We consider an assemble-to-order system where one of the components faces uncertainty in the supply process in which the actual available quantity is equal to some random fraction of the production quantity. Demand is assumed to be price-dependent. We analyze how the supply uncertainty of one component affects the product pricing and production quantities of all the components under the assembly structure. We show that it is profitable for the firm to assemble the product only if the product price exceeds a certain threshold. This price threshold increases as the unit cost of each component or the degree of variability of the supply reliability distribution increases, but it is independent of the underlying demand function and demand distribution. Also, the optimal product price decreases as supply uncertainty decreases. We further show that under deterministic demand, the components can be managed independently such that the production quantity or unit cost of the component with supply uncertainty does not affect the optimal production quantity of the other components, as long as it is profitable to assemble the product. However, when demand is stochastic, the optimal production quantity of each component depends on the supply reliability distribution as well as the unit costs of the other components. For a fixed product price, the optimal production quantities of the components are smaller when the unit product price is low, and they are higher when the unit product price is high as compared to the case with no supply uncertainty.

In this article we analyze the interactions among the assembler and two component suppliers in their procurement decisions under a Vendor-Managed Inventory (VMI) contract. Under the VMI contract, the assembler rst o ers a unit price for each component and will pay component suppliers only for the amounts used to meet the actual demand. The two independent component suppliers then decide on the production quantities of their individual components before the actual demand is realized. We assume that one of the component suppliers has uncertainty in the supply process, in which the actual number of components available for assembly is equal to a random fraction of the production quantity. Under the assembly structure, both component suppliers need to take into account the underlying supply uncertainty in deciding their individual production quantities, as both components are required for the assembly of the nal product. We rst analyze the special case under deterministic demand and then extend our analysis to the general case under stochastic demand. We derive the optimal component prices o ered by the assembler and the corresponding equilibrium production quantities of the component suppliers. . IntroductionSourcing components from a complex global supplier network can lead to a high degree of uncertainty in the supply process. Various supply chain glitches such as unexpected supply disruptions, insu cient supplier capacity, or transportation delays across borders can cause unexpected shortfalls in the required components and halt the assembly of the nal products. At the same time, long procurement lead times in a global supply network make it di cult and expensive to deal with such component shortfalls using emergency orders. Consequently, rms need to e ectively manage supply uncertainty in their component procurement decisions to avoid such potential component shortfalls. This is especially critical for products with rapidly changing technology or short life cycle such as electronic products, as it is expensive to keep safety stock of components due to high obsolescence costs.In this article we analyze the interactions among the assembler and component suppliers in their procurement decisions under a Vendor-Managed Inventory (VMI) contract. Specifically, we assume that the assembler needs to procure two required components from two independent suppliers to assemble the nal product. Under the VMI contract, the assembler rst o ers a unit price for each component and will pay component suppliers only for the amounts used to meet the actual demand. Based on the unit component prices o ered by the assembler, the two independent component suppliers then decide on the production quantities of their individual components and ship the components to the assembler before the actual demand is realized. Here, we assume that the production CONTACT Kut C. So rso@uci.edu Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/uiie. and/or transportation lead times are long,...

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In this paper we use Kuperberg's sl 3 -webs and Khovanov's sl 3 -foams to define a new algebra K S , which we call the sl 3 -web algebra. It is the sl 3 analogue of Khovanov's arc algebra.We prove that K S is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of q-skew Howe duality, which allows us to prove that K S is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group K ⊕ 0 (W S ) Q(q) , to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that K S is a graded cellular algebra. 1 used [1] to give his formulation of Khovanov's link homology, we use Khovanov's [34] sl 3 -foams.We prove the following main results regarding K S .(1) K S is a graded symmetric Frobenius algebra (Theorem 3.9).(2) We give an explicit degree preserving algebra isomorphism between the cohomology ring of the Spaltenstein variety X λ µ and the center Z(K S ) of K S , where λ and µ are two weights determined by S (Theorem 4.8).(Kuperberg [42] proved that W S , the space of sl 3 -webs whose boundary is determined byChoose an arbitrary k ∈ N and let n = 3k. Let us denote by V (3 k ) the irreducible U q (sl n )-module with highest weight 3ω k , where ω k is the k-th fundamental sl n -weight. As the reader will have noticed, we actually use the corresponding gl n -weight (3 k ). This is natural from the point of view of skew Howe duality, as we will explain in the paper.The gl n -weights of V (3 k ) belong to Λ(n, n) 3 , which is the set of n-part compositions of n whose parts are integers between 0 and 3. These weights, denoted µ S , correspond bijectively to enhanced sign sequences of length n, denoted by S as before, and are in bijective correspondence to the semi-standard Young tableaux with k rows and 3 columns.Define the web modulewhere W S is defined as before after deleting the entries of µ S which are equal to 0 or 3. By q-skew Howe duality, which we will explain at the beginning of Section 5, there is anU q (sl n )-action on W (3 k ) such thatIn Section 5 we categorify this result. Let R (3 k ) be the cyclotomic level-three Khovanov-Lauda Rouquier algebra (cyclotomic KLR algebra for short) with highest gl n -weight (3 k ), and letbe its categories of finite dimensional, graded modules and finite dimensional, graded, projective modules respectively. We define grading shifts byfor any M ∈ V (3 k ) and t ∈ Z. We denote the split Grothendieck group of p V (3 k ) bybecomes a Z[q, q −1 ]-module by defining q t [M] = [M{t}] 1 The idea for this 2-representation was suggested by Mikhail Khovanov to M. M. in 2008 and its basic ideas were worked out modulo 2 in the unpublished preprint [47]. 3 jn be the elementary tensor corresponding to (S, J). Interpreting the webs in W S as invariant tensors in V S , we can write any web as a linear combination of elementary tensors. For each state string J ′ , one can consider all flows on a given basis web w = w S J which are compatible with J ′ on the boun...

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