159

1 GENERAL DESCRIPTION OF THE PROBLEM

There is a common opinion that the boxes arriving

within a current party cover the boxes that have

arrived earlier, thus forcing them to ‘sink’ to the

bottom of the stack. Accordingly, to pick a container

one needs to ‘dig out’ the stack, removing the freshly

arrived boxes blocking the access to the designated

one. These redundant moves increase the time and

labor needed to retrieve the required box [1, 2, 3, 4, 5].

The qualitied study of the digging effect needs to

distinguish between individual containers arriving

and departing in random sequences within separate

parties [6, 7, 8, 9]. In order to reveal the inner

mechanisms distorting the pure combinatorics we will

start with simple regular case and determined (not

random) parameters.

2 COMBINATORIAL APPROACH

Let us assume that the size of one party is

150V =

boxes, the dwell time is

dwell

T =

days and the

interval of arrival is

3t =

days. The containers of

any party leave the stack evenly, so the dwell time

dwell

T =

days means that the last containers of the

party would leave in the

disp dwell

TT= ⋅

. In other

words, the container party dispatch time

disp

T =

days, and within this interval

( 1)

disp

VT −

boxes of

the party will leave the stack.

Evaluation of Sinking Effect in Container Stack

A.L. Kuznetsov, A.V. Kirichenko & A.D. Semenov

Admiral Makarov State University of Maritime and Inland Shipping, St. Petersburg, Russia

ABSTRACT: The container yard is the key element of any modern container terminal. The huge amount of

boxes dwelling on the operational areas of the terminals could occupy a lot of space, since one-time storage

capacity of the container mega terminal handling over one million TEUs annually is something around 20 000

TEUs. The ecological pressure imposed on modern container terminal does not permit to allocate for this

storage large land areas, thus forcing the box stacks grow high. The selection of the individual boxes becomes a

complex and time-consuming procedure, demanding a lot of technological resources and deteriorating the

service quality. The predicted combinatorial growth of redundant moves needed to clear the access to the

individual container is aggravated by the well-known and widely discussed ‘sinking effect’, when containers

arrived earlier are gradually covered by the ones arriving afterwards. While the random selection could be

adequately assessed by combinatorial methods, the ‘sinking effect’ allows neither intuitive consideration, nor

any traditional mathematical means. The only practical way to treat this problem today is in simulation, but the

simulation itself causes yet another problem: the problem of model adequacy. This study deals with one

possible approach to the problem designated to prove its validity and adequacy, without which the simulation

has naught gnoseological value.

http://www.transnav.eu

the

International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 14

Number 1

March 2020

DOI: 10.12716/1001.14.01.

160

During the dwell time of a party there will be

dwell

other parties arrived and stored in the

stack, so the average number of boxes in the stack is

dwell

E VT t= ⋅

. Really, since the interval of arrival

parties is

365

tN=

, we could write this

expression as

dwell

E VT t= ⋅

365

dwell

VNT⋅⋅

365

dwell

QT⋅

, which gives us the well-known

Wilson’s formula

365

dwell

EQ T= ⋅

In our case,

150 8 3 400

dwell

E VT t=⋅ =⋅=

boxes. This is the average number of boxes stored in

the stack. The arrival of every next party gives a surge

that will be evenly sent from the stack within the

interval of arrival

. Fig 1 illustrates this dynamics of

the container stack.

Figure 1. The dynamics of the container stack

The container stack occupies the territory which

allows to allocate a limited amount of terminal

ground slots (t.g.s.). If the stack foundation measured

in t.g.s. is

, then the volume of

( )

boxes

forms the stack with the height

( ) ( )

ht Et w

, i.e.

the dynamics of the stack directly determinesits

operational height, as Fig. 2 shows.

Figure 2. The operational height at different stack area

The combinatorics of the selections could be

directly applied to the dynamically changing stack

height, since the most related equations include only

linear member. The theoretical number of moves per

box calculated for the handling systems with top

access by the formula

( 1) 2h +

is given by Fig. 3.

