Comparison of stochastic approaches to the transportation problem: a case study

152 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 41, NO. 2, MAY 1994

to Approaches Comparison of Stochastic the Transportation Problem: A Case Study

Stuart J. Deutsch, Minnie H. Patel, and Antonio J. Dieck Assad, Senior Member, ZEEE

Abstrect-The objective of this case study is to provide insight to practitioners about the methodology of using space-time autoregressive integrated moving average (STARIMA) class of models to formulate stochastic demand of the transportation problem. While providing insight, two other methods--expected value (EV) and stochastic approximation (SA)-are also employed to formulate demand. A comparative evaluation of the methods using brewery data for the distribution of four products from five production pladts to 64 distribution centers is presented. It is shown that the demand characterized by the STARIMA approach results in a lower total cost of transportation. This occurs because the STARIMA approach results in better forecasts. Based upon the case study, the cost analysis indicated that the STARIMA method when used without (with) updating resulted in a 9.49% (10.5%) increase in the company’s net profit as compared with the SA method. Similarly, the STARIMA approach when used without (with) updating resulted in an 11.36% (12.37%) increase in the net profit as compared with the EV method. For the STARIMA approach, computations for a large size problem are shown to be identical to those of the deterministic transportation problem given the demand forecasts. Extra computation effort for producing STARIMA forecasts are easily justified in terms of the increased profit margin.

Index Terms- Transportation Model, Expected Value (EV), SWhastic Approximation (SA), Space-Time Autoregressive In- tegrated Moving Average (STARIMA), Demand Forecasts.

I. INTRODUCTION

RANSPORTATION PROBLEMS frequently involve sys- T tems that vary across time and space. In particular, let zJ (t) for j = 1, 2, . . . , N be the observations of random variables Zj , representing demand, at N fixed points in space. These points may represent the cities of a state, in which ,z3 (t) is the total demand for city j at time t. The N locations are fixed for all t, and the process is observable or observed at discrete points in time with equal intervals between successive observations. It is also possible that the observation at time t represents an accumulation of some quantity from time (t - 1). The system described above is referred to as a space-time system [1]-[6], and the description of such a system with a stochastic model is referred to as space-time or spatial- dynamic modelling.

Several approaches have been advocated for incorporating stochastic demand in the transportation models. These include [7], [8] the autoregressive integrated moving average (ARIMA), exponential smoothing, weighted moving average, expected value (EV), and Wilson’s stochastic approximation (SA) methods. The objective of this paper is to provide insight to practitioners, through an example application, about the methodology and benefit of using the space-time autoregressive integrated moving average (STARIMA) class of models [ 11 to formulate stochastic demand of the transportation problem. While providing insight, two other methods EV and SA are also employed to formulate demand. A case study employing EV, SA, and STARIMA approaches to characterize stochastic demand is performed. For this purpose, the demand data of a Mexican brewery that distributes four products from five production plants to 64 distribution centers is used.

First, we describe the problem in Section 11. In addition, the deterministic formulation of the transportation problem is presented in this section. The three methods of stochastic demand representation are described in Section 111. In Section IV, we compare the forecasts obtained by using each method. For this purpose two separate but related measures, the sum of squares of errors and the sum of absolute deviations of errors, are used. The mean and the standard deviation of these measures are computed for each product. A procedure for obtaining the shipping schedule for each period of the planning horizon with forecasted demand values is given in Section V. In Section VI, we evaluate these methods in terms of their cost performances for each product. We calculate the percentage effective cost over the exact forecast cost, PCT, for different lengths of planning horizons period by period and on a cumulative period basis. The STARIMA approach of demand representation is shown to result in the lowest PCT as compared with the other two approaches. This is because the STARIMA approach takes into account the space- time autocorrelation structure of demand, unlike the other two approaches. In addition, the STARIMA also takes into account the seasonality patterns displayed by the demand data. The overall system analysis for the total cost is performed using each method, and their cost performances are compared in

Manuscript received September 20, 1992; revised March 1993. Review of Section and conclusions are presented this manuscriDt was urocessed bv Editor J. R. Evans. in Section VIII.

S. J. Deutsih is with the Depc. of Industrial Engineering and Information

M. H. Patel is with the DeDt. of Industrial and Svstems Eneineerine. Systems, Northeastem University, Boston, MA 021 15 USA. 11. PROBLEM DESCRIPTION

.zI

University of Wisconsin-Milwaikee, Milwaukee, WI 5<201 USA. .z

Universitv. San Antonio. TX 78228 USA.

The case study application that we consider in this paper concerns a Mexican brewery located in Monterrey, Mex- A. J. D. Assad is with the Dept. of Industrial Engineering, St. Mary’s

IEEE k g Number 9402000. ico, that distributes its four products from five plants to

0018-9391/94$04.00 0 1994 IEEE

DEUTSCH et al.: COMPARISON OF STOCHASTIC APPROACHES TO THE TRANSPORTATION PROBLEM 153

64 distribution centers. Month by month, a shipping schedule on the number of units of each product to be transported from each plant to each distribution center must be made.

A. Formation of Distribution Centers

When this company started its business, it had only a few customers and the shipping quantities for each of its products could be determined very easily. As the number of customers began increasing all around Mexico, the need for the sales people to regionalize the customers became apparent. The company gave a franchise to someone for distributing its products in each region. Thus, the distribution centers were formed. Based on the demand forecasts for each of these distribution centers and the shipping costs from the plants to the distribution centers, the company assigned the best combination of shipments from the plants to meet the demands subject to the capacity constraint of each plant.

The country is divided among 64 registered distribution centers from which one or more areas of customers are covered. The distribution centers are formed in such a way that the aggregated demand of all customer areas covered by each distribution center is to be more or less equivalent. The areas covered by all 64 distribution centers and the locations of the five production plants are shown in Fig. 1. From this figure, it is clear that the geographical area covered by each distribution center in the central and northwestem parts of the country is relatively small since each of them experiences greater demand.

GULF OF MEXICO

Fig. 1. A

B. Information and Data Available

The information and the data provided for this application problem by the company’s officials were the demand data history for the demand centers, the cost information, and plant capacities. This information is given below.

The demand data history was available for four products. These four products are: (1) Lata (355 ml or 12 fl. oz can), (2) Media (355 ml or 12 fl. oz bottle), (3) Caguama (1 liter or 1 quart gallon bottle), and (4) Cuarto (250 ml or 8 fl oz bottle). The demand observations of 43 time periods for each product were available for each distribution center. The demand units are in hectoliters.

The shipping cost information was not provided directly. Instead, the company provided the distances in kilometers from each plant to each distribution center and the transportation cost per hectoliter per kilometer for each product. To obtain the unit transportation cost from each plant to each demand center for each product we need to multiply the transportation cost by the corresponding distance in kilometers. The company officials were asked to estimate the inventorylshortage costs per unit heldshort. The estimates of the inventory costs per unit held were based only on the geographical locations of the distribution centers. The shortage costs per unit short were estimated as a function of the average distance of the distribution center from the five plants. The explana- tion was that it was easier to lose customers farther away from the production plants due to the greater competition likely in those regions. In addition, an image cost of 250.00 pesos per unit short was added. The plant capacities in


hectoliters for each type of product for each plant were supplied.

