%en (y a)imise - ' x1 4 & x2 Su(/ect to constraints5 2 x1 4 x2 6 3"0 A%aila(ility of cutting hours> 3 x1 4 3 x1 6 C10 A%aila(ility of finishing hours> a)imum sales> x2 6 200 on=negati%ity constraint> x1, x2 < 0
Problem 1.4
A company makes two kinds of leather (elts. ?elt A is a high quality (elt, and (elt ? is of lower quality. he respecti%e profits are !e. 0.&0 and !e. 0.30 per (elt. Bach (elt of type A requires twice as much time as a (elt of type ?, and if all (elts were of type ?, the company could make 1,000 per day. he supply of leather is sufficient for only C00 (elts per day (oth A and ? com(ined>. ?elt A requires a fancy (uckle, and only &00 per day are a%aila(le. here are only D00 (uckles a day a%aila(le for (elt ?. Ehat should (e the daily production of each type of (elt@ *ormulate the linear programming pro(lem.
Solution
+et
x1 -
um(er of ?elt A to (e produced x2 - um(er of ?elt ? to (e produced Since the o(/ecti%e is to ma)imise the profit, the o(/ecti%e function is gi%en (y a)imise - .&0 x1 4 .30 x2 Su(/ect to constraints5 2 x1 4 x2 6 1000 otal a%aila(ility of time> otal a%aila(ility of leather> x1 4 x2 6 C00 A%aila(ility of (uckles for (elt A> x1 6 &00 A%aila(ility of (uckles for (elt ?> x2 6 D00 on=negati%ity constraint> x1, x2 < 0
Problem 1.5
r. Fain, the marketing manager of A?9 ypewriter 9ompany is trying to decide on how to allocate his salesmen to the 9ompanyGs three primary markets. arket=1 is an ur(an area and the salesmen can sell, on an a%erage &0 typewriters a week. Salesmen in the other two markets can sell, on an a%erage, 3' and 2# typewriters per week, respecti%ely. *or the coming week, 3 of the salesmen will (e on %acation, lea%ing only 12 men a%aila(le for duty. Also (ecause of the lack of company cars, ma)imum of # salesmen can (e allocated to market area 1. he selling e)penses per week per salesman in each area are !s. C00 per week for area 1, !s. D00 per week for area 2, and !s. #00 per week for area 3. he (udget for the ne)t week is !s. D#00. he profit margin per typewriter is !s. 1#0. *ormulate a linear programming model to determine how many salesmen should (e assigned to each area in order to ma)imise profits. Solution
+et
x1 -
um(er of salesmen to (e allocated to area 1 x2 - um(er of salesmen to (e allocated to area 2 x3 - um(er of salesmen to (e allocated to Area 3 Since the o(/ecti%e is to ma)imise the profit, the o(/ecti%e function is gi%en (y a)imise - 1#0 &0 x1> 4 1#0 3' x2> 4 1#0 2# x3> - ',000 x1 4 #,&00 x2 4 3,D#0 x3 Su(/ect to constraint5 A%aila(ility of total salesmen> x1 4 x2 4 x3 6 12 a)imum salesmen for area 1> x1 6 # C00 x1 4 D00 x2 4 #00 x3 6 D#00 ?udgeted e)penses> on=negati%ity constraint> x1, x2, x3 < 0
Problem 1.
An animal feed company must produce 200 kg of a mi)ture consisting of ingredients H1, and H2 daily. H1 cost !s. 3 per kg and H2 !s. C per kg. ot more than C0 kg of H1 can (e used, and at least '0 kg of H2 must (e used. *ind how much of each ingredient should (e used if the company wants to minimise cost. Solution
+et
x1 -
kg of ingredient X 1 to (e used x1 - kg of ingredient X 2 to (e used Since the o(/ecti%e is to minimise the cost, the o(/ecti%e function is gi%en (y inimise - 3 x1 4 C x2 Su(/ect to constraints5 otal mi)ture to (e produced> x1 4 x2 - 200 a)imum use of x1> x1 6 C0 )2 < '0 inimum use of x2> on=negati%ity constraint> x1 < 0
Problem 1.!