Figure 3. Theoretical number of moves per box

3 BASIC SIMULATION LOGICS

Let us assume that containers from any parties are

selected evenly by numbers but randomly by their

identification number. The introduction of container

identification feature is responsible for the “digging”

or “sinking” effect. The only way to estimate this

effect is to compare the combinatorial results with

those of simulations. In order to be able to reveal all

these hidden mechanisms, it is necessary to increase

the complexity of the simulation models gradually.

Fig. 4 describes the parameters of the process to be

modelled. Each container has a unique identifier

formed as concatenation of the party number and its

number in the party. For example, 2114 is the box

numbered 114 belonging to the party number 2.

Every party of 150 boxes leaves the stack evenly,

with 10 boxes abandon it every day. The containers to

be selected this particular day form a random sample

(10 out of 150, without returning), making it possible

to generate a schedule of container selection by days,

as Fig. 4 shows.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

1067 1081 1054 1106 1088 1004 1030 1105 1094 1118

1018 1091 1003 1117 1135 1053 1039 1114 1119 1045

1092 1107 1131 1050 1059 1060 1046 1061 1127 1137

2073 2025 2041 2030 2054 2064 2106 2009 2013 2031 1009 1087 1124 1122 1079 1049 1016 1073 1110 1130

2059 2044 2120 2015 2069 2116 2022 2006 2057 2083 1047 1150 1005 1044 1035 1125 1014 1100 1007 1032

2085 2071 2076 2033 2003 2024 2141 2029 2018 2008 1068 1139 1029 1056 1098 1023 1099 1006 1008 1113

3006 3044 3038 3148 3067 3132 3128 3050 3146 3047 2077 2143 2098 2036 2043 2102 2049 2078 2108 2010 1148 1090 1017 1042 1065 1145 1149 1027 1086 1071

3112 3083 3149 3033 3015 3057 3049 3114 3018 3089 2048 2012 2112 2028 2092 2063 2129 2132 2014 2051 1133 1069 1104 1076 1078 1121 1024 1116 1147 1132

3066 3045 3079 3127 3073 3117 3123 3108 3118 3005 2042 2096 2082 2121 2104 2055 2087 2135 2123 2124 1080 1077 1112 1063 1062 1142 1134 1010 1036 1064

4084 4013 4081 4078 4115 4126 4134 4091 4014 4057 3063 3069 3092 3052 3134 3011 3003 3147 3072 3097 2093 2026 2052 2053 2039 2149 2079 2117 2105 2002 1025 1085 1109 1074 1022 1143 1012 1115 1040 1021

4027 4130 4068 4075 4133 4066 4079 4110 4139 4051 3048 3076 3074 3120 3121 3129 3096 3028 3041 3135 2004 2011 2127 2050 2109 2017 2114 2131 2137 2001 1001 1072 1041 1002 1026 1037 1013 1096 1033 1034

4080 4043 4144 4073 4137 4124 4056 4002 4083 4042 3125 3030 3110 3007 3104 3091 3058 3027 3107 3078 2080 2007 2097 2062 2100 2142 2058 2111 2027 2148 1070 1043 1097 1082 1031 1028 1015 1101 1095 1083

5078 5022 5041 5086 5110 5070 5012 5028 5030 5025 4045 4119 4096 4082 4095 4001 4064 4054 4108 4024 3144 3070 3102 3101 3082 3126 3010 3059 3075 3031 2084 2118 2140 2134 2081 2056

2136 2065 2005 2019 1120 1051 1019 1084 1111 1126 1128 1055 1123 1052

5095 5116 5128 5066 5133 5082 5113 5136 5112 5035 4121 4111 4020 4145 4053 4030 4149 4117 4125 4040 3109 3020 3055 3060 3017 3032 3061 3137 3056 3115 2095 2040 2068 2070 2066 2122 2067 2130 2107 2090 1093 1141 1075 1140 1108 1058 1020 1089 1138 1066

5109 5048 5002 5130 5097 5146 5119 5092 5123 5003 4148 4138 4092 4129 4008 4055 4147 4114 4088 4021 3087 3051 3088 3024 3025 3103 3065 3130 3054 3081 2047 2144 2146 2128 2150 2133 2139 2091 2089 2020 1146 1011 1136 1048 1038 1103 1144 1057 1102 1129