C. Deterministic Transportation Model

The transportation problem deals with the determination of the number of units of a good to be transported from each supply point to each demand point such that the total transportation cost is minimized and the demand and supply requirements in terms of the number of units of the good are satisfied. The mathematical formulation of the transportation problem with the deterministic demand and supply for product k is given below.

(3) j

x i j k 2 0 for all i and j (4)

where

C i j k is the unit cost of transportation from supply point i

B j k is the number of units of product k demanded at

S i k is the number of units of product k available at supply

and

x i j k is the number of units of product k to be transported

This formulation will be used later for developing the shipping schedules with the deterministic demand replaced by the appropriate forecasts for each method of representation of the stochastic demand considered in this paper.

to demand point j for product k,

demand point j ,

point i,

from supply point i to demand point j .

111. STOCHASTIC REPRESENTATION OF DEMAND In this section, three methods of representing stochastic

demand for solving the stochastic transportation problem are discussed. We describe each of these methods below.

A. The Expected Value Method

This method is the simplest of the methods considered here. In this method, a future realization of the stochastic demand is represented by its expected (mean) value. That is, in the transportation model (1)-(4) the deterministic demand B j k

is replaced by the mean value of the stochastic demand of the demand point j for product k. The transportation model (1)-(4) is then solved with the expected demand as the exact deterministic demand.

B. The Stochastic Approximation

In this method, a future realization of the stochastic demand is represented by the deterministic demand bik for each

product k at each distribution center j , and is determined by satisfying the following equations [ 8 ] :

F j k ( b j k ) = ( s j k - c j k ) / ( s j k + h j k ) for each j and k ( 5 )

for each j and k (6)

where

k,

k,

S j k is the unit shortage cost at demand location j for product

h j k is the unit holding cost at demand location j for product

C i j k is the unit cost of transportation of product k from

F j k is the cumulative distribution function of the demand

and

uj is the total number of supply points 5 minus the number of supply points that cannot reach the distribution center j (infinite cost of shipping from supply point i to distribution center j).

The transportation model (1)-(4) is then solved using b j k

production plant i to distribution center j ,

of product IC at distribution center j ,

as the exact deterministic demand B j k .

C. The STMIMA Method

The third method is a more structured characterization of demand. It gives consideration to spatial and/or temporal dependence of demand. This is done by describing the spatial and/or temporal stochastic demand process by the STARIMA class of statistical models. We briefly describe this class of models below. The STARIMA class of models, denoted by

), is defined in the vector form as [l] S T m M A ( p x l , ~ z , . . . , ~ , ~ qml,m2,... ,m,

lJ XI.

k = l 1 = 1 a m L

k=l1=1

with for a = 0

E[t(t)E(t + a)'] = otherwise

Here, p and q are the autoregressive and moving average orders, respectively; XI, and mk are the spatial orders of the kth order autoregressive and moving average terms, respectively; Adz(t) is the vector of dth order differences of the observations z(t); q!Jkl and d k l are the lcth order time lag and Zth order spatial lag autoregressive and moving average parameters, respectively; W(') is the Zth order weight matrix; and t(t) is a vector of random normal errors. For d = 0, the STARIMA class of models reduce to space-time autoregressive


moving average (STARMA) class of models with z( t ) replaced by %(t), where B ( t ) is the vector of deviation of ~ ( t ) from its mean vector p. For d = 1, we take the differences of the successive observations, for d = 2, we take the differences of the successive differenced observations, and so on. Whenever ~ ( t ) displays a seasonality pattern with a seasonal period s, the general class of multiplicative seasonal STARIMA models [3] must be used.

Future realization of the stochastic demand for this method is represented by the forecast obtained by using appropriate STARIMA model. The transportation model (1)-(4) is then solved by using this forecast as the exact deterministic demand Bjk for each product k.

Product 2:

i j j z ( t + T )

= Z j z ( t + T - 12) + ,h + 0.02797[5j2(t + 7 - 1)

- Z j z ( t + T - 13)] + 0.23267[Zj2(t + T - 2) - Z j z ( t + T - 14)] + 0.22088~j~(t + T - 1) - 0.49024[2jz(t + T - 12) - Z j z ( t + T - 24)] (8)

Product 3:

?j3(t + T ) = fij3 + 0.770342j3(t + T - 1) - 0.32079~js(t + T - 1) + 0.55615Zjs(t + T - 12)

Iv . COMPARISON OF FORECASTS + 0 . 1 6 4 3 4 ~ ~ ~ ~ ~ Z j ~ 3 ( t + T - 12) (9) Among the 43 historical observations of the 64 distribution

centers, 31 observations were used for modelling demand by using the EV, SA, and STARIMA approaches. The last 12 observations were used for testing how well each approach forecasted these 12 observations of the demand. In this section, first we briefly summarize the procedures for obtaining the forecasts for the periods 32 to 43 and then compare the forecasts obtained for each approach with one another.

A. Forecasts Obtained Using the EV Method

The forecasts ijk(31 -t T) for this method are simply the average values of the corresponding demand observations of the time periods 1 to 31 for each distribution center and for each product. That is, the forecast for product k at distribution center j is f i j k (which is the mean value of the demand observations of time periods 1 to 31) for each time period (31 + T ) of the planning horizon. The forecasts j i j k do not change with time.

B. Forecasts Obtained Using the SA Method

The forecasts f j k ( 3 1 + ~ ) for this method for each product k at distribution center j are the values b j k obtained from (5) and (6) for each time period (31 + T ) of the planning horizon. The distribution function F j k used for obtaining bjk is the normal distribution function with mean f i j k and standard deviation & j k

computed from the observations of the periods 1 to 3 1. Notice that in this case also the forecasts do not change with time.

C. Forecasts Obtained Using the STARIMA Method

The forecast functions associated with the fitted STARIMA models obtained by using the procedure discussed in the Appendix are given below for each product:

Product 1 :

fjjl(t + T ) = /;jl + 0.83972Zji(t + T - 1) - 0.43558~jl(t + T - 1)

+ 0 . 0 9 3 8 5 z ~ i j t ~ j 1 l ( t + T - 1)

+ 0.798955jjl(t + T - 12) - 0.13186~jl(t + T - 12)

+ 0 . 1 4 7 8 7 ~ w i j , ~ j i l ( t + T - 12)

j,

(7) j l

j 1

Product 4:

&(t + T ) = fij4 + 0.604385j4(t + T - 1) + 0.23746Zj4(t + T - 2) - 0.39751~j4(t + T - 1) + 0.40899Zj4(t + T - 12)

+ 0 . 2 5 5 2 ~ W ~ j 1 2 j 1 4 ( t + - 12) (10) i f

where

i j k ( t + T ) is the forecasted demand of product k for the T periods ahead of the forecast origin t and is equal to Zjk(t + T ) for T < 0,

j i j k is the mean value of the observations of product k at distribution center j ,

ji is the overall mean of the seasonally differenced series, zjk(t + T - T ) is the deviation of observation of demand

from its mean value at distribution center j for product k in time period t + T - T ,

and

Ejk(t + T - r ) is the error term for product k at distribution center j for period t + T - T. It is equal to zero for T > T .

