Pacific-Asia Joural of Mathematics, Volume 5, No., Jauary-Jue 20 ROLL CUTTING PROBLEMS UNDER STOCHASTIC DEMAND SHAKEEL JAVAID, Z. H. BAKHSHI & M. M. KHALID ABSTRACT: I this paper, the roll cuttig problem uder ucertai demad with ow probability distributios is discussed. This stochastic problem is coverted ito a determiistic problem. The problem turs out to be that of liear programmig uder ucertai demad. A solutio procedure is preseted by exploitig the special structure of the problem i which there is o optimizatio problem i the secod stage as the first stage variables ad the values of the radom elemets uiquely determie the secod stage variables.. INTRODUCTION I this paper, we study the applicatio of stochastic programmig to the roll cuttig problem which is ecoutered quite frequetly i real life situatio. I may situatios a material is required to be cut ito pieces accordig to the demads. Some examples are the problems of cuttig the paper, glass, leather ad log etc. The obective is to miimize wastage while the requiremets are met. I this paper, the roll cuttig problem uder ucertai demad with ow probability distributios is discussed. This stochastic problem is the coverted ito a determiistic oe. The problem turs out to be that of liear programmig uder ucertai demad. A solutio procedure is preseted by exploitig the special structure of the problem i which there is o optimizatio problem i the secod stage as the first stage variables ad the values of the radom elemets uiquely determie the secod stage variables. Several authors have give liear programmig formulatios to this cuttig stoc problem, e.g., Gilmore ad Gomory (96, 963). May heuristic methods are developed to miimize the wastage i cuttig stoc problems such as, Coverdale et al., (976), Golde (976), Roberts (984). 2. STATEMENT OF THE PROBLEM UNDER UNCERTAIN DEMAND Give m rolls of legths l, l 2, l 3, l m. These rolls are to be cut ito stadard sizes of size s, s 2, s 3, s. The cost of the roll of legth s is c. Let the value of the roll of size s cut from i th roll be give by c s. The problem may be stated i the followig form: Keywords: Cuttig plae method, Determiistic equivalet, Ucertaity.
56 Shaeel Javaid, Z. H. Bahshi & M. M. Khalid Problem P : Fid x i, i =, 2,..., m;, 2,..., which maximizes Subect to m S = c s x () i = i s x l i =, 2,..., m (2) i i xi d, 2,..., (3) ad x i o-egative itegers (4) The d demad for the umber of pieces of size s is ot exactly ow, but there is a ow probability distributio p (d ) for each of them. The problem of cuttig the rolls ito stadard legth is formulated ad solved as the apsac problem i Gilmore ad Gomory (96, 963). Here we treat the problem where o each stadard legth there is a ucertai demad with ow discrete probability distributio. The problem turs out to be that of liear programmig uder ucertai demads. For its solutio we give it a two stage liear programmig uder ucertai demad, Madasy (962). Let us deote by x i, the umber of pieces of size s tae from the i th roll. The total supply of the pieces of size s is xi = x (say). Let a uder-supply of a pieces of size s icur a loss of f [e.g. it is purchased from the local maret ad supplied to the customer o loss]. Defie L () d x 0 for d x = f () d xfor d > x. (5) The expected pealty cost from the discrepacies i the pieces of size is the, L () x = {()} E L d {()} x = E f d x. (6) d d
Roll Cuttig Problems uder Stochastic Demad 57 The problem may be ow beig stated as follows: Problem P 2 : Maximize Subect to c x L () x (7) s xi li i =, 2,..., m (8) xi d, 2,..., (9) ad x i itegers 0. (0) Here x i are the first stage variables. These are to be determied before the demad d is actually ow. At the secod stage we do ot have to solve a optimizatio problem, sice the secod stage variables y = d x are uiquely determied for each x, whe x are observed. This fact is explicated i the ext sectio. 3. EQUIVALENT DETERMINISTIC PROBLEM Note that for each the oly possible values of d are itegers. Let these values be 0,,..., d. The expected pealty cost from the uder-supply is the d L ()()() x = f d x p d. () d = x Let us determie L (x ) for, 2,..., ad also the icremetal values L (x ) L (x + ) for each, where x rages from 0 to d. Observe that the sum of the icremetal values startig from x = d ad goig up to x equals L (x ). It is clear from () that L (x ) is decreasig fuctio of x. We may also prove that L (x ) is covex. Sice x are itegers, it is sufficiet to solve (Wager 977) that,