he A?9 rinting 9ompany is facing a tight financial squeee and is attempting to cut costs where%er possi(le. At present it has only one printing contract and, luckily, the (ook is selling well in (oth the hardco%er and paper(ack editions. It has /ust recei%ed a request to print more copies of this (ook in either the hardco%er or paper(ack form. rinting cost for hardco% er (ooks is !s. '00 per 100 while printing cost for paper(ack is only !s. #00 per 100. Although the company is attempting to economise, it does not wish to lay off any employees. herefore, it feels o(liged to run its two printing presses at least C0 and '0 hours per week, respecti%ely. ress 1 can produce 100 hardco%er (ooks in 2 hours or 100 paper(ack (ooks in 1 hours. ress II can produce 100 hardco%er (ooks in 1 hours or 100 paper(ack (ooks in 2 hours. ;etermine how many (ooks of each type should (e printed in order to minimise cost. Solution
+et
x1 -
um(er of hardco%er (ooks per 100> to (e produced x2 - um(er of paper (ack (ooks per 100> to (e produced Since the o(/ecti%e is to minimise the cost, the o(/ecti%e function is gi%en (y inimise - '00 x1 4 #00 x2 Su(/ect to constraints5 2 x1 4 x2 6 C0 inimum running of ress I> inimum running of ress II> x1 4 2 x2 6 '0 x1, x2 < 0on=negati%ity constraint>
Problem 1."
A medical scientist claims to ha%e found a cure for the common cold that consists of three drugs called , S and $. $is results indicate that the minimum daily adult dosage for effecti%e treatment is 10 mg. of drug , ' mg. of drug S, and C mg. of drug $. wo su(stances are readily a%aila(le for preparing pills or drugs. Bach unit o f su(stance A contains ' mg., 1 mg. and 2 mg. of drugs , S and $ respecti%ely, and each unit of su(stance ? contains 2 mg, 3 mg, and 2 mg., of the same drugs. Su(stance A costs !s. 3 per unit and su(stance ? costs !s. # per un it. *ind the least=cost com(ination of the two su(stances that will yield a pill designed to contain the minimum daily recommended adult dosage. Solution
+et
x1 -
Su(stance A x2 - Su(stance ? inimise - 3 x1 4 # x2 Su(/ect to constraints5 ' x1 4 2 x2 - 10 x1 4 3 x2 - ' 2 x2 4 2 x2 - C x1, x2 < 0
!equirement of drug > !equirement of drug S> !equirement of drug $> on=negati%ity constraint>
Problem 1.#
A pu(lisher of te)t(ooks is in the process of presenting a new (ook to the market. he (ook may (e (ound (y either cloth or hard paper. Bach cloth (ound (ook sold contri(utes !s. 2&, and each paper=(ound (ook contri(utes !s. 23. It takes 10 minutes to (ind a cloth co%er, and " minutes to (ind a paper(ack. he total a%aila(le time for (inding is C00 hours. After considera(le market sur%ey, it is predicted that the cloth=co%er sales will e)ceed at least 10,000 copies, (ut the paper(ack sales will (e not more than ',000 copies. *ormulate the pro(lem as a + pro(lem. Solution
+et
x1 -
um(er of (ooks (ound (y cloth x2 - um(er of (ooks (ound (y hard paper Since the o(/ecti%e is to ma)imise the profit, the o(/ecti%e function is gi%en (y a)imise - 2& x1 4 23 x2 Su(/ect to constraints5 10 x1 4 " x2 6 &C,000 otal time a%aila(le in minutes> inimum sales of cloth=co%er (ooks> x1 < 10,000 a)imum sales of hard paper (ooks> x2 6 ',000 on=negati%ity constraint> x2 < 0
Problem 1.1$
A company manufacturing tele%ision sets and radios has four ma/or departments5 chasis, ca(inet, assem(ly and final testing. onthly capacities are as follows5
he contri(ution of tele%ision is !s. 1#0 each and the contri(ution of radio is !s. 2#0 each. Assuming that the company can sell any quantity of either product, determine the optimal com(ination of output. Solution