The forecasts for each product have been obtained by assuming that the historical process remains unchanged in future and taking the conditional expectation of Zjk(t + T )

at time t:

ijk(t + T ) = E[Zjk(t + T) I Z ( t ) , Z ( t - I), . . -1 . For our case the forecast origin t is equal to 31. For each product, the forecasts obtained by using the STARIMA approach have the smaller absolute forecast errors for each period of the planning horizon at each distribution center than that for SA and EV approaches. Note that if no spatial autocorrelation exists among the observations of demand sites, then the STARIMA model reduces to n identical ARIMA models, where n is the number of demand sites. For example, the fitted STARIMA model for product 2 is a seasonal ARIMA model. However, if there is a lagged relationship among the observations over the space, the minimum mean square error

IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 41, NO. 2, MAY 1994

TABLE I SUM OF THE SQUARES OF ERRORS (FIRST ENTRY) AND THE SUM OF THE ABSOLUTE DEVIATION OF ERRORS (SECOND

ENTRY) FOR 1 TO 12 PERIODS OF THE PLANNING HORJZON ON A SINGLE F’ERIOD BASIS FOR EACH PRODUCT

- Period

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 I1 11 12 12 - -

period 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 I1 12 12

STARIMA 4837930.61

9242.96 3430693.72

7958.67 lo660099.60

10819.53 35749.07 6125.19

3125082.24 8066.91

2658849.14 6713.86

4066919.80 6742.1 1

14479551.50 11863.20

7574618.22 11821.97

1968085.64 6074.12

6926625.90 8751.33

2826855.94 7657.99

STARIG 913918.81

4355.13 1567789.47

4988.04 4248158.42

5680.55 3696510.02

5042.48 1088172.00

4540.07 644427.97

3146.10 1181018.41

3668.59 5207280.09

5615.98 1376220.66

5105.50 732946.21

3944.37 3097303.70

5800.08 4229 11 2.75

6231.33

of the forecast function of the STAR

roduct 1 SA

306571222.00 83499.87

300250851.00 74397.74

452576632.00 101962.03

222382753.00 65828.06

509785139.00 93706.14

249639060.00 76268.25

58822609.00 40497.63

1274585%0.00 131206.41

874253837.00 141595.22

69149884.60 47918.37

293634852.00 76625.96

2144221810.00 76318.91

roduct 2 EV

68083172.4 39341.43

61810597.50 37547.96

50245441.70 37974.42

166018535.00 47640.53

136865077.00 48398.01

388079458.00 61743.82

148863434.00 43914.73

199237529.00 48884.60

87446262.40 44257.88

143450827.00 42571.31

55639058.80 36400.67

107375156.00 47206.68

EV 385856405.00

94540.74 369355941.00

84652.07 549478809.00

113939.68 18221805 1 .OO

57000.03 601013669.00

104213.39 210628695.00

67096.58 60313086.70

42724.54 1423406570.00

142916.07 1009092080.00

155372.20 81614965.80

51634.19 342786682.00

82931.78 247848731.00

80467.39

SA 97228294.40

44666.84 88601295.10

42027.84 57273744.90

41549.58 140406009.00

46532.32 120207727.00

48474.77 324937190.00

57060.61 107 16809 1 .OO

39860.58 212668435.00

50171.19 100747976.00

48130.29 138291548.00

44554.87 79474234.60

39710.00 134868893.00

50724.51

rlA model would be less than the average mean square error of the forecast functions of n AFUMA models.

After calculating the forecasts for 1 to 12 periods of the planning horizon (periods 32 to 43), we evaluate the forecast measures of performance. For this purpose two separate but related quantities, the sum of squares of errors (SSE) and the sum of absolute deviations of errors (SAD), are used. These measures of forecast errors are computed for 1 to 12 periods of the planning horizon, period by period. They are presented in Table I for each product. From this table, we observe that for SA, the SSE of each period in the planning horizon is about five to 600 times and for EV, it is about 13 to 500 times that of STARIMA. Similarly, for SA and EV, the SAD of each period in the planning horizon is about six to 20 times that of STARIMA. This shows that the multiplicative STARIMA model is superior than the other two for all four products’ forecasts. The mean and the standard deviation of forecast errors for planning horizons of different lengths, beginning with two periods and ending with twelve periods, are presented in Table 11. These values are very small for the multiplicative

- period

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 - -

period 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 -

STAF

STARIMA 2226160.09

7163.39 1646399.26

6370.06 3751043.76

9329.44 1928475.51

6585.11 2666959.94

7447.37 1095402.36

5589.51 873949.30

5144.03 2954422.32

7830.96

8072.14 2397242.34

7242.03 2901846.23

8099.21 2773635.87

7852.69

STARIMA 684914.66

2260.82 134068.64

1294.77 1025040.75

2666.74 119922.61

1208.09 314996.69

1713.33 359788.11

1665.03 320738.84

1483.53 662028.19

2166.91 686383.73

2121.35 141001.72

1255.69 372761.97

1615.85 254379.29

1490.41

2325720.66

d u c t 3 SA

82270683.50 46485.88

97487224.90 46372.11

182024%8.00 62801.11

299107980.00 81640.18

142844448.00 47022.77

334822445.00 86949.28

196231066.00 65948.87

20247961 12.00 60072.76

171748801.00 64560.77

100480984.00 49834.77

123951345.00 55703.52

97607988.10 52516.02

mdUCl4 SA

16284634.40 16780.75

12655242.80 15512.70

291 12134.10 18298.70

25265144.30 21613.75

68806031.20 25328.67

40126111.30 23617.57

203 15478.90 2n364.73

31496361.90 20621.51

44811081.00 22620.03

48764787.50 232050.7 1

19442940.80 19733.06

33325183.90 22929.86

EV 116941 135.00

58149.78 119667855.00

5581 1.65 202060477.00

68381.29 235392305.00

72123.87 150940950.00

54447.45 230507412.00

67242.55 126741585.00

50922.06 282798882.00

73535.94 251980795.00

83395.81 109808092.00

52377.97 188960391.00

68232.77 162468038.00

63265.71

EV 16403659.50

17031.77 12206447.80

14821.97 30813969.00

18249.42 23246146.40

19856.71 73 175039.60

25828.10 38123726.40

22053.74 18728127.90

19284.52 33528797.40

20483.81 4725145.60

22457.58 50577664.00

22595.78 19067311.30

19239.97 33908753.80

22153.52

YIA model versus the S , and EV models as can be seen from this table. This is expected from the results shown in Table I. The minimum average for both statistics is at period 7 with the standard deviation reasonably small compared to the other planning horizons for all products except product 2. This indicates that the planning horizon consisting of 7 periods is a reasonably good planning horizon with the available information.