58 Shaeel Javaid, Z. H. Bahshi & M. M. Khalid L (( d ))()()(( x + L )) d x L d x L d x i.e. f (( d )) x + 2()(( f )) d x f d x or f 0 which is true. Thus, L () x = {()}()() E L d x = f D x p d = D (2) d D > x beig a covex liear combiatio of L (d x ) is also covex. From that fact that L (x ) is decreasig ad covex, it follows that the icremetal values L (x ) L (x + ) are decreasig. that The maximum possible demad that may occur is d s is greater tha the total availability Let d s li = a0 m i = ai. d s. We may assume. (3) For each piece tae from the slac there is a pealty associated with it. Let us choose the pieces from the slac such that the pealty is miimized. This is the same thig as to miimize. L () y (4) subect to y s = a0. Where y (o-egative iteger) is the umber of pieces of size s tae from the slac. The fact that L () y is covex ad decreasig may be exploited to apply a techique for the apsac problem (Saaty 970), for solvig the problem (4).
Roll Cuttig Problems uder Stochastic Demad 59 Let us put the problem i the followig form: Problem P 3 : Maximize d L d y = 0 (5) where d (6) subect to y s = a0 = 0 ad y = 0 or (7) d = 0 y = y The total umber of variables y is ( ) d + = N, say. We arrage the variables y accordig to the descedig values of s s + L d y L d y = 0 = 0. Problem P 3 may ow be solved by eumeratig the solutio vector i lexicographic orderig of the apsac problem with costraits (6) ad (7) of problem P 3 ad their liear obective fuctio i which the cost coefficiet of y s is {()(( L d ))} s L d s +. Note that i the optimal solutio, the sum of the costs associated with allocatig a 0 (ufilled demad) to the pieces of size s will equal the value of d L d y = 0. Let the optimal distributio of slac amogst various sizes be y,, 2,...,. After distributig the slac i this way we subtract y from d ad the solve the followig problem:
60 Shaeel Javaid, Z. H. Bahshi & M. M. Khalid Problem P 4 : Maximize Subect to m c xi (8) i = s xi = li i =, 2,..., m (9) xi = d y, 2,..., (20) ad x i itegers 0 (2) This is the determiistic problem of cuttig the rolls uder ucertai demad. 4. SOLUTION PROCEDURE FOR EQUIVALENT DETERMINISTIC PROBLEM The parameters of the Problem P 4 ca be arraged i the followig form as: Table Structure of the Problem P 4 Size (s ) s.... s Supply Slac y.... y a o c c l x : : : : : : : : : : c m.... c m l x m x Maximum c m.... c m demad x m x x We call the problem to be balaced if m l = d y ; otherwise we call it ubalaced. i i = The solutio procedure for the above problem P 4, with ow demad is give below, (Bari, 980).
Roll Cuttig Problems uder Stochastic Demad 6 Let us arrage the parameters of the problem P 4 as: l l... l 2 m s s2... s. (22) We fid the solutios to the equatios (9) by tur i =, 2,..., m, which also satisfy (20) ad (2). The solutio to the equatios i (9) are sought through lexicographic orderig. (a lexicographic orderig of a set of solutios is a orderig of the solutios accordig to the first compoets, ad if, there is a tie, the accordig to the secod compoets, ad so o, the solutio with the larger compoets beig larger). The first lexicographic vectors are tae to be the solutios. Note that the first lexicographic vectors may ot maximize the correspodig obective fuctios, but as stated a feasible solutio will serve our purpose ad we eed ot proceed the first lexicographic solutio. For i =, =, 2,..., m, we fid the o-egative itegers x,, 2,...,, such that s x = li (23) x () d y x,, 2,..., m (24) i i = We assume for the preset that the itegers exist which satisfy (23) ad (24). The followig procedure (Sasty 970) may be used for fidig the first lexicographic solutios x,..., x, =, 2,..., m. Step : Allocate the maximum possible to the left most cell, say, th, for which xi () d y. i = Step 2: Allocate the maximum possible of the remaider to the cell ext to the th, say *th, such that x * () * * i d y i =. This process is cotiued for the other cells i the th row util a slac less tha s is left. If the slac is zero, the the first lexicographic solutio to the th row is obtaied.
