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A Heuristic for a Class of Production Planning andScheduling ProblemsKENNETH C. GILBERT a & MANOHAR S. MADAN ba Management Science and Operations Management Department , University of Tennessee ,Knoxville, TN, 37916b Management Department , University of Wisconsin-Whitewater , Whitewater, WI, 53190Published online: 30 May 2007.
To cite this article: KENNETH C. GILBERT & MANOHAR S. MADAN (1991) A Heuristic for a Class of Production Planning andScheduling Problems, IIE Transactions, 23:3, 282-289, DOI: 10.1080/07408179108963863
To link to this article: http://dx.doi.org/10.1080/07408179108963863
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A Heuristic for a Class of ProductionPlanning and Scheduling Problems
KENNETH C. GILBERTManagement Science and Operations Management Department
University of TennesseeKnoxville, TN 37916
MANOHARS.MADANManagement Department
University of Wisconsin-WhitewaterWhitewater, WI 53190
Abstract: This paper describes a solution technique for a general class of problems referred to as aggregate planning and master scheduling problems. The technique is also applicable to multi-item single level capacitated lotsizing problems. The solution technique presented here is a heuristic that is practical for large problems e.g. 9products and 36 periods. We have tested it for problems with varying number of time periods, number of products,setup costs, holding costs, overtime costs and capacity levels. For those problems that we could solve exactly usinga branch and bound algorithm, the solutions produced by the heuristic were all within 1% of optimality. For problems that we could not solve exactly, we are able to compute a lower bound on the optimal cost. Using the boundwe are able to show that our heuristic solutions were within 2.93 % of optimality on the average. Except for thoseproblems having very high setup cost or problems with extreme seasonality, the algorithm produced solutions thatwere within 1% of optimality on average.
• This paper describes a computationally efficient heuristic for a class of production planning and scheduling problems. The problem is one of a general class of problemscommonly referred to as aggregate production planning andmaster scheduling problems.
The problem discussed in this paper differs from the general aggregate planning and master scheduling problem inone important aspect: it considers the aggregate productioncapacity (i.e. the work force size) for each period as an input, rather than a variable to be optimized. With this oneexception, this approach is capable of taking into accountthe major decision variables of intermediate range production planning e.g. setups, monthly or weekly production levels, inventory levels and shortage levels for each product,overtime, subcontracting and other means for temporarilyadjusting capacity. We refer to the problem as MultiproductCapacitated Production Planning Problem (MCPPP). (Aformal description of MCPPP is given in the following section).
MCPPP is found in several manufacturing settings. Plantswith shallow bill of material are good examples, e.g. as-
Received August 1988; revised August 1989. Handled by the Departmentof Scheduling/Planning/Contro1.
sembly processes, stamping processes, bottling operationsor breweries. Although the problem can be formulated asa mixed integer linear programming problem, exact solution approaches have proved impractical except for verysmall problems. The problem is shown to be NP-hard byMadan [17]. Heuristic approaches have been developed forsome special cases.
Many heuristics have been developed for variations ofMCPPP. Linear programming based algorithms were initiated by Manne [18]. Heuristics developed by Dzielinski,Baker and Manne [8], Lasdon and Terjung [16], were basedupon Manne's [18] approach. Bahl [1] and Baht and Ritzman [2] presented heuristics based upon Lasdon andTerjung's column generation algorithm.
Silver and Meal [21] presented a heuristic for a singleproduct problem. Eisenhut [9] developed a dynamic lotsizing heuristic based on marginal analysis of the holding costand setup costs. Lambrecht and Vanderveken [15] furtherextended Eisenhut's [9] heuristic by improving marginal costdetermination and feasibility assurance procedure. Dixonand Silver [6] developed a heuristic similar to the SilverMeal heuristic [21]. Dogramaci, Panayiotopoulos and Adam[7] presented a "greedy" heuristic which starts with a lotfor-lot schedule for each product and then attempts to irn-
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prove the schedule. Karni and Roll [12J presented a heuristic algorithm which started with the optimal Wagner-Whitinsolution [22].