Now that the forecast analysis is performed, the shipping schedule for each period in the planning horizon is determined by using the demand forecast values obtained using each method. This is done in the next section.

v. PROCEDURE FOR OBTAINING THE SHIPPING SCHEDULE FOR EACH TIME PERIOD ( T + T ) OF THE PLANNING HORIZON

For each product k, 2jjk(t + T ) is the forecast of the demand at each distribution center j in time period (t + T )

of the planning horizon. The shipping schedule in time period (t + T ) of the planning horizon is obtained by solving the transportation model for each product k. The transportation model (1)-(4) now can be rewritten for each time period ( t + ~ )


4625.40 858.78

1905713.59 1447771.16

4488.71 863.34

2318409.40 177399.65 4629.62 893.15

2213721.76 1692008.08

4682.50 850.40

2065644.21 1662546.82

4608.68 835.05

2159431.43 1607610.72

4716.99 869.83

233 1904.88 1645 123.3 1

4843.19 937.51

TABLE II MEAN (FIRST ENTRY) AND STANDARD DEVIATION (SECOND ENTRY) OF THE SUM OF SQUARES OF FORECAST ERRORS, AND MEAN (THIRD ENTRY) AND

STANDARD DEVIATION ( F o m ENTRY) OF THE SUM OF ASSOLUTE DEVIATION OF FORECAST ERRORS FOR DIFFERENT PLANNING HOR~ZONS

45441.03 9323.29

145735102.30 116467266.17

45222.99 8530.49

152422905.60 109474354.77

45680.69 8003.10

145203278.58 104669299.16

45522.60 7501.22

145028033.44 98684717.65

45227.47 7133.53

136901763.02 97422816.26

61425.03 7271.97

13444 1212.45 93279258.18

44656.84 6979.89

Roduct 1 Period I STARIMAl SA

2 I 4134312.16 1 303411036.48

6 6 7 7 7 7 8 8 8 8 9 9 9 9 10 10 10

2 2 2 3 3 3 3 4 4 4 4 5 5 5

-

-

-

1704.51 13324.49 4622497.50 300004038.12 2751375.45 149476794.37

7952.75 76594.25 1645.02 20032.63

5854629.25 421826778.16 4316690.62 371318798.69

8441.56 83420.77 2056.93 26772.89

6045739.14 472096451.34 4078393.99 378663840.83

8817.16 89884.59 2229.75 31673.63

5637973.79 431801794.67 4055598.50 379066071.18

8542.85 85687.97

1 1 1 1

5755130.04 419241163.49 3867049.62 36201R5h5 41

of the planning horizon as follows:

where

Minimize

Subiect to

TC = );-P,y,cijkxijk(t + T ) i j

11667590.20

j for i = 1, 2, 3, 4, 5

for all i and j xC;jk(t -k T ) 2 0

87524.12

99573092.26 97710.83

150238265.72 87533.13 23713.45

383091928.17 170566515.24

86907.08 21819.79

336980665.10 197807617.41

80595.29 25992.76

472783903.36 425532979.11

88385.39 32628.07

532373700.71 436350801.89

95828.37 37816.64

487297821.22 435390153.50

91408.95 38295.05

474160450.35 415339168.12

90638.30 36419.68

455301140.42 401363026.42

89790.72 24849.71

C;jk is the unit cost of transportation from production plant i to distribution center j for each product k,

xijk(t + T ) is the number of units of product k to be transported from production plant i to distribution center j in time period (t + T) ,

and

Aik(t + T ) is the plant capacity of plant i for product k in time period (t + T ) .

- ‘eriod 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 9 10 10 10 10 11 I 1 11 1 1 12 12 12 12

_.

-

-

~

_.

-

-

-

-

-

-

-

Pmduct 2

662.93 936.94

EV 92914794.73 6000209.73 43347.34 1865.06

81034444.77 21024614.74

42748.08 1678.81

95817335.80 34291912.66

43694.14 2336.46

10074% 14.03 31628231.72

44650.27 2943.66

138109043.35 95798839.04

46718.66 5709.78

133688907.25 88230443.67

45738.93 5821.25

143561348.20 86326384.14

46292.96 5612.63

138804306.84 820023 10.98

46497.11 5285.74

138753030.92 773 12690.26

46302.89 5021.15

133364049.44 75491583.66

45703.53 5161.61

133489453.04 71979711.71

46121.95 6130.00

For the EV method, . i j k ( t + T ) is b j k for each time period (t + T) , each product k, and each distribution center j. Similarly, for the SA method, i j k ( t + T ) is b j k , and the STARIMA method uses the forecast functions defined by the (7)-(10) depending upon the product number. The transportation model (11)-(14) is then solved by using the forecasts obtained from the EV, SA, and STARIMA methods for each period in the planning horizon and for each product k. The shipping schedules are thus obtained for each product and for each period in the planning horizon. Since we have the actual demand values for each period of the planning horizon, the holding and shortage costs can be calculated. The total effective cost for each product over the planning horizon is the sum of the transportation cost and the corresponding holding and shortage costs over the planning horizon.

VI. COMPARISON OF COST RESULTS USING DIFFERENT METHODS

In this section, the total effective cost of the EV, SA, and STARIMA formulations for each period and on a cumulative basis will be presented. In addition, the distribution centers’ percentage contribution to the total effective cost will be evaluated to classify them by their percentage contribution. We also calculate the percentage effective cost over the exact

158

EV ,

118304495.16 1928081.78

56980.71 1653.31

146223155.69 48375753.92

60780.90 6685.14

168515443.11 59564475.08

63616.65 7871.46

165000544.54 52179666.33

61782.81 7955.17

175918355.85 53790014.72

62692.76 7456.26

168893102.85 52503498.25

61011.24 8131.58

183131325.27 63123931.08

62576.82 8734.13

190781266.33 63350187.25

64890.04 10719.53

182683948.92 64984578.70

63638.84 10853.39

183254534.56 61678822.70

64056.47 10389.17

181522326.50 591 13791.35

63990.07 9908.32


4 4 4 5 5 5

TABLE II (continued)

939009.20 7362.00 1353.73

2443807.71 822718.21

7379.07

Period I STANMA 2 I 1936279.68

4 5 5 5

I 2 I 409952.81

720.17 455788.67 391369.01

1828.75

6766.72

5 6 6 6

I 3 I 1087115.55

1172.99 2219073.49 918980.43

7080.81

7620.96

9 10 10 10

489.50 444888.39 304365.44

1783.63

1278.47

980943.80 6804.13 1377.66

2142851.57 965567.48

693248

905260.28

10 7077.40 1224.51

7170.30 1201.83

7227.16 1162.71

duct 3 SA

89878954.23 10759712.58

46428.99 80.44

120594292.14 53741806.18

51886.37 9452.61

165222713.99 99459733.74

59324.82 16759.78

160747060.75 86714106.91

56861.41 15522.11

189759624.77 105194307.34

61878.55 18536.41

1906841 16.45 96059971.07

62460.03 16991.14

192448115.87 89074096.85

62161.62 15753.37

190148191.98 83606382.76

62428.19 14757.61

181181471.18 83769793.86

61168.85 14472.32

175978732.46

60672.W 13828.18

169447837.09 8077 1327.97

59992.34 13393.22

8 1322776.15

forecast cost for different lengths of planning horizons period by period and on the cumulative period basis. The exact forecast cost is obtained by solving the transportation model (11)-(14) with .2jjk(t + T ) replaced by the exact demand for each time period (t + T ) in the planning horizon. We then add the optimal costs so obtained for each time period to obtain the exact forecast cost over the planning horizon. There are no holding and shortage costs involved since we are using the actual demand.

