Inventory holding costs would include which of the following?

2.3 Mathematical programming model for integrated scheduling and controllability analysis

The major assumptions in the model:

Cyclic production schedule is desired

Demand rate of each kind of product is constant

The following are the parameters of conventional model:

ci = Production cost of product i per unit mass

di = Demand rate of product i

mi = Unit inventory holding cost for product i

ri = Production rate of product i

sij = Transition cost when production is changed from product i to j

τij = Transition time when production is changed from product i to product j

The following are the variables:

H = Length of cycle

Tk = Length of time slot k

Wik = Amout of product i produced during time slot k

Xik = Binary variable to denote assignment of product i and slot k

The binary variables defined as follows:

Xik=1if in time slotkthe productiis assigned0otherwise

(1)min=∑i∑kciriTik−∑j≠iτjiXjk−1XikH+∑i∑j∑ksijXik−1XjkH+∑j∑kmiWik

(2)∑iXik=1,∀k,

(3)H=∑kTk,

(4)Wik=riTk−∑j≠iτjiXjk−1Xik,∀i∀k,

(5)∑kWikH=di,

(6)Xik=01,Tik≥0,Wik≥0,H>0

(7)fiui1ui2…uir=0,

(8)ci=fciui1ui2…uir,

(9)ri=friui1ui2…uir,

(10)τij=fτijui1ui2…uir,

(11)Sij=fsijτij,

(12)g(ui1ui2…uir≥0,

(13)UrMin≤uir≤UrMax,

Since the demand at unit time of each product is constant, so the selling revenue is constant at unit time. Thus the objective function as shown in Equation (1) is to minimizing the inventory cost, transition cost and feedstock cost to maximize the profit at unit time. Equations (1) ~ (6) actually describe the conventional scheduling model of multi-product chemical process. Equation (7) is constraint of operating variables on production point to ensure that the chemical process produces product i. Equation (8) denotes the relationship between operating parameters and production cost. Equation (9) is the functional relationship between operating parameters and production rate. Equation (10) is the constraint of transition time. Equation (11) represents the functional relationship between transition cost and transition time. Equation (12) denotes the constraints of controllability analysis.

15.5.3 Inventory Models

Humanitarian logistics systems are usually forced to keep some of their required relief items and equipment in stock, in order to increase their levels of preparedness against sudden disasters. However, similar to commercial supply chains, high levels of inventory holding costs could be a burden on humanitarian organizations because of their limited funds and operating resources. Therefore, designing effective inventory systems for humanitarian organization can be of great importance.

To demonstrate the modeling of an inventory system for a humanitarian logistics system, we discuss a simple deterministic multiperiod model in detail. Consider a set of CDCs that respond to the demand for a set of relief items, K, ordered by a set of RDCs during various relief operations in multiple consecutive periods of planning, T. Each relief item has a limited supply in each period, skt, k ∈ K, t ∈ T; and the distribution centers have limited capacities, σi, i ∈ CDC. The demand of each RDC for each type of relief item is denoted by djkt, j ∈ RDC, k ∈ K, t ∈ T. Also, consider pkt, hkt, and πkt to be the purchasing, holding, and shortage costs of each relief item in each planning period. The pure inventory level, surplus inventory level, shortage inventory level, and purchasing quantities for each item in each period and CDC are denoted by Ikit,Ikit+,Ikit−,and qkit, respectively. Also, the percentage of the demand of an RDC for a relief item during a period allocated to a specific CDC is denoted by xjkit. Considering the above notations and assuming that the initial inventory level for each item in each CDC is known and equal to Iki,0, the multiobjective mathematical model of such inventory system for a humanitarian logistics system is as follows:

(15.28)Minimize∑t∈T∑k∈K{pk∑i∈CDCqkit+hk∑i∈CDCIkit+}

Minimize

(15.29)maxt∈T{∑k∈Kπk∑i∈CDCIkit−}

subject to

(15.30)Ikit−Iki(t−1)=qkit−∑j∈RDCdjktxjkit∀i∈CDC,k∈K,t∈T

(15.31)∑i∈CDCxjkit≤1∀j∈RDC,k∈K,t∈T

(15.32)∑i∈CDCqkit≤skt∀k∈K,t∈T

(15.33)∑k∈KIkit+≤σi∀i∈CDC,t∈T

(15.34)Ikit=Ikit+−Ikit−∀i∈CDC,k∈K,t∈T

(15.35)xjkit∈{0,1}∀i∈CDC,j∈RDC,k∈K,t∈T

(15.36)qki≥0∀k∈K,i∈CDC

In the above formulation, objective function (15.28) minimizes the total purchasing and holding costs for total planning horizon, whereas objective function (15.29) minimizes the maximum of the shortage costs in every planning period. Though objective function (15.29) is nonlinear, its linearization is fairly straightforward. Constraint (15.30) ensures an inventory balance of each relief item for each CDC in each period. Constraint (15.31) ensures demand for each item from each RDC is at most fulfilled completely. Constraints (15.32) and (15.33) ensure the supply limit of each item and capacity limit of each CDC in each period, respectively. Finally, constraint (15.34) relates the pure inventory level variable to its respective surplus and shortage inventory level variables. The proposed model is a multiobjective, mixed-integer linear programming model that requires efficient multiobjective optimization techniques such as multiobjective evolutionary algorithms for creating nondominated solutions.