6126 6124 6012 6023 6031 6146 6069 6113 6122 6076 5103 5114 5077 5029 5107 5072 5036 5139 5057 5122 4093 4016 4029 4069 4050 4086 4112 4032 4105 4015 3085 3124 3122 3145 3131 3068 3071 3116 3016 3133 2023 2147 2060 2035 2126 2103 2037 2038 2034 2101

6024 6022 6081 6004 6017 6057 6038 6137 6027 6090 5088 5071 5054 5106 5089 5005 5068 5129 5042 5019 4071 4012 4041 4123 4006 4128 4038 4049 4142 4094 3142 3014 3009 3080 3021 3099 3029 3100 3138 3098 2021 2088 2110 2125 2145 2115 2061 2046 2074 2075

6091 6020 6034 6070 6001 6048 6138 6049 6135 6150 5067 5102 5009 5093 5127 5134 5131 5101 5090 5142 4039 4141 4113 4005 4062 4076 4046 4135 4058 4003 3034 3105 3143 3094 3013 3035 3150 3037 3023 3095 2086 2032 2094 2119 2045 2099 2072 2113 2016 2138

7046 7134 7101 7011 7123 7096 7127 7007 7107 7136 6097 6149 6131 6067 6032 6039 6028 6062 6101 6143 5091 5143 5141 5124 5148 5040 5047 5105 5045 5118 4122 4106 4034 4097 4059 4090 4140 4065 4011 4098 3111 3019 3119 3093 3064 3039 3036 3042 3040 3004

7100 7091 7020 7044 7071 7003 7059 7139 7060 7141 6007 6011 6144 6104 6115 6134 6030 6054 6014 6105 5013 5052 5015 5006 5037 5147 5144 5098 5046 5024 4109 4019 4037 4118 4102 4131 4063 4044 4101 4132 3012 3053 3077 3140 3084 3046 3090 3139 3022 3136

7094 7077 7119 7110 7026 7021 7018 7034 7080 7083 6094 6079 6093 6050 6060 6078 6045 6041 6037 6108 5056 5085 5004 5080 5051 5104 5083 5075

5018 5137 4120 4127 4136 4025 4104 4022 4061 4048 4004 4035 3043 3062 3141 3086 3026 3106 3002 3008 3001 3113

8040 8033 8006 8089 8081 8137 8090 8121 8128 8117 7135 7126 7085 7024 7001 7095 7125 7013 7014 7099 6042 6130 6006 6103 6061 6120 6092 6009 6071 6133 5125 5008 5145 5064 5094 5058 5099 5120 5111 5039 4052 4077 4047 4099 4070 4009 4026 4060 4017 4103

8002 8041 8044 8096 8100 8127 8035 8130 8065 8101 7116 7084 7031 7025 7005 7042 7052 7051 7027 7132 6010 6095 6147 6083 6005 6016 6102 6040 6117 6106 5023 5060 5044 5115 5065 5049 5074 5011 5100 5138 4036 4085 4031 4074 4146 4087 4018 4143 4072 4116

8045 8063 8109 8132 8102 8023 8022 8018 8131 8007 7066 7054 7140 7037 7036 7064 7093 7010 7056 7058 6003 6125 6044 6129 6064 6121 6065 6002 6128 6145 5081 5073 5020 5026 5117 5017 5043 5062 5140 5007 4007 4028 4100 4107 4150 4010 4033 4067 4023 4089

9118 9078 9080 9005 9071 9104 9086 9020 9011 9077 8098 8025 8133 8020 8031 8030 8103 8012 8148 8122 7043 7102 7061 7039 7128 7106 7089 7033 7147 7108 6047 6052 6029 6099 6015 6035 6119 6074 6072 6127 5076 5053 5016 5038 5027 5021 5084 5087 5135 5055

9130 9051 9019 9008 9032 9081 9114 9016 9056 9076 8028 8097 8015 8005 8108 8069 8059 8088 8111 8058 7129 7065 7090 7063 7109 7092 7145 7112 7022 7118 6082 6021 6066 6080 6140 6018 6068 6088 6112 6036 5032 5001 5121 5150 5010 5069 5061 5059 5034 5063