The total effective cost obtained by using each method, period by period, is presented in Table 111. They reflect similar results to what has been disclosed in the error analysis of the previous section. That is, the advantages of the STARIMA method of formulation of demand over the other methods in the effective cost are seen from the forecast accuracy.

A. Contribution of the Distribution Centers to the Total Effective Cost

From all the regions involved in the spatial system of this problem application, there are some of the distribution centers that contribute high percentage to the total effective cost, some contribute less, and others contribute uniformly under a given demand characterization. The classification of the distribution centers based on their percentage contribution to the total effective cost is shown in Table IV for each demand characterization and for each product. From this table, it is

409491.65 389506.%

1777.80 683.09

614674.69 449619.91

2074.11 704.78

4 490986.67 4 442681.54 4 1857.61

627.01 439788.58 352238.25

1801.46 564.79

42278 1.47 32468 1.14

1756.05 529.40

452687.31 312270.76

1807.40 511.20

478653.58 302311.30 f 1842.29

497.39

289564.18 1768.37 474.57

1745.21 459.54

ducl4 SA

14469938.57 2566367.42

16140.73 896.65

19350670.40 8646256.45

16864.05 1394.87

20829288.87 7654002.26

18051.47 2633.82

30424637.33 22456432.78

19506.91

3204 1549.65 20472409.02

20192.02 3930.86

303663%.68 19207010.13

20216.7C 3588.96

30507642.34 17786723.17

20267.31 3325.81

32096913.31 17307617Ai

20528.71 3208.34

33763700.7? 17147935.31

20796.31 3141.0:

32461813.4t 16831236.8(

20699.6! 2997.0:

32533761.M 16049889.9:

20885.5: 2929.11

3974.m

EV 14306053.65 2987876.92

15926.87 1562.57

19808025.43 9759725.50

16701.05 1737.50

20667555.67 8152095.65

17489.97 2121.83

31169052.45 24520392.95

19157.59 4157.10

32328164.77 22114723.26

19637.28 3899.46

30385302.37 20832038.33

19586.89 3562.20

30778239.25 19318715.17

19699.00 3313.17

32608673.29 18886915.05

20005.51 3232.72

34405572.36 18691414.09

20264.54 3155.99

3301 1184.99 18325379.72

20171.40 3009.93

33085982.39 17474485.07

20335.57 926.34

clear that when STANMA characterization is used for the demand, there are more distribution centers with contribution to the total effective cost closer to the mean contribution (1.5625%) than when other two methods are used. This is mainly because of the lack of adequate representation of the system demand process by the EV and SA methods which makes the errors very unstable and results in the unstable cost contributions of some system distribution centers. The high contribution at certain distribution centers could be the effect of high demand areas where the number of units short or held obtained by any forecasting method would be high. This would result in higher costs for such distribution centers as compared to other distribution centers. This analysis allows us to identify distribution centers with high contributions to the total effective cost. These distribution centers may be closely monitored and if some other control measures are undertaken, their percentage contribution to the total effective cost may be reduced. Next, we compare effective cost with the exact forecast cost for all three methods.

Effective Cost Versus the Exact Forecast Cost for Different PlanningHorizons

over the exact cost by PCT. It is defined as Let us denote the percentage change in the effective cost

PCT = [(effective cost - exact cost)/exact cost] x 100.


1 2 2

P 3 L 3 A 4 N 4

TABLE III TOTAL EFFECTIVE COST OF THE ST-A, SA, AND EV METHODS FOR DIFFERENT PLANNING HORIZONS ON A SINGLE PERIOD AND ON A CUMULATIVE PERlOD BASIS

18425009.68 1651o900.06 34935909.74 19145179.03 54081088.77 12094786.61 66175875.38

41825825.55 46079921.16 37153323.18 41216471.81 78979148.73 87296392.97 48029818.03 52473739.25 127008966.80 139770132.20 20104520.51 20226078.46 1471 13487.30 159996210.70 N

N I N G

H

4 182510761.40 261931347.50 287702146.70 5 48588735.21 64330032.41 71796171.61 5 231099496.60 326261379.90 35’3498318.30 6 34538528.% 54347843.88 51363766.31 6 265638025.60 380609223.80 410862084.60 7 36783402.98 52342983.44 51091248.41 7 302421428.60 432952207.20 461953333.00

0 N

L 3 A 4 N 4 N 5 1 5 N 6 G 6

7 H 7 0 8 R 8 1 9 z 9

10 10 11 11 12 12

16634559.08 81606523.02 18276765.37 99883288.39 12230916.16 1121 14204.60 14093574.70 326207779.30 20787076.01 146994855.30 20933735.94 167928591.20 18653901.69 186582492.90 21716016.00 208298508.90

30302108.05 132452794.40 34169525.85 166622320.30 24095924.63 190718244.90 23509757.26 214228002.20 36968287.63 251 196289.80 36706493.87 287902783.70 30441203.34 318343987.00 33628352.65 351972339.60

0 N

L 3 A 4 N 4 N 5 1 5 N 6 G 6

7 H 7 0 8 R 8 1 9 z 9

10 10 11 11 12 12

53921778.10 9449346.25 63371124.35 8003838.09 71374962.44 11509189.80 82884152.24 12413222.11 95297374.35 11103061.10 106400438.45 9399354.22

115799792.67

544 15021.72 9192056.34 63607078.06 7879659.00 71486737.06 11652170.45 83138907.51 12555770.73 95694678.24 11111861.14 106806539.38 9444935.52

116251474.90 2241 1479.46 230709988.40

37995378.14 39736214.51 389967718.40 402741328.50

5201862.96 68073185.92

10560183.29 10528634.63 126359975.96 126780109.53

Product 1 Period I STARlMAl SA 1 EV

I I 18425009.681 41825825.551 46079921.16

47277049.98 70965591.95 77918932.95 142660858.90 ux)H)9116.80 225205573.60 I 39849902.5A 61422230.721 62496573.111

N d/ 5 1 84247635.681 18071760.30 1920243%.80/ 44910909.53 10~~3364.30l 49047153.60

12207611.99 21417898.84 21453013.60 G W55247.67 213442295.60 2304%377.90

13872602.24 19096531.61 21495371.51 110327849.90 232538827.20 251991749.40 i 1 21388250.761 60085069.531 64486774.991 353091956.60 50670528.041 509542594.00 76590386.801 548600883.00 86647549.951

52683389.21 81992093.18 94234300.62 405775345.80 591534687.20 642835183.60 45737010.75 61242164.04 67248761.62

316478524.40 69491012.30 385%9536.70 26731240.07 412700776.80 38792172.33 45 1492949.10 38424578.27 489917527.40

