Distress among disaster-affected populations: delay in relief provision

Abstract

Central to humanitarian logistics is the minimization of distress among impacted populations in the aftermath of a disaster. In this paper, we characterize two levels of distress, termed criticality and destitution, with respect to the delay provision of relief items. Delay in provision of a relief item will lead to destitution for a tolerable number of days, beyond which it will lead to criticality. We develop a mixed-integer goal program that quantifies these two metrics with respect to the number of days without provision of each of a set of relief items. The model determines the allocation of resources and the distribution of available relief items in a manner that minimizes criticality and destitution in affected population segments. The use of the model is demonstrated for the aftermath of a catastrophic earthquake in Istanbul, expected to occur by 2030.

Keywords:

• humanitarian logistics
• planning
• distribution
• integer program

“Relief material for Jammu & Kashmir lies in neglect on railway platforms.”—The Indian Express, 9-24-2014.

Introduction

The stark commonality in reports of disasters is the distress among affected populations due to delays in provision of relief items, ranging from essential infrastructural items, such as tents or medical devices, to consumables, such as water or toiletries. After Cyclone Nargis hit Myanmar in 2008, provision of water, medication, food, and shelter was delayed by more than a week. After the 2014 flood in the Kashmir valley, transport of relief materials was delayed by over a fortnight due to lack of trucks and manpower. The greater the delay, the greater the distress due to lack of one or more of the relief items. Delays of a couple of hours in provision of medical supplies can prove to be critical, of a few days in provision of water, of a week in provision of food.

The defining characteristic of a disaster, as noted by the International Federation of Red Cross and Red Crescent Societies, is the insufficiency of immediately available resources. The Federal Emergency Management Agency in the United States formalizes proactive planning in two stages of disaster management, namely response and preparedness. Much of the literature on disaster relief focuses on response, i.e., last mile provision of supplies to affected population segments. The literature that focuses on strategic planning of relief provision, i.e., preparedness, has addressed shortfalls in supplies of relief items in meeting demand. More recently, the relationship of such shortfalls to the distress faced by populations has received attention. However, shortfalls are not only due to insufficient supply of relief items. In the immediate aftermath of a disaster, deteriorated relief provision can be due to, beyond the lack of timely available relief items, limited transport capability and compromised manpower/handling capacity.

In this paper, we address the allocation of resources, as they become available, for the distribution of relief such that overall distress among different segments of the population is minimized. Avoiding, or at least minimizing, shortfalls in relief provision due to delays in the availability of relief items, transport vehicles, and handling capacity entails decisions about the timing and amounts of replenishment of different relief items for each of several distinct population segments. Further, such decisions should be in concert with the need for specific relief items in each population segment. The need, i.e., level of distress in a segment of the population depends on the extent of delay in provision of one or more of the relief items and changes dynamically over time as relief items are provided. Distress begins when replenished provisions are depleted. It becomes greater with the passage of days, during which the population is increasingly destitute, up to a tolerable delay beyond which criticality, the extreme level of distress, sets in.

In Section 2, we review the disaster management literature that addresses shortfalls in meeting the needs for relief items. In Section 3, we formalize the relationship between delays in provision of supplies and distress. In Section 3.1, the relief states that a population segment can be in are defined with respect to the time since replenishment by relief items as being provided, destitute, or critical. In Section 3.2, we formalize the transitions of population segments through different relief states with respect to the replenishment by different relief items. Section 3.3 presents the mixed-integer goal programming model which distributes relief items while minimizing criticality with priority one and destitution with priority two. In Section 4, we demonstrate the use of the developed model in relating delays in relief provision—due to insufficiency in relief items, vehicles, and handling capacity—on criticality and destitution among affected populations in the aftermath of a catastrophic earthquake in the greater Istanbul area which is predicted to occur by 2030 (Parsons, 2009). Section 5 concludes the paper with summarizing comments.