+et
x1 -
um(er of tele%isions to (e produced x2 - um(er of radio to (e produced Since the o(/ecti%e is to ma)imise the profit, the o(/ecti%e function is gi%en (y a)imise - 1#0 x1 4 2#0 x2 Su(/ect to constraints5 3 x1 4 x2 6 &,#00 a)imum capacity in chasis department> C x1 4 x2 6 C,000 a)imum capacity in ca(inet department> 2 x1 4 x2 6 &,000 a)imum capacity in assem(ly department> 3 x1 4 x2 6 ",000 a)imum capacity in testing department> on=negati%ity> x1, x2 < 0
Problem 1.11
A tim(er company cuts raw tim(er=oak and pine logs into wooden (oards. wo steps are required to produce (oards from logs. he first step in%ol%es remo%ing the (a rk from the logs. wo hours are required to remo%e (ark from 1,000 feet of oak logs and three hours per 1,000 feet of pine logs. After the logs ha%e (een de(arked, they must (e cut into (oards. It takes 2.& hours for cutting 1,000 feet of oak logs into (oards and 1.2 hours for 1,000 feet of pine logs. he (ark remo%ing machines can operate up to '0 hours per week, while the cutting machine are limited to &C hours per week. he company can (uy a ma)imum of 1C,000 feet of raw oak logs and 12,000 feet of raw pine logs each week. he profit per 1,000 feet of processed logs is !s. 1,C00 and !s. 1,200 for oak and pine logs, respecti%ely. Sol%e the pro(lem to determine how many fee t of each type of log should (e processed each week in order to ma)imise profit. Solution
+et
x1 -
*eet of logs in 1000 feets> of oak to (e produced x2 - *eet of logs in 1000 feets> of pine to (e produced Since the o(/ecti%e is to ma)imise the profit, the o(/ecti%e function is gi%en (y a)imise - 1C00 x1 4 1200 x2 Su(/ect to constraints5 2 x1 4 3 x2 6 '0 a)imum machine hours for (ark=remo%ing> 2.& x1 4 1.2 x2 6 &C a)imum machine hours for cutting>
x1 6
1C x2 6 12 x1, x2 < 0
a)imum a%aila(ility of raw oak> a)imum a%aila(ility of raw pine> on=negati%ity constraint>
Problem 1.12
Jpon completing the construction of his house, r. Sharma disco%ers that 100 square feet of plywood scrap and C0 square feet of white pine scrap are in usa(le form for the construction of ta(les and (ook cases. It takes 1' square feet of plywood and 1' square feet of white pine to construct a (ook case. It takes 20 square feet of plywood and 20 square feet of white pine to construct a ta(le. ?y selling the finished products to a local furniture store, r. Sharma can realise a profit of !s. 2# on each ta(le and !s. 20 on each (ook=case. $ow can he most profita(ly use the left=o%er wood@ Solution
+et
x1 -
um(er of ta(les to (e produced x1 - um(er of (ook cases to (e produced Since the o(/ecti%e is to ma)imise the profit, the o(/ecti%e function is gi%en (y a)imise - 2# x1 4 20 x2 Su(/ect to constraints5 20 x1 4 1' x2 6 100 a)imum plywood scrap a%aila(le> 20 x1 4 1' x2 6 C0 a)imum white pine scrap a%aila(le> on=negati%ity constraint> x1, x2 < 0
Problem 1.13
A ru((er company is engaged in producing three different kinds of tyres A, ? and 9. hese three different tyres are produced at the companyGs two different plants with different production capacities. In a normal C hours working day, lant 1 produces #0, 100 and 100 tyres of type A, ? and 9, respecti%ely. lant 2, produces '0, '0 and 20 0 tyres of type A, ? and 9, respecti%ely. he monthly demand for type A, ? and 9 is 2,#00, 3,000 and D,000 units, respecti%ely. he daily cost of operation of lant 1 and lant 2 is !s. 2,#00 and !s. 3,#00, respecti%ely. *orm + odel to determine the minimum num(er of days of operation per month at two different plants to minimise the total cost while meeting the demand. Solution