EV 37920022.26 379uXn2.26 35501762.23 73421784.49 34747662.62 108169447.10 30518356.15 138687803.30 34723531.71 173411335.00 23258888.22 196670223.20 22881624.51 219551847.70 37876934.41 257428782.10 38530288.54 295959070.70 31737086.81 3276%157.50 35308956.55 363005 114.00

45 15 12356.60 65277685 1.20 7 10083945.u) NI iy I 52731853.821 72873777.081 82391744.261 504244210.40 725650628.30 792475689.50

12 52323149.55 72068160.53 81325940.08 12 556567359.90 797718788.80 873801629.60

12

Pmduct 2

4379071.64 8831817.21 8539886.54

7359406.54 14857641.58 15215473.86 ~2627636.321 3wi36.521 391~547.861

29987042.86 4736872.54 34723915.40 4465069.38 39188984.78 6452927.30 45641912.08 6347752.00 51989661.08 5609698.38 57599362.46 5271960.50 62871322.96

hat for each product the PCT pattern is not explained in any way by either the SA or the EV formulations. Thus, the STARIMA model is very flexible characterization with its parameters. It includes parameters only when they are statistically significant to the demand process. On the other hand, the SA and EV formulations always include their parameters and their values change only when the process mean, standard deviation, or costs change. Their parameter values do not change due to the changes in the correlation structure of the observations. Also, another advantage of the STARIMA formulation is the consideration of the spatial relationships within the model. This was found to be significant for three out of the four products. The spatial relationship is very important because it reflects the effect of the physical relationships of the distribution centers within the spatial system boundaries, and it should be accounted for in order to have a better representation of the real situation whenever it exists.

rig. L grapuciuiy iiiusuaes and the cumulative PCT for the STARIMA formulation are always close to zero percent. In addition, the variability of the PCT in a single period and the cumulative PCT for the SA and EV are higher than the STARIMA formulation. This shows that the better resolution of the demand characterization in this problem application has an impact on the solution. This illustrates the advantages of the STARIMA formulation.

C. Different Methods of Demand Formulations and Their Pe$ormances

Different formulations used here performed differently ow- ing to the differences in the extent to which they used information from the demand process. The SA formulation uses the mean and standard deviation of the demand process and estimates of the related costs. The EV formulation uses only the mean value of the demand process. The STARJMA formulation may use as many parameters as necessary depending directly on the type of the demand process under study. The fitted STARIMA models have at least four parameters with at least one of them reflecting the seasonality pattern of the demand process whenever it exists. This seasonality

VII. OVERALL SYSTEM ANALYSIS FOR THE TOTAL COST

The E T ’ S for the overall system for each method are summarized in Table V. From this table, we can see that the STARIMA performs about 10 times better for the overall

350

303

2SO

2 0 0

Pcr 1 SO

1 0 0

50

0

1 0 0

PCT 50

0

-50

100

* 50

0

-50

150

1 0 0

Pcr 50

0

-50


7 0 0

I S 0

CUM 1 0 0 m 50

0

Roduct 1

150

1 0 0 CUM PCT

50

+ ----. -

12 STAFUMA u

0

Product 2

A 100

CUM 50

Ev SA

- STARIMA 6 12

Ev SA

ST" 6 12

EV SA /

I . . STARTMA 6 12

Product 3

EV SA

STARIMA

EV SA

12

Period Roduct 4 Period

Fig. 2. Percentage of the effective cost over the exact forecast cost on a single period and on a cumulative period basis for four products.

system than the other two formulations. The total savings per year that can be realized using the STARIMA formulation versus the other two are summarized in Table VI. In Table VII, we provide the net expected profit based upon the company's estimate of the expected profit of 750 pesoshectoliter and the actual number of hectoliters of each product sold for the year.

From the company sources, it was found that the real net profit per year was 6 762 OOO 000.00 pesos. So, if we compare the expected value with the real value, we see that the actual net profit is 1379759250 pesos less than the net expected profit. This amount in dollars is $459919.75 at the exchange rate of 1$ = 3000 pesos.

We know that the number of units sold during the year under consideration was 10 855 679 hectoliters. If we divide the difference in the expected net profit and the real net profit in pesos (dollars) by the number of hectoliters sold, we get the amount of net profit decreased in pesos (dollars) per hectoliter.

This when computed gives 127.10 pesos (4.24 cents) per hectoliter.

The estimated average distribution cost per hectoliter was 100 pesos by the company's officials. This cost is not consis- tent with the numbers presented above since they have realized less net profit than the expected net profit. Therefore, the above estimate of the average distribution cost is incorrect. The actual average distribution cost must be 227.10 pesos per hectoliter. In Table VIII, we present the average distribution cost per hectoliter over the planning horizon for the three methods considered in this paper. These averages take all the products into account. Using the above average distribution cost, increase in the net profit for various demand formulations from that of company's method is summarized in Table IX. The percentage increase in net profit of the STARIMA formulation over the EV and SA formulations are given in Table X. From the information given in Table X, it is clear that

DEUTSCH et al.: COMPARISON OF STOCHASTIC APPROACHES TO THE TRANSPORTATION PROBLEM

EV SA

STARIMA (no updating) STARIMA (updating)*

Company's Method

161

(in Pesos) Per Hectoliter. 172.89 161.22 102.10 95.8

227.10

TABLE IV DISTRIBL~TION CENTERS AND THEIR PERCENTAGE CONTRIBUTION

TO THE TOTAL EFFECTIVE COST FOR EACH PRODUCT

STARIMA EV SA

Distribulion Centus

Cost For the Overall System 8.4%

81.0% 94.0%

Roduct3 I EV I SA I STARIMA (1)>10 I 12 I 1.2 I 1.2 (2) 4 10 10 I 5.4064 I 5.8.40.64 I5.40,64

5 I d4.43.47

6.11,12,13.14,25~~2638.29.

TABLE V PERCENTAGE TOTAL EFFECTIVE COST OVER EXACT FORECAST COST, m, FOR THE OVERALL SYSTEM OBTAINED USING THE STARIMA, SA, AND EV METHODS

Percentage Total Effective used I Cost Over Exact Forecast I

TABLE VI SAVINGS OF THE STARIMA VERSUS THE

EV AND SA METHODS FOR EACH PRODUCT

I Product I Savings of STARIMA I Savines of STARIMA I

251,487,988.00

4 58,286,790.00 58.706.924.00

TABLE VI1 THE NUMBER OF HE(ST0LITERS SOLD AND

NET EXPECTED PROFIT FOR EACH PRODUCT

the STARIMA formulation is considerably more advantageous for the application problem considered than the other two methods as well as the one that the company is presently using. In fact, the STARIMA formulation would result in

TABLE VIII AVERAGE DISTRIBUTION COST PER HECTOLITER FOR EV, SA, STARIMA (No

UPDATING), STARIMA (UPDATING), AND THE COMPANY'S METHOD

I Formulation I Average Distribution Cost I

TABLE M NET EXPECTED PRoFl? IINCREASE FOR EV, SA, S T U A (NO UPDATING),

AND STARIMA (UPDATING) OVER THAT OF THE COMPANY'S METHOD

I Formulation I Increase in Net I

TABLE X PERCENTAGE INCREASE IN NET PROFIT OF THE STARIMA

FORMULATION OVER THE EV AND SA FORMULATIONS

Percentage Increase in Net Profit

11.36

12.37 10.50

a 20.06% increase in net profit versus the demand forecast method that the company is currently using. In addition, it is evident from Table X that the STARIMA method when used without (with) updating results in a 9.49% (10.5%) increase in the company's net profit as compared with the SA method. Furthermore, the STARIMA method when used without (with) updating results in an 11.36% (12.37%) increase in the net profit as compared with the EV method. There may also be other intangible factors that should be considered when the STARIMA formulation is used versus the other two; for example, the sensitivity of the STARIMA model to the changes in the demand pattern and the computational effort used for running the problem solution. However, the extra computational effort for producing STARIMA forecasts are easily justified in terms of the increased profit margin.