62 Shaeel Javaid, Z. H. Bahshi & M. M. Khalid Step 3: If the slac, say β, is ot zero, the the positive allocatio of the right most cell, say i th, is reduced by s t ad s t = β is ow distributed as i step ad step 2 i the cells other tha the i th. If a slac, say β 2, is still left the reduce the allocatio of the i th cell by 2s t ad 2s t = β 2 is ow similarly attempted to be distributed ad so o. The cotributio of the purpose i step 3 will yield a solutio, if oe exists. This is because the solutio of the i th row, which is the first lexicographic, gives more freedom for the solutio of the i + I st row as compared to the freedom give by ay other solutio. I step 3 of the above procedure oe may ot fid zero slac for some. I this case, i order to obtai a feasible solutio to the origial problem, we bac trac ad chage the solutio of the ( ) th row to the secod lexicographic solutio ad proceed with the remaiig rows as i step to 3 above. Every time we ecouter a solutio of a o zero slac i ay row we will have to bac trac ad proceed with the ext ad proceed with the remaiig rows lexicographic solutio i the previous row. Let the total umber of zero slac lexicographic solutios i the i th row be z. A exhaustive search for explorig a feasible solutio i the th row will require z z 2... z permutatios of the lexicographic solutios i the previous ( ) rows. However, the umber of zero slac lexicographic solutios is geerally ot large. Further, i the search of a feasible solutio for the + st row oe eed ot explore those permutatios of the lexicographic solutios i the rows which have already bee explored durig the search for a feasible solutio of the th row. If a feasible solutio exists the above search beig (implicitly) exhaustive, must yield it. For the maximum, oe should go through the other solutios i lexicographic orderig. However, as stated above, a feasible solutio suffices for the balaced case. Thus we may use the first lexicographic solutios. 5. NUMERICAL EXAMPLE Cosider the problem with the followig data. m = 3, = 4, s = (, 7, 4, 2) ad a = (a, a 2, a 3 ), where s ad a are vectors. The correspodig cost is give i the followig Table 2. Table 2 Costs of the Stadard Rolls 8.4 6.0 4.8 3.6 7.7 5.5 4.4 3.3 7.0 5.0 4.0 3.0
Roll Cuttig Problems uder Stochastic Demad 63 The demad d, d 2, d 3 ad d 4 are ucertai with their probability distributios as: We have P ( d = )( = P d2)( = = 3)( P d = 4) = P d = = 4 P ( d2 = )( = P d2)( 2 = = 3) P d2 = = 3 P ( d 3 = 4) =, P ( d3 = 5)( = P d6) 3 = = 2 4 P ( d 4 = 6) =, P ( d 4 = ) =, 2 2 m d s l = 0. i i = Let the losses from uder supply be as: f = f 2 =, = = 2. By usig equatio (3) we fid L (x ), ad the L (x + ) for all, 2,...,. We write values i the followig tabular form i Table (3): Table 3 Calculated Values of L (x ), ad L (x + ), ( ) x L (x ) L (x ) L (x + ) L 2 (x 2 ) L 2 (x 2 ) L 2 (x 2 + ) 0 5/2 f f 2 f 2 f 2 2 6/4 f 3/4 f f 2 2/3 f 2 3 2 3/4 f /2 f /3 f 2 /3 f 2 4 3 /4 f /4 f 0 0 5 4 0 0 0 0 6 5 0 0 0 0 7 6 0 0 0 0 8 7 0 0 0 0 x L 3 (x 3 ) L 3 (x 3 ) L 3 (x 3 + ) L 4 (x 4 ) L 4 (x 4 ) L 4 (x 4 + ) 0 9/4 3/2 2 5/4 /2 3 2 /4 9/2 4 3 7/4 7/2 5 4 3/4 /2 5/2 6 5 /4 /4 3/2 7 6 0 0 /2 /2 8 7 0 0 0 0
64 Shaeel Javaid, Z. H. Bahshi & M. M. Khalid Now we solve the problem Maximize d L d y = 0 subect to y s = a0 d = 0 ad y = 0 or The pealty fuctios are arraged i ascedig order of magitude from left to right. The allocatios i followig table for solvig the problem are doe by usig the rules for ordiary Kapsac problem. f f f 2 3 4 2 3 4 f f 2 2 3 f 3 4 f f 2 3 2 f s 7 7 4 7 4 4 Allocatio 4 4 4 4 f s 4 4 4 2 2 2 2 2 2 2 Allocatio 2 The solutio is 6 5 6 3 = 3 = 4 = y y y Now the solutio to the full problem ca be obtaied as show i Table (4) Table 4 Equivalet Determiistic Problem s 7 4 2 Supply Slac 0 0 2 0 8.4 6.0 4.8 3.6 20 0 7.7 5.5 4.0 3.3 30 2 0 2 0 7.0 5.0 4.0 3.0 43 2 2 5 Max. demad 4 3 6 7 The first lexicographic solutio with zero slac solutio i all the rows is show i the Table (5).