The relevant disaster management literature

The body of literature in the area of disaster operations management that has evolved over the past two decades has been surveyed comprehensively by Altay and Green (2006) and Natarajarathinam et al (2009) and, more recently, by Galindo and Batta (2013). Models that address the minimization of demand shortfalls encompass a variety of contexts, primarily using modeling constructs similar to those used in retail distribution, vehicle routing, and facility location models. Ozdamar et al (2004) minimize unsatisfied demand over a planning horizon in determining optimal quantities and types of loads picked up. Dessouky et al (2006) minimize unmet stochastic demand when locating facilities for distribution of medical supplies using optimized vehicle routes with stochastic travel times. Jia et al (2007) include the possibility of lack of supplies and possible shortfall in meeting demand in the distribution of medical supplies for large-scale emergencies. Tzeng et al (2007) include the maximization of satisfied demand while addressing two other objectives: minimizing the total cost and minimizing the total travel time in locating facilities in a two-echelon network in which relief items are transferred from relief collection points to candidate depots and then to demand locations. Yi and Ozdamar (2007) addresses unsatisfied demand for medical equipment and manpower in the context of evacuation of wounded persons in locating temporary and permanent emergency facilities. Rawls and Turnquist (2010) penalize unmet stochastic demand in a model that addresses costs of facilities, procurement, inventory, and shipment in locating facilities and distribution. Huang et al (2011) capture response time and equity in service, by penalizing the percentage of unsatisfied demand at demand points using a convex nondecreasing step function. Rawls and Turnquist (2012) include the reliability of meeting specific levels of demand in the distribution of supply to persons evacuated to shelters.

Departing from addressing logistics costs, Yushimito et al (2010) minimize a social cost function when locating distribution centers that maximize coverage of affected regions. Holguin-Veras et al (2013), pointing out that the impact of shortfalls has not been addressed, minimize deprivation which is expressed as a nonlinear convex function that is monotone with respect to the time since last delivery. Perez Rodriguez (2011) minimize deprivation, travel time, and handling costs for the allocation and distribution of critical items.

The model that we develop determines the replenishment amounts of different relief items for different segments of the population over time by an allocation of resources over time, accounting for the days since last replenishment with respect to each item. The necessarily complex constraints that are developed afford an objective that captures two levels of distress, namely criticality and destitution, taking into account the timing and amounts of replenishments to different populations segments.

Planning relief replenishment with visibility of distress

In this section, we formalize the relationship between the lack of timely provision of relief items to, and distress among, affected populations. For each relief item (henceforth simply item) , we define the days-of-provision, denoted , to be the number of days for which the replenished amount is adequate for a single person, and the days-of-destitution, denoted , to be the number of days for which a person can survive without the relief item. An infrastructural item, such as a tent, is supplied only once and is categorized to be Type I. A consumable item, such as water, which requires replenishment is categorized as Type C. A Type I item has the days-of-provision defined, for modeling purposes, to be 0, i.e., . For Type C items the demand recurs every days. Section 3.1 introduces relief states and their classification with respect to the time since last replenishment for each relief item. In Section 3.2, we describe how population segments transition from one relief state to another after replenishment with different mixes of items. The multiperiod, integer goal programming model that we develop is presented in Section 3.3.

Relief states and distress

The relief state of a population segment with respect to a single relief item is defined to be the number of days since replenishment. The days-of-provision and days-of-destitution allow a classification of the states as either provided, destitute, or critical. When a population segment is replenished, it is in a provided state. The lack of a relief item leads to destitute states for a population segment which, when not replenished, eventually enters a critical state. States are provided, states are destitute, and state is critical. For modeling purposes, is defined to be larger than the number of days in the planning horizon for items with respect to which criticality cannot occur. A Type C item is assumed to provide a population enough supplies to last through the provided states. For example, for a Type C item with days-of-provision and days-of-destitution, the relief states range from 1 to 6 where states 1, 2, and 3 are provided, states 4 and 5 are destitute, and state 6 is critical. A Type I item, with days-of-provision defined to be 0, has a single provided state, and states are destitute. The relief state of a population segment with respect to the entire set of relief items is denoted by a K-tuple , where is the relief state with respect to item k and the set of all relief states is denoted by . The state of a population segment is provided if the state with respect to all items is provided. The state is destitute or critical if the state with respect to at least one item is such.