VIII. SUMMARY AND CONCLUSIONS

In this section, we summarize the results presented in this paper and give the conclusions. The space-time autocorrelation functions had shown individual regular patterns for each of the products. In addition, they had exhibited seasonality patterns at lag 12 for all products. This indicated that the multiplicative STARIMA model may be appropriate and the quantitative advantages of the STARIMA model versus the other two may be greater. The forecast functions and the error measures developed in the paper indicated that the STARIMA model was indeed superior to the EV and SA models. The appropriate


planning horizon for our application based on the minimum average values of the errors as a measure of performance was seven periods for the three out of four products.

With respect to the total effective cost results, we found that this cost reflects the better performance of the STARIMA formulation due to the high differences that this cost shows versus the SA and EV formulations on a single period basis. From the analysis of the percentage contribution to the total effective cost of the individual distribution centers, we observed that the tendency of the STARMA formulation was to have more distribution centers in the average range as compared to the other two formulations. However, the high contributions may also be present because of the high demand distribution centers. This is because the high demand distribution centers may have a large number of units shipped as well as the large number of units short or held. The percentage of the total effective cost over the exact forecast cost, PCT, for the STAFUMA model formulation was found to be lower than the other two formulations. In addition, higher stability in the PCT was achieved when the STARIMA formulation was used than the SA and EV formulations. Sometimes, the PCT curves were below zero when the STARIMA formulation was used. This may be due to the following reason. For a certain period, we may have units short and the shortage cost is smaller than the shipping cost of the extra units shipped under the exact formulation. In this case, the PCT may become negative. This, however, does not happen when the SA or EV formulation is used.

In conclusion, the present results and analysis have shown throughout the paper that a more structured demand characterization is indeed appropriate because of the better representation of the spatial demand characteristics. The STARIMA formulation affords a systematic way to describe spatial and/or temporal considerations of demand and results in a more accurate representation of the real situation. Furthermore, this case study application illustrates the practical consequences in optimal minimum cost solutions of the transportation model due to its sensitivity on its demand representation. From the results obtained here, we can see that large cumulative losses occur when space and/or time considerations are ignored with seasonal demand processes. This occurs because the cyclic demand variation of the process is not explained by the other problem formulations.

The results presented in this paper are obviously specific with respect to the brewery data. A computation simulation experiment [9] which varied magnitude of spatial and temporal autocorrelation in demand data compared EV and STARIMA approaches over a greater domain than the specific brewely data. This experiment indicated that the STARIMA representation of the demand data resulted in a lower total cost over the EV method whenever the space-time dependencies existed. The same conclusions about the need for the better forecasting models for the stochastic demand because of the sensitivity of the optimal solution of the transportation problem on the stochastic demand because of the sensitivity of the optimal solution of the transportation problem on the stochastic demand representation were reached in the experimental study as were reached in the case study presented in this paper.

Ix. APPENDIX

In this Appendix, we describe the procedure to obtain an appropriate STARIMA model for each product of the brewery. We first need to determine the weight matrices W1 and W2. For developing these weight matrices, the demand system must be spatially ordered. That is, with respect to each distribution center the remaining distribution centers must be spatially ordered. This is done to incorporate systematic dependence between observations at each distribution center and their neighboring distribution centers within the physical boundaries of the demand center locations. We use the boundary information of the distribution centers to develop a list of neighbors with different spatial lags. The first- and second- order neighbors are important. The higher-order neighbors are assumed to have a negligible effect on the observations of the distribution centers. This is not a serious assumption since in practice low-order STARIMA models are known to adequately describe the space-time dependent processes. The first-order neighbors are those distribution centers that have a significant length of common boundary with respect to the demand center of interest. Here, significant length refers to the length of the boundary that is better than that of one point boundary or a little longer. The second-order neighbors are those distribution centers that have a nonsignificant amount of boundary in common with the distribution center of interest or have their boundaries in common with the first-order neighbors.

A. Development of Weight Matrices

The scaled weights were selected for the spatial system with the first- and second-order neighbors shown in Table XI. The scaled weights are such that the sum of the elements in each row is equal to one for each order weight matrix. The (i, j)th element, wl. , of the first order weight matrix, W1, is zero if the distribution center j is not the first-order neighbor of the distribution center i. Otherwise, it is equal to l/Nl(i), where N1 (i) is the number of first-order neighbors of the distribution center i. Similarly, the (i, j)th element, w$, of the second- order weight matrix, W 2 , is equal to zero if the distribution center j is not the second-order neighbor of the distribution center i, and is otherwise equal to l/Nz(z), where N2(i) is the number of second-order neighbors of the distribution center i. Now, we proceed to the three-stage iterative STARIMA model building procedure for each of the four products.

%3.

B. Three-Stage Iterative Model Building Procedure for STARIMA

Below, we describe the three-stage iterative model building procedure, first described by Box and Jenkins [lo] for fitting the autoregressive moving average model and later extended by Pfeifer and Deutsch [ l ] to accomodate the space-time dimensions, for the general class of STARIMA models. This procedure is used to fit the appropriate STARIMA model to the demand data (historical demand observations for periods 1 to 31) of each product.