A basis for the heuristic presented here is our observationthat MCPPP can be formulated as a type of fixed chargetransportation problem. The heuristic is based on a primalsimplex algorithm for the transportation problem. The pivot rule chooses the nonbasic variable whose entrance intothe basis brings about the largest decrease in the objectivefunction (which includes the fixed charges associated withsetups). The algorithm converges to a solution that is locallyoptimal in the ·sense that no further improvement in the objective function can be achieved by replacing a single variable in the basis.
Problem Description and Formulation
K = source of production capacity available;
T = number of time periods in the planning horizon;
DJ" = forecasted demand (expressed in units of capacity)ofproductj in period n. j=I, ....J, n=I, ... ,T;
S, = setup cost for product}. i- I , .. •J;
H, = holding cost in dollars per unit of productj per period, j=I •... ,J;
Pk = cost per unit of capacity source k = I , ... •K;
Ck ... = units of capacity source k available in period m,k=I, ....K, m=I, ... ,T;
Minimize
J K T T
+ E E E E [n-ml*[Xjkm"*Hll (I)j-I k-I ...-I.-...
J T J K T T
E E [SJ*lJ",] + E EE E [Xjk....*Pklj-I m--t jol l=-l m-Incm
lJ... = a binary variable where Yj ... = I and li... =0 signifyrespectively presence and absence of setup for jin period m, j=I, ....J. m=I, ... ,T.
(3)
(4)
(2)
m=I, ... ,T
m=I .... ,T
n=I, ...• T
k=l .... ,K
k=I, ... ,K
j=I, ... ,J
> 0
11 if t EXJkm"
k-I II sz m
o otherwise
J T
E E XJkm" :$ c....j-III-m
K •
E E XJk..." = Dink~1 ...~I
subject to:
Let Xjk..." be the quantity (expressed in units of capacity)of product} produced using capacity source k in period mused to meet the demand in period n, where, j=I .... ,J,k=I, ... ,K, m=I, ... ,T, and n=I, ... ,T.
The problem discussed in this research will hereafter bereferred to as PI. The fixed charge transportation formulation of the problem PI is given below.
In the Multiproduct Capacitated Production Planning Problem it is assumed that the planning horizon is divided intoT equal time periods (typically weeks). There are J different products. The demand DJ" , j = I •... ,J, n = I , ... ,T, forproduct j in period n is assumed to be known.
The products can be produced using k different sourcesof capacity (e.g. regular time and overtime). C«.... k= I , ... ,K,m= I , ... ,T, denotes the number of units of capacity (typically machine hours or labor hours) of source k availablein period m.
It is assumed that the number of units of capacity usedin the production of a given product is directly proportionalto the quantity produced. Thus for purposes of expositionwe will express all quantities in units of capacity required.For productj in period m. j=I .... ,J, m=I, ... ,T, the demand, the production level, and the end of period inventorylevel, will be expressed in units of capacity, i.e. one unitof a product is the amount that can be produced using oneunit of capacity.
The cost of capacity source k, k= 1, . .. .K, is P; dollarsper unit. PI :$ P~ :$ ... S Pk • For purposes of discussion,it will be assumed that backordering is not permitted. Thedevelopment for the case when backordering is permittedis, with the obvious changes, the same. Each product i.j= I , ... •J, has a holding cost of HJ (e.g. dollars per unitper period). Holding cost is based on the end of period inventory levels.
Each product requires a setup for each period in whichit is produced. The cost of a setup for productj is 5J dollarsper setup.j = I , ... ,J. It is assumed that downtime consumedby the setup operation is negligible.
Let} index product. k index source of production capacity, m index period in which product is produced and n index period in which product is demanded, j= I .... ,J,k=l, ... ,K, m=I, ... ,T. n=I, ... ,T.
Let,
J = number of products; .
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A Discussion of the Heuristic
The heuristic works with basic feasible solutions to thetransportation problem (1), (2), (3) and (5). (The associatedvalues of the setup variables are determined implicitly byconstraints (4).) The heuristic differs from standard primal
The problem (1)-(5) is a variation of the fixed charge transportation problem. It differs from most fixed charge transportation problems in that each fixed charge is associatedwith a group of cells, rather than a single cell or a singlerow or column. The supplies in the transportation tableauare denoted by C.... the quantity of source k production capacity available in period m, k=I, ... ,K, m=I, ... ,T. Thedemands are denoted by DJ• quantity ofproductj demandedin period n, j= I , ... ,J, n= 1, ,T (see Figure I). The rowsare ordered as (l, I), ... ,(K, I), ,(I ,T), ... ,(K, T). The col-umns are ordered as (I,I), ... ,(J,I), ... ,(I,T), ... ,(J,T).