9013 9141 9135 9149 9024 9133 9075 9091 9093 9088 8107 8011 8110 8134 8066 8138 8051 8068 8056 8149 7098 7117 7124 7122 7019 7002 7076 7111 7004 7148 6033 6056 6109 6114 6096 6053 6063 6087 6026 6025 5132 5126 5014 5108 5031 5149 5096 5050 5033 5079

10035 10047 10134 10022 10010 10056 10097 10105 10081 10144 9047 9079 9038 9035 9082 9115 9033 9131 9054 9060 8077 8095 8080 8125 8013 8036 8092 8129 8123 8142 7048 7130 7028 7017 7008 7079 7137 7097 7035 7074 6013 6055 6116 6110 6084 6148 6077 6139 6046 6059

10038 10017 10111 10059 10066 10116 10048 10061 10092 10082 9119 9043 9069 9052 9139 9074 9083 9128 9107 9117

8008 8064 8052 8136 8043 8049 8082 8061 8054 8115 7082 7142 7069 7029 7049 7105 7088 7070 7104 7073 6123 6073 6043 6132 6058 6098 6111 6051 6086 6089

10139 10133 10110 10078 10114 10046 10093 10024 10084 10135 9084 9010 9123 9068 9025 9029 9058 9097 9121 9070 8026 8139 8104 8093 8024 8116 8112 8146 8072 8039 7103 7045 7078 7150 7006 7016 7086 7143 7057 7067 6118 6141 6136 6107 6019 6142 6008 6075 6100 6085

11068 11143 11017 11148 11052 11022 11013 11034 11051 11040 10090 10012 10031 10075 10027 10083 10036 10067 10074 10063 9132 9037 9111 9039 9110 9034 9085 9031 9067 9143 8124 8073 8143 8001 8003 8140 8070 8050 8038 8076 7055 7038 7032 7053 7040 7030 7121 7146 7015 7050

11104 11004 11037 11059 11063 11081 11114 11074 11009 11027 10087 10136 10029 10042 10117 10141 10098 10020 10018 10058 9012 9042 9147 9053 9127 9018 9122 9096 9142 9099 8032 8083 8105 8075 8071 8113 8055 8144 8060 8126 7081 7113 7012 7149 7133 7047 7115 7009 7023 7144

11082 11096 11066 11146 11133 11134 11120 11103 11132 11021 10149 10138 10123 10137 10085 10104 10125 10014 10091 10089 9045 9065 9144 9061 9014 9003 9062 9048 9073 9100 8042 8135 8016 8010 8014 8147 8087 8086 8120 8019 7131 7120 7041 7068 7072 7062 7087 7114 7138 7075

12081 12130 12123 12036 12040 12057 12125 12070 12146 12041 11097 11086 11087 11111 11078 11150 11084 11035 11029 11116 10051 10145 10088 10146 10050 10142 10015 10127 10101 10131 9044 9059 9126 9116 9120 9001 9015 9006 9021 9007 8053 8021 8094 8119 8009 8037 8099 8114 8062 8048

12126 12025 12080 12094 12064 12145 12131 12124 12119 12133 11128 11105 11069 11109 11139 11080 11079 11033 11100 11136 10003 10102 10112 10103 10045 10122 10033 10013 10068 10079 9055 9134 9101 9023 9040 9136 9148 9108 9030 9022 8118 8106 8004 8150 8141 8017 8079 8047 8046 8029

12048 12082 12009 12086 12044 12101 12099 12066 12138 12072 11014 11094 11002 11015 11053 11088 11045 11123 11064 11076 10148 10053 10150 10005 10032 10055 10128 10034 10023 10108 9138 9145 9098 9028 9137 9057 9004 9140 9066 9036 8078 8074 8145 8091 8057 8084 8085 8027 8067 8034

13137 13050 13014 13032 13100 13115 13051 13047 13140 13049 12002 12105

12084 12051 12110 12112 12034 12006 12097 12096 11010 11025 11124 11011 11075 11099 11144 11070 11073 11032 10095 10124 10096 10044 10057 10025 10021 10129 10130 10001 9150 9087 9129 9103 9050 9113 9041 9009 9072 9095