Figure 1 Destitution due to lack of water and shelter

We define destitution due to the lack of provision of a relief item k as a convex piece-wise linear monotone increasing function truncated over days since replenishment. The population is provided for the first days since replenishment, and hence there is no destitution. We distinguish lack of replenishment or provision beyond as being critical. The destitution faced by populations can be more severe with respect to some items, and further, more severe in some districts. Figure 1 displays destitution functions, for one district, for a consumable item, water, for which and , and an infrastructural item, tents, for which and , where T is the number of days in the planning horizon. The slopes of the piece-wise linear function are defined to be constants that quantify the destitution faced by the population in district j for which days have lapsed since replenishment of item k. The constants, thus, satisfy , where . The destitution faced by a population in state r, denoted , is defined to be the largest of the item destitution levels, i.e., .

Relief state transitions

A population segment in a provided or critical state does not need replenishment. Populations in destitute states need to be replenished by one or more items. Different mixes of relief items, referred to as bundles, are distributed to different segments of the population. The specific mix of relief items in any bundle is dynamically constituted, taking into account available amounts of relief items, available levels of resources, and different needs in population segments, on any day. The unique state that a population segment transitions to depends on the current state and the specific items, if any, they have been replenished with.

Figure 2 Transitions of population segments in four states after replenishment

Figure 2 illustrates transitions of 540 persons in one of four states, three of which are destitute. The three Type C items with , , days-of-provision, , , days-of-destitution, and supplies of, respectively, 245, 160, and 505 units. The 100 persons in provided state [2,2,1] do not need replenishment and transition to state [3,3,2]. The 120 and 140 persons in destitute states [4,1,5], [5,1,4] are in need of items 1 and 3 since the time since replenishment for both items 1 and 3 is greater than 3. A total of 245 units of bundle is distributed to 110 persons in state [4,1,5] and to 135 in state [5,1,4]. Thus, 110 persons from state [4,1,5] and 135 from state [5,1,4] transition into state [1,2,1]. The days since replenishment for the 10 persons in state [4,1,5] that are not replenished increase by one for each item and they transition, on the next day, to state [5,2,6], which is critical due to item 3. Similarly, the 5 persons in state [5,1,4] that are not replenished transition to state [6,2,5], which is destitute. The 180 persons in destitute state [1,4,5] are in need of items 2 and 3. Of these, 160 transition to provided state [2,1,1] after replenishment by bundle and the remaining 20 transition to the critical state [2,5,6].

Figure 3 Possible Transitions Into a State After Replenishment by Type I Items

Figure 3 illustrates possible transitions to a unique state from multiple states after replenishment by different bundles that are subsets of a mix of Type I items. In this example, there are three Type I items, , and one Type C item, , with , , , days-of-provision, and , , , days-of-destitution. Transition into state [4,0,5,0] is possible from multiple states with 3 and 4 days since replenishment of, respectively, item 1 and 3, after replenishment by bundle , and its subsets, i.e., . Namely, states [3,1,4,1], [3,1,4,2],... [3,6,4,8] after replenishment by , states [3,1,4,0], ... [3,6,4,0] after replenishment by , states [3,0,4,1], ... [3,0,4,8] after replenishment by , and destitute state [3,0,4,0] after no replenishment.

Model

The stochastic instance-based model that we present in this section determines the mixes of relief items to distribute based on the needs of segments of the population in different states using resources available, on each day, to minimize criticality and destitution over the entire planning horizon. The availability of resources is reflected by stochastic parameters specifying the amounts of the items, the number of vehicles, and the manpower, made available on each day. The model tracks the states of specific segments of the population with respect to replenishment by different bundles of items across the planning horizon, and a multicriteria objective captures the extents of criticality and destitution in populations over the planning horizon. The planning horizon is defined in days, where each day is further divided into time slots to better reflect the time windows during which vehicles transport items. The model ships items from supply sites, assumed to have unlimited capacity, to distribution sites in vehicles of differing capacities over the planning horizon. Vehicles return, after delivery, to any of the supply sites for subsequent trips. The time for a vehicle to get from a supply time to a distribution site accounts for travel. However, the return time for a vehicle to travel from a distribution site to a supply site includes, additionally, the time taken for bundling of items and the distribution of bundles. Vehicles that are not deployed on a day remain at either a supply or a distribution site. Before presenting the model, the index sets, parameters, and variables are defined.