The cell in row (k,m), 1 :s k :s K, 1 :s m :s T, andcolumn (j,n), I -s j -s J, 1 :s n :s T, will bedenoted «k,m),(j,n». The flow assigned to this cell corresponds to the variable XJb ••• and the value of the setup variable lj.. is implicitly determined by relationship (4). Hence a feasibletableau solution corresponds to a feasible solution to (1)-(5).Since backordering is not permitted, any cell «k,m), (j,n»with m > n will be infeasible.
transportation simplex algorithm only in the criterion usedfor choosing variables to enter the basis.
The heuristic starts with a basic feasible solution. At eachiteration it brings into the basis the nonbasic cell that willgive the maximum decrease in the objective function (taking into account the effect on both the continuous and thefixed charge portions of the objective function). The heuristic stops when there is no nonbasic cell that will improvethe objective function. The resulting solution is locally optimal in sense that it is better than any adjacent extreme point.
Computationally. the implementation of the heuristic issimilar to the primal network algorithm of Bradley, Brown,and Graves [4]. It differs only in the procedure for selectinga cell to enter the basis.
Consider a basic feasible solution to (1)-(5) and assumethat dual variables associated with the continuous portionof the objective function have been computed (temporarilyignoring the fixed charge portion of the objective function).
Next consider any nonbasic cell in this solution. If thiscell is brought into the basis, the rate of change, say W,of the continuous portion of the objective can be computeddirectly using the dual variables associated with the cell'srow and column. Furthermore the value (flow), say e, thatthe cell would assume if it were made basic, can be determined via the ratio test. The continuous portion of the objective function will change by w*e if the given cell isbrought into the basis.
The effect of the pivot on the fixed charge portion of theobjective function can also becomputed directly. When thecell enters the basis it will drive a nonbasic cell to zero.The pivot may also cause basic cells to go from positiveto zero or from zero to positive. These cells can be identified concurrently with the performance of the ratio test.A cell that becomes positive will require an additional setup
(5)
n=I, ... ,T
m=I, ... ,T
k=I, ... ,K
j=I, ... ,J
Product Demands in Periods DJ>
Period#1
Period#2
Period113
J J J
PeriodCapacity #1Available
inPeriods K
C•• Period113
K
· · ·· · ·· · ·
'Denote cells sharing a fixed charge
Figure 1. Tableau Associated with Fixed Charge Transportation Problem Pl.
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unless the setup is already required (i.e. there are positiveflows in one or more other cells associated with productionof the same product in the same period). On the other handan existing setup may be eliminated if all the cells sharingthat setup are driven to zero. Let M'denote the sum of thecosts of the new setups required minus the cost of the setupseliminated. M' is the change that will occur in the fixedcharge portion of the objective function if the given cell isbrought into the basis.
The steps of the heuristic are summarized in Algorithm I.
Algorithm 1
Step I: Begin with a basic feasible solution to (1)-(5).
Step 2: For each nonbasic cell compute:
fJ.Z = M' + W*8
where fJ.Z is the change in the objective function(I) that will occur if the given cell is brought intothe basis.
Step 3: If the value of fJ.Z is nonnegative for all nonbasiccells, stop. Otherwise bring into the basis the nonbasic cell which has the most negative value fort1Z. Go to step 2.
Procedure for Identifying a Starting Basis
The quality of the starting basis may affect the qualityof the final solution as well as the number of iterations required. The procedure for finding the initial feasible solution is essentially Vogel's Approximation method except thatthe "cost" of a cell is defined so as to take into accountthe fixed charge portion of the objective function.
Consider a transportation problem tableau having a partially completed basis as in the implementation of Vogel'sApproximation Method. Let i index the rows and letj indexthe columns. Let C1 represent the unused supply (capacity)for row i and let Dj represent remaining unmet demandfor column j. Let Au represent the coefficient of cell (i,j)in the continuous portion of the objective function. In addition, let Sij denote the fixed charge setup cost associatedwith cell (iJ).