Roll Cuttig Problems uder Stochastic Demad 65 Table 5 Zero Slac Solutio i The rows 7 4 2 8.4 6.0 4.8 3.6 20 0 7.7 5.5 4.0 3.3 30 2 0 2 0 7.0 5.0 4.0 3.0 43 2 2 5 4 3 4 6 The total retur for this solutio is 82.2. The obective value for the zero slac solutios with the other orderig A K s of are listed below. It is observed that the obective value goes o decreasig as we move from the order A, A 2, A 3. Table 6 Optimal Solutio of the Problem S. No. Order of A K s Obective values for the zero slac A, A 2, A 3 82.2 2 A, A 3, A 2 82.8 3* A 2, A 3, A 583.2 4 A 2, A, A 3 80.8 5* A 3, A, A 2 83.2 6* A 3, A 2, A 83.2 The orders with asteris (*) sig are the discarded oes by some precedig orders. It may be oted that oly 3 of the above orders eed to be eumerated. The optimal solutio is the zero slac solutio obtaied from the order A, A 2, A 3 as preseted i Table (6). REFERENCES [] Bari A., Slicig Optimally the Rolls of Give Legths Ito the Stadard Sizes, Pure Applied Math. Scieces, 2(-2), (980), 8. [2] Gilmore P. C., ad Gomory R. E., A Liear Programmig Approach to the Cuttig Stoc Problem Part I, Operatios Research, 9, (96), 849 859. [3] Gilmore P. C., ad Gomory R. E., A Liear Programmig Approach to the Cuttig Stoc Problem Part II, Operatios Research,, (963), 863 888. [4] Golde B. G., Approaches to the Cuttig Stoc Problem, AIIE Trasactios, 9, (976), 265 274. [5] Coverdale I., ad Wharto F., A Improved Heuristic Procedure for a No-Liear Cuttig Stoc Problem, Math. Prog., 2, (976), 8-25.
66 Shaeel Javaid, Z. H. Bahshi & M. M. Khalid [6] Madasy A., Methods of Solutio of Liear Programs Uder Ucertaity, Operatios Research, 0, (962), 463-47. [7] Roberts S. A., Applicatio of Heuristic Techiques to the Cuttig Stoc Problem for Worshops, Joural of Operatioal Research Society, 35(5), (984), 369 377. [8] Saaty T. L., Optimizatio ad Related Exteral Problems, McGraw Hill Boo Compay, New Yor, (970). [9] Wager H. M., Priciples of Operatios Research with Applicatios to Maagerial Decisios, Pricetice Hall of Idia Pvt. Ltd., New Delhi, (977). Shaeel Javaid * & M. M. Khalid ** Departmet of Statistics & Operatios Research A.M.U., Aligarh, U.P., INDIA, E-mails: shaeel.d@operamail.com * mmhalid2007@yahoo.com ** Z. H. Bahshi Departmet of Statistics, Meelle Uiversity, Ethiopia, Africa. E-mail: bahshistat@gmail.com