Index sets:

	=	Set of instances indexed by
	=	Set of supply sites,
	=	Set of distribution sites,
	=	Set of days of planning horizon,
	=	Set of time slots in each day
	=	Set of vehicle types
	=	Set of relief items
	=	Subset of relief items of Type I,
	=	Subset of relief items of Type C,
B	=	Subset of items that a population is replenished with.
	=	Superset of item bundles,
	=	Superset of bundles of items that include item k
	=	Subset of bundle ,
	=	Superset of subsets of bundles of items in ,
	=	Set of relief states r
	=	A vector of relief states with respect to each item, , where can be if or can be if
	=	Subset of provided states where and,
	=	Subset of destitute states where
	=	Subset of critical states where
	=	Subset of states that the population can transition into by receiving a bundle or its subset on the previous day, where ; ; ; .
	=	Subset of states that the population can transition from by receiving a bundle or its subset on the previous day, where ; ; ; .
	=	Relief state, , that a population segment in state r transitions to, if not replenished
	=	Bundle of items that allows a population segment in r to transition to a provided state
	=	State of a population segment that transitions to state r if not replenished

Parameters:

	=	Supply of item available on day t in instance
	=	Handling capacity, in number of vehicles, at available on day t in instance
	=	Number of available vehicles of type made available on day t in instance
	=	Population at distribution site
	=	Number of time slots taken by a vehicle to travel from to
	=	Number of time slots taken by a vehicle for distribution at and return from to
	=	Vehicle capacity in lbs,
	=	Days-of-destitution
	=	Days-of-provision
	=	The destitution level for a population segment in district j in state r
	=	Initial state for replenishment for item , defined to be
	=	Preemptive priorities corresponding to two objectives

Variables:

	=	Amount of item allocated to supply site on day
	=	Inventory for item at on day
	=	Inventory for mix at distribution site on day
	=	Amount of flow of item from to on day in time slot
	=	Number of people in destitute state at on day satisfied by bundle
	=	Number of people in relief state at on day
	=	Total criticality at due to shortage of item k on day
	=	Units of bundle containing at on day
	=	Units of bundle at on day
	=	Vehicles of type made available at in time slot on day
	=	Vehicles of type en route from to in time slot on day
	=	Vehicles of type en route from to in time slot on day
	=	Idle vehicles of type at in time slot on day
	=	Idle vehicles of type at in time slot on day
	=	Total criticality over the planning horizon
	=	Total destitution over the planning horizon

Model: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 The objective minimizes criticality at priority one and destitution at priority two. The two goals are defined in constraints 22 , and 33 . Constraint 44 allocates supply to supply sites in each time period. Constraints 55 and 66 enforce the inventory balance for each relief item, at each supply site, in each time period. Constraints 77 and 88 create bundles of relief items and their inventory balance is enforced in Constraints 99 and 1010 at distribution sites. Constraints 1111 and 1212 initialize the population at each distribution site to the first destitute state, , for each item. Constraints 1313 –1515 track the progression of the population with respect to both time and relief state. Constraint 1313 accumulates transitions into each state for all bundles B, namely population segments in (i) state which are replenished with a bundle or its feasible subsets ; (ii) the population segment in state that are not replenished by any bundle. Each state can be transitioned into from states , when replenished by the items in specific subsets . The subsets, , of the bundle B that need to be taken into account must necessarily contain all the Type C items and can contain none or several of the Type I items. Constraint 1414 moves the portion of the population in state that is not replenished to a unique state where if , and if . Constraint 1515 , which is definitional, tracks the population in the critical states. Constraint 1616 ensures that no supply is allocated to a relief state if it has a zero population.