For each nonbasic cell we wish to define R,j , a cost thatreflects the rate at which the objective function will increaseas the flow in cell (i,j) is increased when it is added to thebasis. If the setup cost associated with the cell has alreadybeen incurred (i.e. some cell sharing this setup is alreadybasic) this cost is simply:
On the other hand, if no cell sharing a setup with cell (i ,j)is basic, then when cell (i,j) is added to the basis there will
September 1991, lIE Transactions
be an increase of SIj to the fixed charge portion of the objective function. Since the value cell (i,j) will assume willbe the minimum of C/ and Dj the average rate of increase ofthe fixed charge portion of the objective function will beSljlMin (C"DJ. The average rate at which the objectivefunction increases when cell (i ,j) is added to the basis isthen in this case:
The costs R1j are used in a modified Vogel's Approximation Method and are updated at each iteration. Examination of these costs will reveal that they have someproperties that are desirable from an intuitive standpoint:(a) The costs tend to be smaller for cells for which setupalready exists, thus the algorithm will tend to take advantage of existing setups, (b) the cost of incurring setups tendto be smaller when D, is large; thus the algorithm will tendto do setups in periods of high demand.
One minor modification is made to the setup costs to takeadvantage of apriori knowledge about mandatory setups. Anyproduct that has a nonzero demand in period I will requirea setup in period I, assuming backordering is not permitted. Hence, the setup cost for these products in period 1is set to zero. This prevents the algorithm from schedulingsetups for the products in period 2, for example, withoutconsidering that a setup must occur in period 1.
Assuming backordering is not permitted, certain cells willbe inadmissible. These cells will be given very large costcoefficients. Should the algorithm terminate with any of theseinadmissible cells in the basis, (a condition that never occurred in our computational tests) the conventional transportation algorithm is used to drive these cells from the basis.A failure to drive these variables from the basis would bean indication of infeasible problem.
Using the costs Rij as described above, Vogel's Approximation Method is used to find a starting basis. The stepsof the method are summarized in Algorithm 2.
Algorithm 2
Step 1: Initially all cells are nonbasic. All rows and columnsare unmarked. For each column initialize C/ tothe associated initial supply (capacity) and for eachcolumn j initialize Dj to the associated initial demand.
Step 2: Compute the cost RIj for each nonbasic cell that isin an unmarked row and an unmarked column.
Step 3: For each unmarked column compute the penaltyas the difference in costs between the nonbasic cellsin that column having the smallest cost and the second smallest cost. (Ignore cells in marked rows.)Similarly compute the penalty for each column.
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Step 4: Identify a row or column having the largest penalty.In that row or column choose the cell (iJ) havingthe lowest cost. This cell will be added to the basis.AUocate a flow of XIJ = Min (CIoDj ) to thecell. Set C/ to C/ - XIJ and Dj to Dj - XIJ. EitherC/ or Dj will become zero. Mark row i if C/ o or mark column j if Dj = O.
Step 5: If all rows and columns are marked, stop. Otherwise go to step 2.
Design of Computational Experiments
The computational study consisted of the solutions of 243test problems. The test problems differ with respect to (1)the number of products, (2) the number of time periods,(3) the seasonality of demand, (4) the available regular timecapacity (as a percentage of average demand), and (5) thesetup. Each of these five problem parameters were variedover three values to generate 35 = 243 different parametercombinations on which the problems were based. The testdesign is similar to that of Graves [11].
The number of products for any problem can be 3, 6 or9. The number of time periods can be 12, 24, or 36. Theproduct demands for any problem can have either no seasonality, moderate seasonality or extreme seasonality. Thesetup cost for each product can assume three values-low,medium and high. The low cost is one fifth of the mediumcost, which is one fifth of the high cost. The holding costsare the same for each problem. The level of regular timecapacity can assume three values such that it can cover 80 %,100% or 120% of total demand over the planning horizon.A detailed description of the specification of the test problems is given in Appendix n.
Summary of Results
The test problems were solved on the IBM 4381 system.The computation time for our heuristic ranged from .94c.p.u. seconds to 189.01 c.p.u. seconds with the averagetime per problem being approximately 30 c.p.u. seconds.We discuss the performance of our heuristic with respectto (l) seasonality in demand, (2) setup costs, (3) tightnessof capacity, (4) number of products, and (5) number of periods.