13055 13078 13039 13120 13135 13079 13064 13074 13095 13033 12050 12011 12089 12143 12030 12062 12134 12018 12021 12095 11091 11047 11145 11042 11054 11024 11122 11067 11125 11057 10118 10126 10054 10039 10140 10099 10119 10077 10016 10040 9094 9089 9027 9092 9109 9090 9105 9026 9112 9125

13061 13059 13040 13114 13090 13003 13123 13077 13096 13006 12091 12027 12139 12083 12118 12107 12004 12013 12140 12024 11046 11102 11129 11030 11115 11065 11138 11119 11126 11058 10100 10072 10008 10086 10107 10064 10069 10052 10043 10065 9049 9046 9063 9146 9124 9017 9102 9002 9064 9106

14071 14088 14068 14101 14110 14034 14095 14090 14117 14145 13080 13131 13121 13062 13101 13024 13015 13093 13109 13145 12015 12104 12010 12149 12115 12019 12071 12005 12003 12049 11149 11092 11026 11090 11031 11118 11137 11020 11039 11007 10076 10004 10121 10080 10106 10049 10002 10062 10009 10060

14121 14057 14102 14146 14055 14131 14054 14005 14013 14008 13076 13068 13133 13035 13087 13041 13094 13036 13102 13045 12106 12141 12077 12022 12016 12061 12093 12144 12087 12113 11117 11062 11142 11043 11071 11098 11085 11018 11060 11001 10113 10007 10094 10026 10011 10132 10147 10070 10006 10071

14022 14119 14099 14069 14085 14074 14111 14105 14136 14058 13116 13124 13048 13148 13132 13042 13027 13004 13070 13125 12103 12068 12136 12039 12045 12037 12092 12122 12026 12035 11147 11131 11108 11028 11041 11130 11077 11093 11083 11048 10030 10120 10143 10073 10041 10028 10115 10109 10019 10037

15109 15085 15038 15081 15051 15005 15102 15100 15010 15091 14079 14123 14129 14030 14112 14125 14006 14036 14086 14016 13139 13108 13117 13029 13098 13144 13069 13052 13043 13141 12047 12074 12132 12108 12055 12127 12014 12029 12117 12129 11008 11056 11036 11005 11089 11135 11127 11112 11055 11106

15095 15135 15016 15066 15105 15058 15019 15097 15002 15148 14049 14118 14033 14029 14028 14116 14134 14060 14083 14066 13126 13016 13011 13009 13112 13083 13012 13067 13026 13025 12100 12120 12056 12008 12135 12098 12043 12075 12058 12012 11113 11141 11121 11110 11006 11049 11101 11023 11016 11072

15079 15017 15067 15026

15029 15103 15030 15052 15035 15099 14065 14127 14025 14072 14144 14108 14135 14073 14045 14003 13107 13018 13028 13099 13088 13113 13082 13072 13044 13030 12033 12085 12102 12063 12114 12088 12017 12116 12032 12111 11107 11019 11044 11140 11003 11038 11095 11050 11012 11061

16047 16147 16136 16130 16091 16010 16075 16111 16003 16089 15059 15070 15131 15098 15078 15111 15042 15106 15125 15007 14019 14042 14046 14048 14107 14040 14091 14038 14018 14126 13031 13017 13081 13023 13118 13054 13021 13134 13106 13020 12090 12121 12137 12142 12073 12128 12060 12076 12023 12028

16139 16099 16101 16026 16141 16125 16107 16011 16062 16119 15141 15033 15055 15090 15104 15074 15015 15032 15001 15043 14059 14063 14103 14093 14089 14070 14076 14143 14041 14109 13105 13122 13073 13007 13147 13136 13084 13065 13103 13091 12046 12007 12150 12078 12079 12031 12053 12067 12059 12052

16032 16048 16037 16061 16071 16113 16016 16020 16109 16051 15094 15022 15127 15144 15114 15050 15054 15108 15020 15128 14122 14035 14114 14104 14032 14139 14138 14124 14147 14020 13010 13058 13149 13092 13060 13104 13019 13142 13097 13063 12065 12038 12042 12054 12148 12109 12020 12147 12069 12001

17137 17053 17107 17028 17145 17125 17009 17047 17090 17142 16072 16085 16024 16080 16054 16056 16131 16133 16128 16132 15024 15036 15136 15003 15065 15028 15119 15088 15138 15045 14098 14002 14140 14050 14023 14026 14037 14084 14015 14077 13119 13138 13046 13057 13001 13085 13013 13127 13128 13056

Figure 4. Daily tasks of stack operations

Every strings of this table represents the daily

tasks of the stack operations, i.e. the lists of individual

boxes from different parties that leave the stack in this

day. The order in which these boxes have to be picked

is also random, so the elements in every string should

be shuffled as a playing card deck.