Constraint 1717 allocates vehicles made available in specific time periods to supply sites. Constraint 1818 initializes the vehicle inventory at each supply site. Constraint 1919 initializes the vehicle inventory at the distribution sites. Constraints 2020 and 2121 enforce the conservation of flow and inventory balance of vehicles at, respectively, each supply and distribution site in each time slot. Constraint 2222 ensures that the vehicle capacity is not exceeded. Constraint 2323 ensures that the handling capacity at each supply site is not violated. Finally, Constraints 2424 –2626 specify nonnegativity and integrality of the variables.

The model minimizes criticality and destitution for the levels of supply, vehicles, and manpower specified in instance . Delays are reflected in the specific values given to the parameters for the supply, , of item , the number of vehicles, , and the handling capacity at supply site i, , in each time period t.

Illustrative examples

In this section, we present two examples of the transition of population segments from state to state.

Figure 4 Transitions of population after replenishment by a single type C item

In the first example, a single Type C item with days-of-destitution and days-of-provision has a supply of 20, 40, 60, 80, 100, and 120 units over the first six days. The replenishments for a population of 200 at a single distribution site over a planning horizon of 12 days are illustrated in Figure 4. The distribution of the item is spread over the planning horizon, minimizing criticality. Over the 12 days, criticality is 80 and destitution is 400 accounting the persons not replenished when in a destitute state. Only 20 persons are replenished on the first day and 180 persons transition to state [5], of which 140 transition to state [6] on day 3 and, finally 80 transition to state [7] on day 4. From the 4th day onwards, there is excess supply and much of it remains in inventory. There are 80 units in inventory at the end of the 4th day, and 160, 240, 180, 180, 160, 120, 60, and 60 units at the end of, respectively, days 5 through 12. Had the 80 units of supply on day 4 been available a day earlier, there would be no criticality.

Figure 5 Transitions of population after replenishment by three type C items

In the second example, there are three items, the first of Type I and the latter two of Type C, with and . Supplies of , that are available at the outset are distributed to a population of 200 in destitute state [1,4,5]. Replenishments over a 12-day horizon in bundles that minimize destitution are illustrated in Figure 5. On the first day, 28 units of bundle {1, 2}, 58 of {1,3}, and 114 of {1, 2, 3} are distributed. Replenishments of bundle {2} on days 5 and 9 and of {3} on days 6 and 11 keep 114 in provided states. The resulting total destitution is 372: 28 persons remain destitute on days 1–5 due to nonreplenishment of item 3; 58 persons remain destitute on days 1–4 due to nonreplenishment of item 2. The total criticality is 86 (58 on day 5 and 28 on day 6).

Planning relief replenishment for Istanbul

In this section, we demonstrate the use of the developed model to examine the impact of delays in availability of relief items, vehicles, and handling capacity on the distress faced by affected populations. The study uses the four scenarios of damage, namely A, B, C, and D, to the North Anatolian Fault documented in a report published by Japan International Cooperation Agency (JICA) in collaboration with Istanbul Metropolitan Municipality (JICA et al, 2002; Gormez et al, 2011). Of the four scenarios, Scenario A is documented to be the most probable, and Scenario C to be the worst-case. The estimated number of persons affected in the 30 districts of Istanbul, computed using data on the average number of persons per building and the numbers of heavily, moderately, and partially damaged buildings, for Scenarios A–D are, respectively, 1367550, 1139900, 1489800, and 1007400. The supply sites in the study are the seven largest harbors in Turkey which, we assume, can deploy 50% of their total capacities for relief operations. In the model, the 14 more severely affected districts can be replenished only by small capacity vehicles while the other 16 districts by large capacity vehicles.

Table 1 Days-of-provision and days-of-destitution for six relief items

	Relief Item
	Tents	Medical devices	Water	Food	Toiletries	Medical kits
Days-of-provision,	0	0	3	5	6	5
Days-of-destitution,	†	2	4	7	†	6
Weight in lbs,	25	25	33	3	19	3

† Days-of-destitution is larger than the number of days in the planning horizon

We consider provision of subsets of items listed in Table 1 that are selected to be representative of need in differently affected population segments.