The performance of the heuristic is evaluated by comparing the objective function value of each heuristic solutionwith lower bound on the optimal objective function value.Two relaxations for computing lower bound for the problem are described in Appendix I.
Tables I through 5 illustrate the effects of each of the 5parameters. In each table we have classified the 243 problems into 3 sets of (81 problems each) representing the 3settings of the given parameter. For example, in Table 1we classify the test problems by setup costs. Each point inTable I represents an average of 81 problems having the
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same setup cost, but covering all combinations of the otherfour parameters.
The average percentage difference (a.p.d.) between theheuristic solution value and the lower bound is computedfor each problem parameter. The a.p.d. across all 243 problems was 2.93 %. The percentage difference ranged from0.00% to 14.67%.
Not surprisingly the most significant effect on the difference between the heuristic solution and the lower bound wasdue to the setup cost parameter as shown in Table 1. Asthe setup cost increases the problem becomes more of a purecombinatorial problem, and is therefore more difficult tosolve. The lower bound, which depends heavily on the continuous portion of the objective function could also expectto get looser as the setup cost increases. The second mostsignificant effect on the difference between the heuristic solution and the lower bound was due to the seasonality parameter as shown in Table 2 as seasonality increases theaverage percent difference increases. We suspect that muchof this increase is due to increased looseness of the boundas seasonality increases. Tables 3, 4 and 5 illustrate thatthere are minor effects of number of products, number ofperiods or capacity restrictions on the difference betweenthe heuristic solution and the lower bound.
Table 1. The Effect of Setup Costs on AveragePercent Difference (A.P.D.)
Setup Cost
Low" Medium High··
A.P.D. .43 1.59 6.53
"Low Setup Cost = (Medium Setup Cost)/5··High Setup Cost c (Medium Setup Cost)·5
Table 2. The Effect of Seasonality on AveragePercent Difference (A.P.D.)
Seasonality
None Moderate -Extreme
A.P.D. 1.62 1.98 4.94
Table 3. The Effect of Tightness of Capacity on AveragePercent Difference (A.P.D.)
Tightness of Capacity
80% 100% 120%
A.P.D. 2.74 3.43 2.37
Table 4. The Effect of Number of Products on AveragePer~ent Difference (A.P.D.)
Number of Products
3 6 9
A.P.D. 3.34 2.67 2.53
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Table 5. The Effect of Number of Periods on AveragePercent Difference (A.P.D.)
Number of Periods
12 24 36
A.P.D. 2.82 2.87 2.65
The simultaneous effect of the two most important parameters, seasonality and setup cost, is shown in Table 6. Inthis table we have classified the problems into groups according to the nine different combinations of setup cost andseasonality. Each celJ in the table represents an average of27 problems sharing the same setup cost and seasonality butvarying over all combinations of the other three parameters.Table 6 suggests the following:
I. Problems with low setup cost can be solved to near optimality (within 1%) by the heuristic.
2. Problems with medium setup cost can be solved to nearoptimality unless there is extreme seasonality.
3. For problems with high setup cost, there is a large gapbetween the optimal solution and the lower bound.
Table 6. The Effect of Interaction of Setup Costs andSeasonality on Average Percent Difference (A.P.D.)
Setup Cost Seasonality
None Moderate Extreme
High 4.40 4.76 10.00Medium 0.46 0.84 3.45Low 0.00 0.39 0.97
Comparison of the Heuristic Solutions withExact Solutions
Obviously, the difference between the heuristic solution. and the lower bound is a pessimistic measure of performance. This difference is due to the looseness of the bound,as well as the suboptimality of the heuristic solution.
To gain a feel for the amount of this pessimistic bias thelooseness of the bounds would cause, we attempted to solveour smallest (3 products and 12 periods) test problems using an exact branch and bound algorithm. We were ableto solve 5 of the test problems exactly. The results are shownin Table 7. For these problems the heuristic solution differedat most 0.3% from the optimal solution. The percent difference between the heuristic solution and the lower boundwas at least double the percent difference between the heuristic solution and the optimal solution. Also the loosenessof the lower bound increased as the seasonality increasedand as setup cost increased. This suggests that the performance of the heuristic is considerably better than the resultsbased on comparisons with the lower bound would indicate.