All the boxes arriving to the stack come into the

stack model, all the operation to retrieve boxes are

161

conducted over this model. The stack model is

represented by the rectangular table which reflects the

cross-section of the stack: the strings correspond to

tiers, the columns refer to t.g.s. in the stack. This is an

analogue of the bay-plan in container vessels and

container yard layout plans, as Fig. 5 shows.

Figure 5. Bay plan of the 3D stack

Fig. 6 shows a beginning stage of the simulation

when the first party has already arrived.

0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150

1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100

1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050

Figure 6. Initial state of the stack

Next stage removes boxes from the stack in

accordance with the daily schedule, and place the

fresh party on top of the stack surface when it arrives.

The boxes on top of the one to be selected are moved

into lowest free positions (cells) in the stack. Fig. 7

shows the intermediate stage of the simulation when

only two parties dwell in the stack, while Fig. 8 shows

a more advanced stage of the simulation.

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0

0 0 0 0

0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0

0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2131 0 2133 2134 2135 2136 2137 2138 2139 2140 2141 2142 2143 2144 2145 2146 2147 2148 2149 2150 2114 1123 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2081 2082 2083 2084 2085 2086 2087 2088 2089 2090 2091 2092 2093 2094 2095 2096 2097 2098 2099 2100 2101 2102 1099 2104 2105 2106 2107 2108 0 2110 2111 2112 2113 2053 2115 2116 0 2118 2119 2120 2121 2122 0 2124 2125 0 2127 2128 0 2130

1130 1080 2033 2034 2035 2036 2037 2038 2039 2040 2041 2042 2043 2044 2045 2046 2047 2048 2049 2050 2051 2052 2067 2054 2055 2056 2057 2058 0 2060 2061 2062 2063 2103 2065 2066 2079 2068 2069 2070 2071 2072 1149 2074 2075 2024 2077 2078 0 2080

1101 1102 2008 2009 2010 2011 2012 1108 2013 2014 2015 1112 1113 2016 1115 1116 2017 2018 2019 1120 1121 1122 2076 1124 1125 1126 2020 1128 0 2021 2022 1132 1133 1134 2023 1136 2129 2025 2026 1140 2027 2028 1143 2029 2030 2117 1147 1148 1129 2032

1051 1052 1144 1110 1055 1056 1057 1058 1146 1109 1096 1062 1063 1064 1065 1066 1103 1100 1069 1070 1071 1072 2126 1074 1075 1076 1077 1078 0 1104 1139 1082 1083 1084 1085 1086 2123 1089 1095 1090 2002 2003 1093 2004 2005 2006 1097 1098 2059 2007

1001 1002 1068 1111 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1142 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1138 1031 1032 1033 1034 1035 1036 1037 1038 1145 1040 1041 1042 1043 1044 1141 1150 1047 1048 2109 2001

Figure 7. The state of the stack with only two parties

dwelling

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 4145 0 0 4148 4149 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 4085 4086 4002 2148 0 4052 0 3070 0 0 0 0 4095 0 0 4098 4099 0 0 0 0 0 0 4106 4107 4108 0 0 0 0 0 0 0 0 0 0 4119 0 0 4122 0 0 0 0 4127 0 0 0

1052 0 4041 4042 4005 3120 3141 4096 4094 4053 0 1084 0 1103 4051 4023 2122 4054 4055 4141 1051 3017 0 4003 1136 4061 4062 4063 2090 0 0 0 0 0 0 4083 4120 3100 4073 0 4037 4075 3084 3142 1108 2103 4080 4017 2081 0