+et
x1 -
o of days of operation in lant 1 x2 - o of days of operation in lant 2 Since the o(/ecti%e is to minimise the cost, the o(/ecti%e function is gi%en (y inimise - 2#00 x1 4 3#00 x2 Su(/ect to constraints5 #0 x1 4 '0 x2 - 2,#00 !equirement of type A>
100 x1 4 '0 x2 - 3000 100 x1 4 200 x2 - D000 x1, x2 < 0
!equirement of type ?> !equirement of type 9> on=negati%ity constraint>
Problem 1.14
wo products A and ? are to (e manufactured. A single unit of product A requires 2.& minutes of punch press time and # minutes of assem(ly time. he profit for product A is !s. 0.'0 per unit. A single unit of product ? requires 3 minutes of punch press time and 2.# minutes of welding time. he profit for product ? is !s. 0.D0 per unit. he capacity of the punch press department a%aila(le for these products is 1,200 minutesKweek. he welding d epartment has an idle capacity of '00 minutesKweek and assem(ly department has 1,#00 minutesKweek. 1. *ormulate the pro(lem as linear programming pro(lem. 2. ;etermine the quantities of products A and ? so that total profit is ma)imised. Solution
+et
x1 -
um(er of product A to (e manufactured x2 - um(er of product ? to (e manufactured Since the o(/ecti%e is to ma)imise the profit, the o(/ecti%e function is gi%en (y a)imise - .'0 x1 4 .D0 x2 Su(/ect to constraints5 2.& x1 4 3 x2 6 1200 A%aila(ility of punch press time> # x1 6 1#00 A%aila(ility of assem(ly time> 2.# x2 6 '00 A%aila(ility of welding time> on=negati%ity constraint> x1, x2 < 0
Problem 1.15
he HL company during the festi%al season com(ines two factors A and ? to form a gift pack which must weigh # kg. At least 2 kg. of A and not more than & kg. of ? should (e used. he net profit contri(ution to the company is !s. # per kg. for A and !s. ' per kg. for ?. *ormulate + odel to find the optimal factor mi). Solution
+et
x1 -
gs of roduct A x2 - gs of roduct ? Since the o(/ecti%e is to ma)imise the profit, the o(/ecti%e function is gi%en (y a)imise - # x1 4 ' x2 Su(/ect to constraints5 otal weight of the pack> x1 4 x2 - #
x1 <
2 x2 6 & x1, x2 < 0
inimum requirement of A> a)imum requirement of ?> on=negati%ity constraint>
Problem 1.1
An ad%ertising agency wishes to reach two types of audiencesM customers with annual income of more than !s. 1#,000 target audience A> and customers with annual income of less than !s. 1#,000 target audience ?>. he total ad%ertising (udget is !s. 2,00,000. Nne programme of O ad%ertising costs !s. #0,000 and one programme of radio ad%ertising costs !s. 20,000. *or contract reasons, at least 3 programmes ha%e to (e on O and the num(er of radio programme must (e limited to #. Sur%eys indicate that a single O programme reaches &,#0,000 customers in target audience A and #0,000 in the target audience ?. Nne radio programme reache s 20,000 in target audience A and C0,000 in the target audience ?. ;etermine the media=mi) to ma)imise the total reach. Solution E%%e&ti'e E()o*ure
+et
x1 -
um(er of ad%ertisement on ele%ision x2 - um(er of ad%ertisement on !adio Since the o(/ecti%e is to ma)imie the total audience, the o(/ecti%e function is gi%en (y a)imise - #,00,000 x1 4 1,00,000 x2 Su(/ect to constraints5 #0,000 x1 4 20,000 x2 6 2,00,000 otal amount a%aila(le> inimum ad%ertisement on O> x1 < 3 a)imum ad%ertisement on !adio> x2 6 # on=negati%ity constraint> x2 < 0
Problem 1.1!