September 1991, lIE Transactions
Table 7. Comparison of Solution Value to the HeuristicSolution Value for Problems Solved Optimally
(3 product 12 period cases)
Difference Differencebetween between
Season. Setup Capacity Heuristic Heuristicand Optimal and Lower
Solution Bound Value(percent) (percent)
None Medium 100% 0.00 0.634None Medium 80% 0.18 0.583Moderate Low 100% 0.02 0.797Moderate Low 80% 0.00 0.017Moderate Medium 80% 0.30 0.751
Appendix I: Lower Bound
Two procedures for computing a lower bound for the problem PI is described. The lower bounds are computed bysolving relaxations of the problem.
Procedure #1 for Computing a Lower BoundThe objective function (1) of problem (1)-(5) is to min
imize the sum of setup, holding and production costs. Alower bound on total cost for the problem, is computed bysolving two separate relaxations of the problem. In the first,a lower bound on the sum of setup and holding costs is foundand in the second a lower bound on production costs is found.The lower bound on the sum of setup and holding costs isdetermined by solving a relaxation of the problem with onlysetup and holding costs in the objective function. The second computes a lower bound on the production costs solving the problem with only production costs in the objectivefunction. ~, a lower bound on the objective value for theproblem, can be obtained by summing the lower bound onthe setup and holding costs and the lower bound on the production costs.
Lower Bound on Setup and Holding CostsThis is found by solving a relaxation of the problem (1)-(5).
If cost of production is set to zero the objective functionof the problem becomes one of minimizing the sum of setupand holding costs. The problem then reduces to a multiproduct capacitated lotsizing problem. Further if the production capacity constraints (3) are relaxed the problem reduces to several single-product uncapacitated lotsizingproblem. The Wagner-Whitin algorithm [22] can be usedto solve these single-product uncapacitated lot sizing problems optimally. A lower bound on the objective value forthe above relaxation can be obtained by minimizing the sumof setup and holding costs separately for each product usingthe Wagner-Whitin algorithm and summing the individualobjective values. This lower bound will be a lower bound(Z') to the objective of the relaxed problem and also alower bound on the sum of setup and holding costs of problem (1)-(5).
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Setup Cost: For each product a range (~. s) is specified as given below:
where ff,. is the multiplicative seasonality factor (see Table8) for product j in period n in problem set q. (Problem set#1 consists of problems with no seasonality in product demands, set #2 consists of moderate seasonality in productdemands and set #3 consists of extreme seasonality in product demands.)
Available Regular Time: e,m, the regular time capacityavailable in period m (for a twelve period problem) and for100% resource coverage is given by,
For each product j we set SJ by drawing x random numbers(x = 5) over the range ~,S) where SJ = ;:s.; SJ is themedium setup cost for product j for all time periods for alltest problems. To get the "low" setup cost we divide SJby five, while to get the "high" setup cost we multiply SJby five.
Product Demand: For each product j we draw one random number (dp ) each from the five ranges specified in Table 8. The demand for product j in period t is given by,
#3
100.0
200.0
m=I, ... ,T.
200.0
100.0
#1
50.0
150.0
Products
Table 8. Demand Ranges and Setup CostRanges for Products.
Product #1
a, 20 40 15 25 80
d" 40 60 25 65 120
s, 50 50 50 50 50- 150 150 150 150 150s,
Product #2
a, 80 15 40 40 55
d. 120 25 60 60 85
~ 100 100 100 100 100- 200 200 200 200 200s,
Product #3
~ 20 20 30 30 20 30 30 50 60 40
d,. 40 40 50 50 40 70 50 100 80 80
~ 20 20 30 30 20 30 30 50 60 40
- 40 40 50 50 40 70 50 100 80 80s,
Lower Bound on Production CostIf the setup costs and holding costs are set to zero in the
problem (1)-(5), it reduces to a standard transportationproblem. The objective of the reduced problem is to minimizethe production costs. The optimal objective value for thereduced problem provides a lower bound ~) on the production costs for the problem.