• Injured Relief items required by injured population segments (Item 1: Tents, Type I; Item 2: Medical Devices, Type I; Item 3: Medical Kits, Type C).

• Uninjured Relief items required by uninjured population segments (Item 1: Tents, Type I; Item 2: Water, Type C; Item 3: Food, Type C)

• All Relief items required by the entire affected population. (Item 1: Toiletries, Type C; Item 2: Water, Type C; Item 3: Food, Type C).

Baseline for required supplies and transport capacity

To provide a baseline for the required levels, and timing, of distribution of different subsets items and determine the consequent vehicle deployment, we create twelve instances, three each for the four scenarios, A–D, using the three above subgroups of differently affected populations. The model in Section 3.3 is used to compute the minimum number of preallocated vehicles and amounts of supplies of each item required for zero criticality and destitution for a planning horizon of days. Specifically, the parameters, and are converted, respectively, to variables and , and the criticality and destitution levels, and set to zero in, respectively, constraints 22 and 33 . The objective is altered to and the optimal values of these variables are retained. In a subsequent optimization, these levels of supply are used to determine the minimum number of vehicles by altering the objective to . The solution is then used to compute the minimum required supply for each item and the minimum required number of vehicles . The obtained results for the twelve instances are reported in Table 2 in which an instance is identified by the type of subgroup (‘i’ for Injured, ‘u’ for Uninjured, and ‘a’ for All) and the scenario (A, B, C, D).

Table 2 Required vehicles and supplies for zero criticality and destitution

Instance	Population	Supply Amount^†			Vehicles			Trips
		Item 1	Item 2	Item 3	Large	Small	Total	Total	/vehicle	Day 1 (%)
iA	115250	115250	115250	230500	253	618	871	935	1.07	93
iB	95000	95000	95000	190000	217	495	712	812	1.14	88
iC	135100	135100	135100	270200	295	724	1019	1098	1.08	95
iD	83700	83700	83700	167400	180	460	640	744	1.16	86
uA	1252300	1252300	3756900	2504600	2858	7990	10,848	23703	2.19	46
uB	1044900	1044900	3134700	2089800	2453	6556	9009	21035	2.33	43
uC	1354700	1354700	4064100	2709400	3119	8597	11716	25268	2.16	46
uD	923700	923700	2771100	1847400	2096	5920	8016	17104	2.13	47
aA	1367550	2735100	4102650	2735100	2829	7838	10,667	28722	2.69	37
aB	1139900	2279800	3419700	2279800	2430	6415	8845	23355	2.64	38
aC	1489800	2979600	4469400	2979600	3111	8496	11607	30873	2.66	38
aD	1007400	2014800	3022200	2014800	2068	5805	7873	20897	2.65	38

^†(1, 2, 3): i(Tents, Medical Devices, Medical Kits); n(Tents, Water, Food); a(Toiletries, Water, Food)

For zero criticality and destitution, the minimum required amount of a relief item is dictated by the days-of-provision which dictates the required number of replenishments. For example, for instance nC with a population of 1354700, exactly that many units of the first item, tents, are required, three times as many units of the second item, water (since ), and twice as many units of the third item, food (). Fewer large vehicles are required not only because they have greater capacity but, more so, because the number of persons in the districts served by large vehicles is a smaller portion of the total population. For example, of the total of 11716 vehicles required for instance uC, 3119 are large, serving 509800 persons in 16 districts, and 8597 are small, serving 844900 persons in 14 districts.

Figure 6 Vehicle requirements for twelve instances

Figure 6 displays the number of vehicles, number of trips, and percent trips on the first day for the twelve instances. The total number of vehicles and the number of trips depend on the number of affected persons as well as the mix of relief items. Both are highest for Scenario C and lowest for Scenario D because of the associated weight of items that need to be transported. Further, the number of trips is higher when the required number of replenishments is higher. The opposite is true when fewer replenishments are required with consequently higher percentage of trips on the first day. For example, for the mix of items in the ‘injured’ instances the trips per vehicle is lower and the percentage trips on the first day is higher, whereas the opposite is true for the ‘all’ instances. For the ‘injured’ instances, the brunt of the weight is in Type I items, which must be distributed on the first day to avoid criticality and destitution.