1 ) Identijication Stage: Identification is the process by which one subclass of the general STARIMA model class is chosen that exhibits theoretical properties most closely

DEUTSCH et al.: COMPARISON OF STOCHASTIC APPROACHES TO THE TRANSPORTATION PROBLEM

8 9 10

163

5.6.7.17.19 4,27.32.34,39 3,4.10,12,14,17 5,11,18,19 9,11,13,14 4,5,12,15,17,18,19

TABLE XI

Distribution First Order Neighbors Second Order Neighbofi FIRST- AND SECOND-ORDER NEIGHBORS OF THE 64 DISTRIEIUTION cENTJ5RS

2,3,6 22,23,25,28 5,20,26 1,23 24 1,4 5 9

11 12 13

10,15,16,18 . 9,13,19,60 9.14 3,4,10,13 10,14 11.12

21 22 23

11.60

19 8,17,18,32

20 23,25,28 1.23,25 2.24.28 1,2,22,24 20,21.24,25

62 63 64

matching those estimated from the data. The first step is to determine whether or not the space-time series of observations are stationary. This can be determined by computing the space-time autocorrelation function (AF) and partial space- time autocorrelation function (PAF) using the data. If the AF and PAF decay at a very slow rate from their values, the space-time process is nonstationary. Otherwise, it is stationary. For a stationary space-time process, a STARIMA model is identified whose theoretical AF and PAF closely resemble that of the observed space-time AF and PAF, respectively. For a nonstationary space-time process, we take the differences of the successive observations and compute the AF and PAF of these differences. If the computed AF and PAF of the differences show the nonstationarity behavior, we take the second-order differences (differences of the differences)

63 46.6 1,64 46,61,62,64 4736 61,63 46,62

TABLE XI (continued)

of the observations, and compute the AF and PAF again. We continue this procedure until the AF and PAF display stationarity. Once this is achieved, a STARIMA model is identified whose theoretical AF and PAF closely resemble the AF and PAF of the differenced observations that have displayed stationarity. This identification is based on either AF or PAF tails off and the other cuts off, or both tail off.

2. Estimation of the Parameter Stage: After a candidate model from the STARIMA model family has been chosen, it is necessary to estimate its parameters. The best estimates of Q, and 0 are the maximum likelihood estimates (MLE). For moderate to large number of time periods used in developing the STARIMA model, these MLE's become approximately equal to the least squares estimates and can be obtained easily with great savings in computational efforts. For small number of time periods, however, the least squares estimates are inappropriate and we need to employ exact MLE procedures in order to estimate the parameters of the identified STARIMA model.

3. Diagnostic Checking Stage: After a candidate model has been selected and its parameters estimated, the model must be subjected to diagnostic checks to determine if the model does adequately represent the data. The model can fail in two important ways. First, the model may insufficiently represent the observed correlation of the process, This inadequacy surfaces in the form of a significant correlation among the residuals of the fitted model. Secondly, the model may be unduly complex. In this instance, estimated parameters will prove to be statistically insignificant. The first phase of the diagnostic checking stage is the examination of the residuals from the fitted model. If the fitted model adequately represents


TABLE XII FITTED STARIMA MODELS AND THFJR

PARAMETER ESTIMATES FOR FOUR PRODUCTS Product Regular Seasonal Regular Parameter Seasonal Parameter

STARIMA STARIMA Estimates Estimates Model Model

1 (1n.0.11) (ln,O.li) $10=0.83972 $10=0.79895 s=12 elo= 0.43558 eln= 0.13186

-” e , = -0.09385 el, = -0.14787

2 (2o,O,lo) (ln.l.0) $10 = 0.02797 $10 = -0.49024 S=12 $20 = 0.23267

the data, these residuals should be white noise. If the residuals do not form a white noise, the procedure once again returns to the estimation stage after reidentifying the model. The second phase of the diagnostic checking stage involves checking the statistical significance of the estimated parameters. This is done via the confidence regions for the parameters [l]. Any estimated parameter that proves to be statistically insignificant is removed from the model. The model building procedure then moves once again to the estimation stage.

The three-stage modeling procedure continues through these stages until at some point the model at hand passes the diagnostic checking stage. The final form of the model must then evidence parameters that are all significant and residuals that can effectively be considered to be white noise. The above three-stage iterative procedure when applied to each of the products of the brewery yields the STARIMA models as shown in Table XII. The parameter s appearing in this table is a seasonal period. Note that for products 1, 3, and 4, the fitted STARIMA models are actually the seasonal STARMA models (d = 0), and for product 2 the fitted STARIMA model is seasonal ARIMA model.

REFERENCES

P. E. Pfeifer and S. J. Deutsch, “A three-stage iterative procedure for space-time modelling,” Technometrics, vol. 22, pp. 35-47, 1980. S. J. Deutsch and J. A. Ramos, “Space-time modelling of vector hydrologic sequences,” Water Resources Bull., vol. 22, pp. 967-981, 1986. P. E. Pfeifer and S. J. Deutsch, “Seasonal space-time ARlMA modelling,” Geographical Analysis, vol. 13, pp. 117-133, 1981. P. E. Pfeifer and S. J. Deutsch, “Identification and interpretation of first order space-time ARMA models,” Technometrics, vol. 22, pp. 397-408, 1980. P. E. Pfeifer and S. J. Deutsch, “Variance of the sample space-time correlation function of contemporaneously correlated variables,” SIAM

J. Appl. Math., VO~. 40, pp. 133-136, 1981. P. E. Heifer and S. J. Deutsch, “Variance of sample space-time autocorrelation function,” J. R. Statist., vol. 43, pp. 28-33, 1981. R. B. Chase and N. J. Aquilano, Production and Operations Manuge- ment: A Life Cycle Approach, fifth ed. D. Wilson, “A mean cost approximation for transportation problems with stochastic demand,” Naval Res. Logistics Q., vol. 22, pp. 181-187, 1975. S. J. Deutsch, M. H. Patel, and A. J. Dieck, “The effect of space-time demand processes on the solution of transportation problems,” accepted for publication in Computers & Industrial Engineering, 1993. G. E. P. Box and G. M. Jenkins, lime Series Analysis, Forecasting and Control. San Francisco, CA: Holden-Day Inc., 1976.

Homewood, IL: Irwin, 1989.

Stuart J. Deutsch author biography and photograph unavailable at time of publication.

Minnie H. Patel received the Ph.D. degree in industrial and systems engineering and the M.S. degree in operations research from Georgia Institute of Technology in 1988 and 1984, respectively.

She has been an Assistant Professor in the De- partment of Industrial and Systems Engineering at the University of Wisconsin-Milwaukee, since 1990. She has also held a faculty position at the University of Tennessee, Knoxville in the Management Science Program from 1987-1990. At UWM, she is a Fac- ulty advisor to the Society of Women Engineers. Her

current research interests are in the modeling and optimization of problems arising in transportation and distribution systems, transportation of hazardous materials and optimal location of obnoxious facilities causing atmospheric pollution, and spherical and network location problems. She has published in six different technical refereed journals.

Dr. Patel is a full member of ORSA and a member of IIE.

Antonio J. Dieck Assad received the Bachelor’s degree in industrial and systemas engineering from Monterrey Tech in Monterrey, MCxico. He also holds Master of Science degrees in systems engineering and industrial engineering and a Ph.D. degree in industrial engineering from Georgia Tech.

He has been a faculty in Industrial Engineering at The University of Missouri-Columbia and St. Mary’s University of San Antonio. He is currently a faculty of Industrial Engineering at Monterrey Tech in Monterrey, Mtxico. He is also an associate for

the consulting firms Sintec and Asesoda Econ6mica Estratkgica y Financiera. He has done academic, research and consulting work for several companies in the United States and Mtxico. He has served as President for the Institute of Industrial Engineers (E) in Columbia, Missouri, San Antonio, Texas and currently in Monterrey, Mkxico. His research interests are manufacturing, logistics, concurent engineering, applications of expert systems in manufacturing as well as productivity in banking, govemment and health care systems. Dr. Dieck Assad is a senior member of IIE and IEEE.

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