Hence, the summation of the lower bounds on the objective values of the two reduced problems gives a lower boundon the objective of the original problem, i.e,
Procedure #2 for Computing a Lower BoundAnother approach to computing a lower bound on total
cost for the problem again uses two separate relaxations ofthe problem. In the first a lower bound on the total fixed(setup) cost is computed and in the second a lower boundon the total variable (production and holding) cost is computed. The lower bound on the total setup cost is found bydetermining a minimum number of setups required for feasibility. The lower bound on total variable cost is computedby solving the relaxation of the problem with only production and holding costs in the objective function. ~, a lowerbound for the problem is computed by summing the lowerbound on total setup cost and the lower bound on the totalproduction and holding costs.
Regular Time and Overtime Costs: For all test problemsthe regular time cost, P" is 40.00 and the overtime cost,P3' is 60.00.
Holding Cost: The inventory holding cost is 4.00 forproduct], 4, and 7, 7.00 for products 2,5, and 8, and 6.00for products 3, 6, and 9.
z = Z' + Z".- -
Appendix H: Specifications for Test Problems
Lower Bound on Total Production and Holding CostsIf the setup costs are set to zero, problem (1)-(5) reduces
to a standard transportation problem. The objective of thereduced problem is then to minimize the total productionand holding costs. The optimal objective value for the reduced problem provides k' a lower bound on total variable(production and holding) costs for problem (1)-(5).
Hence another lower bound on total cost for problem PIcan be obtained by summing up ~ and !:,.
Lower Bound on Total. Setup CostFor each product a setup is required for each period in
which it is produced. Hence a lower bound on the numberof setups required is the smallest integer that exceeds thequotient of the total demand for the product and the capacity available per period. The lower bound on setup cost (~is obtained by sum over all products j the product of thesetup cost SJ and this lower bound on the number of setupsrequired for product j.
288 lIE Transactions, September 1991
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Table 9. Multiplicative Seasonality Factor
Problem Set#1 #2 #3
Time Product # Product # Product #Period 1 2 3 1 2 3 1 2 3
1 1.0 1.0 1.0 1.0 0.8 1.0 1.0 0.8 0.32 1.0 1.0 1.0 1.0 0.8 1.0 0.6 0.6 0.53 1.0 1.0 1.0 1.0 0.7 1.0 0.0 0.3 0.64 1.0 1.0 1.0 1.0 0.5 1.2 0.0 0.0 1.05 1.0 1.0 1.0 1.0 0.7 1.3 0.0 0.0 1.26 1.0 1.0 1.0 1.0 1.0 1.5 0.0 0.6 1.57 1.0 1.0 1.0 1.0 1.0 1.3 0.0 1.2 2.08 1.0 1.0 1.0 1.0 1.2 1.0 0.8 1.5 2.29 1.0 1.0 1.0 1.0 1.3 0.9 1.6 2.0 1.3
10 1.0 1.0 1.0 1.0 1.5 0.7 3.0 2.0 1.011 1.0 1.0 1.0 1.0 1.2 0.6 3.0 1.5 0.512 1.0 1.0 1.0 1.0 1.0 0.8 2.0 1.2 0.2
(C'mfor 24 and 36 period problems are computed similarly.)For the case of 120% coverage we multiply C,mby 1.20,while for 80% resource coverage we multiply C'mby 0.80.
AvailableOvertime. Cl m , the overtime capacity available in period m is given by,
m=I, ... ,T.
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Kenneth C. Gilbert is Chairman of the Management Science and Operations Management Program at the University of Tennessee. He holds aB.S. degree in Mathematics from Berea College. a M.S. in Mathematicsfrom the University of Tennessee, and a Ph.D. in Management Sciencefrom the University of Tennessee. His publications have appeared in lIETransactions. Applications ofManagement Science, Journal of the Amer
ican Society of Civil Engineers. Decision Sciences, Journal for the lassociation of Computing Machinery, Journal of Engineering Computing andComputer-Imegrated Matlu!aeturing Review. He is an associate editor ofNaval Research Logistics Quanuly.
Manohar S. Madan is an assistant professor in Management Departmentat the University of Wisconsin-Whitewater. He holds a bachelor's degreein Mechanical Engineering from M. S. University, India, a master's degree in Engineering Management from the University of Detroit. and aPh.D. in Operations Management from the University of Tennessee.
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