Impacts of delays

To examine the impact of delays in the availability of relief items, vehicles, and handling capacity on criticality and destitution, we focus on the worst-case scenario, namely Scenario C. For each of the three population subgroups (injured, uninjured, and all), two levels of delay are defined for each of the three resources. In the low level of delay in relief item availability, 50% of the required amount of an item (obtained in the base instance) is available at the outset, with 25% becoming available on days 1 and 2. In the high level of delay, 25% is available at the outset with 25% becoming available on days 1, 2, and 3. In the low-level delay for vehicle availability, one-sixth of the required number of vehicles are available on day 1, two-sixths on day 2, and three-sixths on day 4. In the high-level delay, two-sixths are delayed further by a day to day 3. The extent to which handling capacity is compromised is reflected by lowered percentages of capacity allocated for relief operations. For the both low- and high-level delay, the capacities of the three ports in the greater Istanbul area are lowered to 0% and of the other four ports lowered to 40% for days 1–3 only. For the high-level delay, the three Istanbul ports have 40% available on day 4 with full capacity recovered as of day 5.

Table 3 Delayed relief provision instances—Scenario C

Instance	Days-destitute/capita^†				Destitution/capita^†				Criticality		Percent trips
	Item 1	Item 2	Item 3	Total	Item 1	Item 2	Item 3	Total	Total	Percent	Day 1	Days 1–3
i100	0.3	0.3	0.0	0.5	0.3	0.3	0.0	0.5	0	0	70	95
i010	1.7	0.8	0.0	2.4	6.6	0.8	0.0	7.4	0	0	16	95
i001	0.0	0.0	0.0	0.0	0.0	0.0	0.0	.0	0	0	95	95
i111	2.0	0.8	0.0	2.8	16.5	0.8	0.0	17.3	0	0	16	77
i200	0.8	0.8	0.0	1.5	2.6	2.8	0.0	5.3	33775	25	65	95
i020	2.0	0.9	2.0	4.9	9.4	3.2	9.7	22.3	37020	27	22	95
i002	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0	0	95	95
i222	2.0	1.0	2.0	5.0	10.4	3.8	9.7	24.0	45420	34	25	93
Average	1.1	0.6	0.5	2.1	5.7	1.5	2.4	9.6	14527	10.75	50	92
u100	0.3	0.3	0.0	0.5	0.3	0.3	0.0	0.5	0	0	33	54
u010	0.7	1.7	0.0	2.4	0.7	9.9	0.0	10.6	0	0	8	44
u001	3.7	2.3	0.0	6.0	106.4	27.1	0.0	133.5	0	0	4	12
u111	3.7	2.3	0.0	6.0	106.4	27.1	0.0	133.6	0	0	4	12
u200	0.8	0.1	0.0	0.8	2.6	0.1	0.0	2.7	0	0	33	51
u020	1.0	2.4	0.0	3.4	3.3	21.5	0.0	24.8	0	0	8	38
u002	4.2	2.0	0.0	10.2	138.0	21.9	116.4	276.3	12962	1	4	12
u222	4.2	2.1	4.0	10.3	138.0	22.7	116.4	277.0	12962	1	4	12
Average	2.3	1.6	1	5	62	16.3	29.1	107.4	3240	0.24	12	29
a100	0.0	0.4	0.0	0.4	0.0	0.4	0.0	0.4	0	0	28	46
a010	0.7	1.6	0.0	2.3	0.7	8.9	0.0	9.5	0	0	6	36
a001	4.0	2.2	1.9	8.1	115.2	24.3	56.2	195.7	0	0	3	10
a111	4.0	2.2	1.9	8.1	115.2	24.6	56.2	195.9	0	0	3	10
a200	0.0	0.4	0.0	0.4	0.0	0.4	0.0	0.4	0	0	28	44
a020	0.9	2.3	0.0	3.2	2.3	20.3	.0	22.5	0	0	6	31
a002	4.0	2.2	4.0	10.2	109.8	29.3	116.4	255.5	126422	8	4	11
a222	4.0	2.3	4.0	10.3	109.8	30.4	116.4	256.6	126422	8	4	11
Average	2.2	1.7	1.5	5.4	56.6	17.3	43.1	117.1	31605	2.12	10	25