N! *eed 9ompany markets two feed mi)es for cattle. he first mi), *ertile), requires at least twice as much wheat as (arely. he second mi), ultiple), requires at least twice as much ( arley as wheat. Eheat costs !s. 1.#0 per kg., and only 1,000 kg. are a%aila(le this month. ?arley costs !s. 1.2# per kg. and 1,200 kg. are a%aila(le this month. *ertile) sells for !s. 1.C0 per kg. up to "" kg. and each additional kg. o%er "" sells for !s. 1.'#, ultiple) sells at !s. 1.D0 per kg. up to "" kg. and each additional kg. o%er "" kg. sells for !s. 1.## ?harat *arms will (uy any and all amounts of (oth mi)es of N! *eed 9ompany. Set up the linear programming pro(lem to determine the product mi) that results in ma)imum profits. Solution
+et
x1 -
8uantity of wheat to (e mi)ed in *ertile) upto "" kg x2 - 8uantity of (arley to (e mi)ed in *ertile) upto "" kg x3 - 8uantity of wheat to (e mi)ed in *ertile) for a(o%e "" kg x& - 8uantity of (arley to (e mi)ed in *ertile) for a(o%e "" kg x# - 8uantity of wheat to (e mi)ed in ultiple) upto "" kg x' - 8uantity of (arley to (e mi)ed in ultiple) upto "" kg xD - 8uantity of wheat to (e mi)ed in ultiple) for a(o%e "" kg xC - 8uantity of (arley to (e mi)ed in ultiple) for a(o%e "" kg
C+l&ul+tion o% &ontribution )er unit
Since the o(/ecti%e is to ma)imise the profits, the o(/ecti%e function is gi%en (y a)imise - .30 x1 4 .## x2 4 .1# x3 4 .&0 x& 4 .20 x# 4 .&# x' 4 .0# xD 4 .30 xC Su(/ect to constraints5 78uantity mi) constraint: x1 < 2 x2 78uantity mi) constraint: x3 < 2 x& 7Sales constraint: x1 4 x2 6 "" 78uantity mi) constraint: x' < 2 x# 78uantity mi) constraint: xC < 2 xD 7Sales constraint: x# 4 x' 6 "" 7Eheat supply constraint: x1 4 x3 4 x# 4 xD 6 1000 7?arley supply constraint: x2 4 x& 4 x' 4 xC 6 1,200 7on=negati%ity constraint: x1, x2, x3, x&, x#, x', xD, xC < 0 Problem 1.1"
A company has three operational departments wea%ing, processing and packing> with cap acity to produce three different types of clothes namely suitings, shirtings and woollens yielding the profit !s. 2. !s. & and !s. 3 per meter respecti%ely. Nne=meter suiting requires 3 minutes in wea%ing, 2 minutes in processing and 1 minutes in packing. Similarly one meter of shirting requires & minutes in wea%ing. 1 minute in processing and 3 minutes in packing while one meter woollen requires 3 minutes in each department. In a week, total run time of each department is '0, &0 and C0 hours for wea%ing, processing and packing departments respecti%ely. *ormulate the linear programming pro(lem to find the p roduct mi) to ma)imise the profit. Solution
+et x1, x2 and x3 denote the num(er of meters produced of suitings, shirtings and woollens respecti%ely. he gi%en data can (e ta(ulated as (elow5
Since the o(/ecti%e of the company is to find the product mi) to ma)imise the profit, the o(/ecti%e function is gi%en (y a)imise Z - 2 x1 4 & x2 4 3 x3 Su(/ect to the constraints5 3 x1 4 & x2 4 3 x3 6 3'00 2 x1 4 x2 4 3 x3 6 2&00 x1 4 3 x2 4 3 x3 6 &C00 where x1, x2 and x3 < 0
a)imum wea%ing time> a)imum processing time> a)imum packing time> on=negati%ity constraint>