It is noteworthy since humanitarian logistics systems mostly belong to nonprofit organizations, considering shortage cost and possibility of not satisfying an RDC are somehow questionable and against humanitarian principles. However, the proposed multiobjective model can be useful in reducing the negative effects of such assumptions. Also, it must be noted that in uncertain disaster environments, shortages can and will happen. Therefore, using various modeling techniques such as stochastic programming and robust optimization can be of great help in extending the above model to embrace such complex situations. Demonstration of such extension is beyond the scope of this chapter, and the reader is referred to [58,59] for an overview of the aforementioned modeling techniques.

At the end, it is noteworthy that research regarding effectively designing inventory systems for humanitarian logistics systems is still limited in comparison with other aspects of humanitarian logistics systems, and much research is needed. The reader is referred to [60] for a discussion of an advanced inventory model in the context of integrated logistics models (see Section 15.5.4).

Step 2: Modeling the Cost Equation

After developing equations that relate customer-service level and inventory, a cost equation was developed to derive the cost associated with a given service level:

(11.1)C(SL)=inventoryunitsperiod×inventoryholdingcostunit+expectedlostsalesunitsperiod×profit margin/unit

Inventory holding costs include warehousing costs, the decrease in the value of products from the time they are manufactured until they are sold, the opportunity cost of investing in inventory, and the scrapping of obsolete products.

The cost of lost orders was determined by asking customers to indicate their reactions to stock outs. Interviews revealed that 80% of customers would wait for the desired product or accept a substitute from the same manufacturer; the other 20% would buy the desired product from a competitor.

The developed cost equation is shown in Table 11.5.

Table 11.5. The Notations of Cost Equation

tPeriod indexiProduct indexSLitService level (percent of orders delivered when requested) for product i in period tWOIitUnits of inventory of product i held during period t divided by weekly forecasted demandFitAverage weekly forecast for product i during period t (units)DitAverage weekly demand for product i during period t (units)PiVariable production cost for product iRiRevenue for one unit of product iHitInventory holding cost as a percent of variable production cost for product i in period t

The inventory holding cost of customers was determined to be:

(11.2)IHCit=HitPiave(WOIit)Fit

Considering Eqn (11.2), the overall cost equation is:

(11.3)C(SLit)=IHCit=HitPiave(WOIit)Fit+0.2(1−SLit)Dit(Ri−Pi)

Other Considerations

2.5 Objective Function

We propose a solution procedure to look for the optimal solution of the MINLP model. It consists on sequentially solving a pair of mathematical programming formulations: (a) an MILP continuous-time model that is a tight approximation of the MINLP rigorous formulation, and (b) an NLP model that includes the nonlinear terms arising in the rigorous expressions of the key component mass balances, and accurate inventory carrying cost calculation. The NLP model is obtained from the original MINLP formulation by fixing the binary variables.

The MINLP problem goal is given in Eq. (17), where Cwvt stands for vessels demurrage costs; Cuvt accounts for the cost of crude unloading; Csett considers setup costs for inter-tank crude transferring; Cchtt represents changeover costs incurred by switching CDU feeds; and the last two terms are the inventory holding costs of crude mix in storage tanks (Cinvkt) and charging tanks (Cinvmt) throughout the planning horizon.

(17)Minz=Cwvt+Cuvt+Csett+Cchtt+Cinvkt+Cinvmt

In the MILP objective function the last two terms are computed by linear expressions based on estimated average inventory costs. For reference, Eq. (18) estimates the average inventory levels in charging tanks (Ainvmtm). The sum of the inventory in every control point is divided by the total number of inbound |IKnew| and outbound |IM| flows (i.e., the total number of time events where inventory levels are controlled).

(18)Ainνmtm=1|IKnew|+|IM|(∑ik∈IKnewIm(ik)+∑im∈IMIm(im))

By multiplying Ainvmtm by the horizon length and by the unit inventory holding cost we obtain the total estimated inventory cost in charging tanks (Cinvmt). A similar approach is used for determining Cinvkt. In contrast, the NLP model comprises a simple nonlinear computation of the exact inventory holding costs. The last two terms in Eq. (17) are generically accounted for by expression (19). Parameter cinvt stands for the inventory cost in tank t, iit represents initial crude inventory in tank t. Indices in and out account for generic inbound and outbound lots, respectively. Q, S and C represent the lot size, start time and completion time of every discharge.

(19)cinvt∑tiitH+∑inQinH−Sin+Cin2−∑outQoutH−Sout+Cout2