4045 0 4016 2134 4048 2110 3091 4146 3062 4097 3102 3090 3081 3064 2089 4104 3140 4020 4021 4109 4116 3136 4035 4019 3107 3099 4024 2101 2097 1126 3071 4027 2072 2147 4089 3078 1101 4034 4032 1102 2150 2139 3053 4059 4049 2140 4036 4046 2100 4074

2020 0 4004 3012 4092 1128 3119 3077 3025 4147 2142 2080 3131 2056 2125 4077 2130 2136 4006 4025 4070 3086 3031 1057 2133 2037 3022 3013 2027 3029 3032 2068 3065 2019 4011 4072 2047 2034 2091 2119 4100 3039 4043 2070 4093 2062 4012 4090 3110 4033

3030 0 2095 3046 4142 4060 4128 1144 2038 4137 3113 2099 4121 4007 3023 4124 2032 3021 1095 1075 4030 4010 3016 3095 4056 3034 3104 3059 3054 4015 3008 2046 4050 4022 4067 4118 2086 2021 3126 3004 4150 3080 4087 3002 4143 4113 3061 4140 3014 1129

3026 0 3111 3087 2023 4105 4064 2084 2138 3001 4026 2074 3041 4028 3060 4125 2007 3058 3036 1120 4009 4031 4008 3145 4101 3075 2066 1097 2045 4040 2126 3051 4102 4058 2040 3074 4088 1123 2113 3020 4038 3130 4136 2016 2075 2146 3024 2058 2088 3068

3122 3040 2114 3137 2065 3109 3133 2035 1072 3055 4065 4076 2060 1083 3106 3085 1140 3103 3042 1070 3037 4071 4029 4135 4138 3125 3088 3139 3101 2145 2067 3094 4039 4103 3150 3124 4144 1089 3093 4018 4082 3082 1093 2061 2005 3019 2094 2118 1066 3007

4114 3009 3056 1111 2115 2128 1082 1055 4111 3098 1011 4123 1013 4112 1015 3135 3043 3035 1019 1020 4131 4117 4069 3027 4132 1026 3138 1028 1146 1138 1031 3144 4001 4044 2107 1058 3105 1038 3143 4047 4129 3010 1043 2111 1141 3116 2144 1048 2109 3115

Figure 8. The state of the stack with four parties dwelling

4 RESULTS

The simulation experiments include the shuffling of

container parties and selection of the boxes from the

stack with different area of blueprint or area

measured in t.g.s., i.e. different values of parameter

. In its turn, the value of parameter

determines

the operational height of the stack

, responsible for

the number of moves per box. The results received in

the serial of experiments are represented in Tab. 1.

Table 1. The results of the experiments

w 450 400 350 300 250 200 150 100 50 25

h 0,9 1,0 1,1 1,3 1,6 2,0 2,7 4,0 8,00 16,00

theor

0,9 1,0 1,1 1,2 1,3 1,5 1,8 2,5 4,5 8,5

N sim 1,0 1,1 1,2 1,4 1,5 1,7 2,0 2,7 4,7 8,7

Difference 0,06 0,12 0,17 0,19 0,19 0,21 0,20 0,21 0,22 0,17

Fig. 9 represents the same results in graphical

form.

Figure 9. The results of the experiments

These results show that under assumed conditions

both the ‘sinking effect’ and ‘digging’ of the stack do

exist, but they are not significant by the value. This

could be explained by the fact that digging of first hot

boxes to a great extend re-shuffle the whole stack,

thus reestablishing its random combinatorial

structure.

Still, this is a conclusion derived from just one

sample of the simulation experiment with very simple

and regular parameter. Certainly, this hypothesis

should be proved by much larger modelling

experiments.

5 CONCLUSIONS

1 The ‘sinking effect’ could play a very important

negative role in container handling operations,

since there are no theoretical instruments to access

the size of its influence.

2 The representative statistical data to use in its

evaluation are very difficult to acquire and reflect

to many affecting factors at once, excluding the

possibility to separate the one under study.

162

3 The paper offers a regular procedure which

introduces a simplified effect of box parties’

arrivals and storage on top of each other, that

makes it possible to identify the consequences and

make numeric assessments by comparison with

combinatorial calculations.

4 The procedure exploits the simulation model

described in the paper and provides rather

interesting results which need to discuss with the

expert society.

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