^†(1, 2, 3): i(Tents, Medical Devices, Medical Kits); n(Tents, Water, Food); a(Toiletries, Water, Food)

Table 3 reports the results individually for the eight stochastic instances for each of the three population segments considered. The model for each instance is defined by the values for the parameters , , and and solved with criticality being minimized at priority one and destitution at priority two. Each instance is identified by four digits: The population subgroup (i, u, or a), followed by three digits that reflect the level of delay in, respectively, the available supplies, the available vehicles, and the available handling. No delay is represented by ‘0,’ low-level delay by ‘1,’ and high-level delay by ‘2.’ The impacts of delays in distribution are reflected in the Days-Destitute/Capita, Destitution/Capita, Total and Percent Criticality, and Percent Trips on Day 1 and Days 1–3.

The model determines distribution mixes that minimize criticality and destitution by prioritizing items with respect to days-of-destitution (). We see that the Days-Destitute/Capita tend to be lower for Medical Devices () for the injured mix of items and for water () in the uninjured and all mixes of items. For the injured instances, the possibility of entering criticality due to lack of provision of Medical Devices is greater. For this mix, criticality occurs for high-level delays in availability of relief items or vehicles (i200, i020, i222). For the ‘uninjured’ and ‘all’ instances (u002, u222, a002, a222), a high-level delay in handling capacity leads to criticality due to compromised distribution of water due to its weight. For these four instances, the total days-of-destitution per capita is above 10 and the destitution per capita is above 250. For both the ‘uninjured’ and ‘all’ instances, we see that the degradation in destitution per capita due to heavier delays in handling capacity is significant. For example, destitution per capita deteriorates from 133.5 for u001 to 276.3 for u002 when manpower availability is further delayed. Since the volume associated with the items in the ‘all’ instances is larger than that for the ‘uninjured’ instances, the criticality is also much higher, i.e., 8% for a002 and a222 as opposed to 1% for u002 and u222. A low- or high-level delay in availability of relief items (i100, i200, u100, u200, a100, a200) leads to some destitution only since the mix of items distributed is dynamically prioritized to avoid criticality.

The Percent Trips on Day 1 affords a means of gauging the deterioration in provision of relief. When vehicle and handling capacity are not compromised, a greater percentage of the trips can be made on the first day. The baseline recorded for zero criticality and destitution, depending on the mix of relief items and the number of affected persons, for this metric is 95% for instance iC, 46% for uC, and 38% for aC. The brunt of the items for the ‘injured’ instances needs to be delivered forthwith as opposed to the ‘all’ and ‘uninjured’ instances in which there is recurrent replenishment over the planning horizon. The percent trips on the first day does not deteriorate for the ‘injured’ instances—criticality and destitution remain 0 (i001, i002), primarily because the overall volume of items to be distributed is relatively low. Quite the opposite is seen for both the ‘uninjured’ and ‘all’ instances: The percentage of trips required for the first day drop to about 4% (u001, u002, a001, a002).

Summary and conclusions

The model developed in this paper distributes dynamically constituted mixes of relief items, simultaneously accounting for needs in different segments of disaster-affected populations, the days since last replenishment, the days-of-destitution and weight of relief items, and the available resources. The model distinguishes between two levels of distress, termed criticality and destitution, both of which are sought to be minimized. It is used to relate the extent of delay in availability of supplies, transport capacity, and handling capacity to the extent of criticality and destitution among affected populations. The computational study in the paper pinpoints the distinguishing characteristic of humanitarian logistics: That the requirements for resources are greatest at the onset of distribution. As such, strategic planning of disaster relief, beyond the consideration of infrastructure for, and storage of, relief items, must include the establishment of protocols for immediate deployment of transport and manpower capacity.